A Lie algebra attached to a projective variety

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Affine Algebras, Langlands Duality and Bethe Ansatz

AFFINE ALGEBRAS, LANGLANDS DUALITY AND BETHE ANSATZ
EDWARD FRENKEL
arXiv:q-alg/9506003v3 Sep 1999
In memory of Claude Itzykson
1. Introduction. By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a ﬁeld F , and homomorphisms from the Galois group Gal(F /F ) to the Langlands dual group GL . It was originally introduced in the case when F is a number ﬁeld or the ﬁeld of rational functions on a curve over a ﬁnite ﬁeld [1]. Recently A. Beilinson and V. Drinfeld [2] proposed a version of Langlands correspondence in the case when F is the ﬁeld of rational functions on a curve X over C. This geometric Langlands correspondence relates certain D–modules on the moduli stack MG (X ) of principal G–bundles on X , and GL –local systems on X (i.e. homomorphisms π1 (X ) → GL ). A. Beilinson and V. Drinfeld construct this correspondence by applying a localization functor to representations of the aﬃne Kac-Moody algebra g of critical level k = −h∨ , where h∨ is the dual Coxeter number. The localization functor assigns a twisted D–module on MG (X ) to an arbitrary g– module from a category O0 . The ﬁbers of this D–module are analogous to spaces of conformal blocks from conformal ﬁeld theory. In fact, the D–module, which corresponds to the vacuum irreducible g–module of level k ∈ Z+ , is the sheaf of sections of a vector bundle (with projectively ﬂat connection), whose ﬁber is dual to the space of conformal blocks of the Wess-Zumino-Witten model. The analogy between conformal ﬁeld theory and the theory of automorphic representations was underscored by E. Witten in [3]. It is at the critical level where this analogy can be made even more precise due to the richness of representation theory of g. The peculiarity of the critical level is that a completion of the universal enveloping algebra of g at this level, U−h∨ (g) = U (g)/(K + h∨ ), contains a large center Z (g). This center is isomorphic to the classical W–algebra W(gL ) associated to the Lie algebra gL , which is Langlands dual to g [4]. Recall that W(gL ) consists of functionals on a certain Poisson manifold, which is obtained by the Drinfeld-Sokolov reduction from a hyperplane in the dual space to gL [5]. Elements of this Poisson manifold can be considered as connections of special kind on a GL –bundle over a punctured disc called gL –opers in [6]. For example, sl2 –opers are the same as projective connections. Thus,

Lie algebroid structures and Lagrangian systems on affine bundles

a r X i v :m a t h /023178v1[mat h.DG ]18Mar22Lie algebroid structures and Lagrangian systems on aﬃne bundles Eduardo Mart´ınez †,Tom Mestdag ‡and Willy Sarlet ‡†Departamento de Matem´a tica Aplicada Universidad de Zaragoza,Mar ´ıa de Luna 3,50015Zaragoza,Spain ‡Department of Mathematical Physics and Astronomy Ghent University,Krijgslaan 281,B-9000Ghent,Belgium Abstract As a continuation of previous papers,we study the concept of a Lie algebroid structure on an aﬃne bundle by means of the canonical immer-sion of the aﬃne bundle into its bidual.We pay particular attention to the prolongation and various lifting procedures,and to the geometrical construction of Lagrangian-type dynamics on an aﬃne Lie algebroid.1Introduction Since the book of Mackenzie [12],the mathematics of Lie algebroids (and groupoids)has been studied by many authors;for a non-exhaustive list of references,see for example [3,5,6,9,10,13,18].The potential relevance of Lie algebroids for applications in physics and other ﬁelds of applied mathematics has gradu-ally become more evident.In particular,contributions by Libermann [11]and Weinstein [21]have revealed the role Lie algebroids play in modelling certain problems in mechanics.The concept of ‘Lagrangian equations’on Lie algebroids certainly deﬁnes an interesting generalisation of Lagrangian systems as known from classical mechanics,if only because of the more general class of diﬀerentialequations it involves while preserving a great deal of the very rich geometrical structure of Lagrangian (and Hamiltonian)mechanics.One of us [16],in particular,has produced evidence of this rich structure by showing that one can prolong a Lie algebroid in such a way that the newly obtained space has all the features of tangent bundle geometry,which are im-portant for the geometrical construction of Lagrangian systems.That is to say,the prolonged Lie algebroid carries a Liouville-type section and a vertical endomorphism which enables the deﬁnition of a Poincar´e -Cartan type 1-form,associated to a function L ;the available exterior derivative then does the rest1for arriving at an analogue of the symplectic structure from which Lagrangian equations can be derived.In[20],we have started an investigation on the possible generalisation of the concept of Lie algebroids to aﬃne bundles.Our principle motivation was to create a geometrical model which would be a natural environment for a time-dependent version of Lagrange equations on Lie algebroids,as discussed for example in[21]and[16].Since classical time-dependent mechanics is usually described on theﬁrst-jet bundle J1M of a manifold Mﬁbred over I R(see e.g. [4,14]),and J1M→M is an aﬃne bundle,it looks natural to build up a time-dependent generalisation in such a way that J1M→M is the image bundle of the anchor map of a Lie algebroid structure on some aﬃne bundle E→M. An additional indication that such a set-up is well suited came from a naive calculus of variations approach,which gives a clue on the analytical format one should expect for such time-dependent Lagrange equations(see[19]).Lie algebroids on vector bundles are known to give rise,among other things, to a linear Poisson structure on the dual bundle,as well as a coboundary opera-tor on its Grassmann algebra;in fact,these properties equivalently characterise the Lie algebroid structure.One of the features of the approach to aﬃne al-gebroids adopted in[20]was our speciﬁc choice to develop,in a direct way, a consistent theory of forms on sections of an aﬃne bundle and their exterior calculus.By contrast,however,in the context of brieﬂy mentioning the related Poisson structure in the concluding remarks,we did announce a forthcoming diﬀerent approach,which would be based on the fact that an aﬃne bundle can be regarded as an aﬃne sub-bundle of a vector bundle,namely the dual of its extended dual.This is the line of reasoning we will develop here;it could be termed‘indirect’because it makes use of an imbedding into a larger bundle, but it has some marked advantages,such as the fact that proving a number of properties becomes much easier and that new insights come to the forefront.As a matter of fact,one readily recognises via this approach that much(if not all) of the theory of aﬃne Lie algebroids can be developed without needing an extra ﬁbration of the base manifold over I R.We will accordingly start our present analysis in this more general set-up and brieﬂy come back to the special case appropriate for time-dependent systems in the concluding remarks.Note:while the very last editing of this paper was being done,we have been informed of similar investigations on aﬃne algebroids,which have been carried out by Grabowski et al[7].The reader mayﬁnd it instructive to compare the two simultaneous developments,which are related to each other up to and including our Section6.The scheme of the paper is as follows.A fairly detailed description of purely algebraic aspects of the theory is given in Sections2to4;it involves the introduc-tion of the concept of a Lie algebra over an aﬃne space and aspects of exterior calculus.A Lie algebroid structure on a general aﬃne bundleτ:E→M is deﬁned in Section5:essentially,it comes from a classical Lie algebroid on the dual˜E of the extended dual E†of E,with the property that the bracket of sections in the image of the inclusion i:E→˜E,lies in the image of the under-lying vector bundle.Equivalent characterisations of this property can be found2in the subsequent section on the exterior diﬀerential and the associated Poisson structure.Section7presents a number of simple examples of aﬃne algebroids. The important concept of prolongation of an algebroid is discussed in Section8: starting from a general construction on vector algebroids,it is shown that the prolonged bundle inherits the aﬃne structure coming from E when the vector algebroid is the one on˜E.For the speciﬁc case of interest,it is further shown (see Section9)that there is a canonical map which gives rise to a‘vertical en-domorphism’on sections of the prolonged bundle.Natural constructions which are then available are complete and vertical lifts;they play a role in the ge-ometric deﬁnition of Lagrangian systems on aﬃne Lie algebroids,presented in Section10.2Immersion of an aﬃne space in a vector space Let A be an aﬃne space modelled on a vector space V,and let A†=Aﬀ(A,I R) be the extended dual of A,that is,the vector space of all aﬃne maps from A to the real line.We consider the bidual˜A of A,in the sense˜A=(A†)∗.It is well known that in the case of a vector space V,the bidual˜V=(V∗)∗is isomorphic to V.In the case of an aﬃne space,the bidual includes‘a copy’of A,as it is shown in the following statement.Proposition1.The map i:A→˜A given by i(a)(ϕ)=ϕ(a)is an injective aﬃne map,whose associated vector map is i:V→˜A given by i(v)(ϕ)=ϕ(v) Proof.If a∈A and v∈V,then for allϕ∈A†,i(a+v)(ϕ)=ϕ(a+v)=ϕ(a)+ϕ(v)=i(a)(ϕ)+i(v),from which it follows that i is an aﬃne map whose associated linear map is i.To prove that i is injective,it suﬃces to prove that i is injective,which is obvious since if v is an element in the kernel of i then i(v)(ϕ)=ϕ(v)=0for allϕ∈V∗,hence v=0.The vector space˜A is foliated by hyperplanes parallel to the image of i. Every vector z∈˜A is either of the form z=i(v)for some v∈V or of the form z=λi(a)for someλ∈I R\{0}and a∈A.Moreover,λ,a and v are uniquely determined by z.The image of the map i consists of the points for whichλ=1. To understand this description in more detail,we will prove that we have an exact sequence of vector spaces0−→V i−→˜A−→I R−→0.To this end we consider the dual sequence.Proposition2.Let l:I R→A†be the map that associates toλ∈I R the con-stant functionλon A.Let k:A†→V∗the map that associates to every aﬃne function on A the corresponding linear function on V.Then,the sequence of vector spaces0−→I R l−→A†k−→V∗−→0is exact.3Proof.Indeed,it is clear that l is injective,k is surjective and k◦l=0,so that Im(l)⊂Ker(k).Ifϕ∈A†is in the kernel of k,that is the linear part ofϕvanishes,then for every pair of points a and b=a+v we have thatϕ(b)=ϕ(a+v)=ϕ(a)+ϕ(v)=ϕ(a),that isϕis constant,and hence in the image of l.The dual map of k is i,since for v∈V we havek(ϕ),v = ϕ,v = ϕ,i(v)The dual map j of the map l is given by j(αi(a)+iv)=α.Indeed,for every λ∈I R we havej(z)λ= z,l(λ) = αi(a)+i(v),l(λ) =α i(a)),l(λ) + i(v),l(λ) =αλIt follows thatCorollary1.If A isﬁnite dimensional,then the sequence0−→V i−→˜A j−→I R−→0is exact.Note that in this way we can clearly identify the image of V as the hyperplane of˜A with equation j(z)=0,and the image of A as the hyperplane of˜A with equation j(z)=1,in other wordsi(V)=j−1(0)and i(V)=j−1(1).Note in passing that if we have an exact sequence0−→Vα−→W j−→I R−→0, then we can deﬁne A=j−1(1);it follows that A is an aﬃne space modelled on the vector space V and W is canonically isomorphic to˜A.The isomorphism is the dual map ofΨ:W∗→A†,Ψ(φ)(a)=φ(i(a)),where i:A→W is the canonical inclusion.We now discuss the construction of a basis for A†.Let(O,{e i})be an aﬃne frame on A.Thus every point a has a representation a=O+v i e i.The family of aﬃne maps{e0,e1,...,e n}given bye0(a)=1e i(a)=v i,is a basis for A†.Ifϕ∈A†,and we putϕ0=ϕ(O)andϕi=ϕ(e i),then ϕ=ϕ0e0+ϕi e i.It is to be noticed that,contrary to e1,...,e n,the map e0does not depend on the frame we have chosen for A.In fact,e0coincides with the map j.Let now{e0,e1,...,e n}denote the basis of˜A dual to{e0,e1,...,e n}.Then the image of the canonical immersion is given byi(O)=e0i(e i)=e i4from which it follows that for a=O+v i e i,we have i(a)=e0+v i e i.If we denote by(x0,x1,...,x n)the coordinate system on˜A associated to the basis {e0,...,e n},then the equation of the image of the map i is x0=1,while the equation of the image of i is x0=0.Coordinates in˜A∗=A†associated to the basis above will be denoted by (µ0,µ1,...,µn),that isµα(ϕ)= eα,ϕ for everyϕ∈A†.3Lie algebra structure over an aﬃne space Implicit in our previous paper[20]is the following deﬁnition of a Lie algebra over an aﬃne space.Deﬁnition1.Let A be an aﬃne space over a vector space V.A Lie algebra structure on A is given by•a Lie algebra structure[,]on V,and•an action by derivations of A on V,i.e.a map D:A×V→V,(a,v)→D a v with the propertiesD a(λv)=λD a vD a(v+w)=D a v+D a wD a[v,w]=[D a v,w]+[v,D a w],•satisfying the compatibility propertyD a+v w=D a w+[v,w].Incidentally,it is suﬃcient to require in theﬁrst item that the bracket on V is I R-bilinear and skew-symmetric,since the Jacobi identity then follows from the requirements on D a.If we use a bracket notation[a,v]≡D a v,then the conditions in the deﬁni-tion above read[a,λv]=λ[a,v][a,v+w]=[a,v]+[a,w][a,[v,w]]=[[a,v],w]+[v,[a,w]][a+v,w]=[a,w]+[v,w].This allows us to deﬁne a bracket of elements of A by putting[v,a]=−[a,v]and,if b=a+v,then[a,b]=[a,v].This bracket is skew-symmetric by construction and also satisﬁes a Jacobi-type property.5Theorem1.A Lie algebra structure over an aﬃne space A is equivalent to a Lie algebra extension of the trivial Lie algebra I R by V.Explicitly,it is equivalent to the exact sequence of vector spaces0−→V i−→˜A j−→I R−→0being an exact sequence of Lie algebras.Proof.If the exact sequence is one of Lie algebras,we of course have a Lie algebra structure on V and the map D determined by D a v=[i(a),i(v)]satisﬁes all requirements to deﬁne a Lie algebra structure on A.Conversely,assume we have a Lie algebra structure on the aﬃne space A.If weﬁx an element a∈A,then every element z∈˜A can be written in the form z=λi(a)+i(v).We can deﬁne a bracket of two elements z1=λ1i(a)+i(v1) and z2=λ2i(a)+i(v2)by[z1,z2]=i([v1,v2]+λ1D a v2−λ2D a v1)This bracket is clearly bi-linear and skew-symmetric,and a straightforward cal-culation shows that it satisﬁes the Jacobi identity.Moreover,the deﬁnition does not depend on the choice of the point a;if a′is another point in A,then a′=a+w for some w∈V and the compatibility condition implies that the result is independent of that choice.Finally,it is obvious that the maps i and j then are Lie algebra homomorphisms.Notice that the only condition for a Lie algebra structure on˜A to be an extension of I R by V is that the bracket takes values in V,symbolically:[˜A,˜A]⊂V.Once we have chosen an aﬃne frame on A,we have that the bracket on˜A is determined by the brackets of the associated basis elements.These must be of the form[e0,e0]=0[e0,e j]=C k0j e k[e i,e j]=C k ij e k,since all brackets must take values in the image of the map i.It is well known that a Lie algebra structure on a vector space deﬁnes, and is deﬁned by,a linear Poisson structure on the dual vector space.In the light of the results of the previous section we have a Poisson bracket on V∗and one on A†.Furthermore,˜A being an extension of I R by V,we have that the Poisson structureΛA†is an extension byΛV∗ofΛI R=0(see[2]for the details on Poisson extensions and their relations to Lie algebra extensions). Therefore,once we haveﬁxed a point a∈A,we have a splitting of the sequence 0−→V i−→˜A j−→I R−→0given by h(λ)=λi(a),and the Poisson tensorcan be written asΛA†=ΛV∗+X a∧X Da ,where X Dais the linear vectorﬁeld associated to the linear map D a∈End(V),and X a is the constant vector corresponding to i(a).Moreover,we have that L XD aΛV∗=0.In the coordinatesµ0,µ1,...,µn on A†associated to the basis{eα},we have {µ0,µ0}=0{µ0,µj}=µk C k0j{µi,µj}=µk C k ij,6where C k0j,C k ij are the structure constants introduced above.Therefore,the Poisson tensor readsΛ†A=1∂µi∧∂∂µ0∧∂Then,conditions1and2in the deﬁnition above are trivially satisﬁed.More-over,if weﬁx a0∈A and write a i=a0+v i,then by skew-symmetry of˜ωwe haveω(a1,...,a k)=˜ω(i(a1),...,i(a k))=˜ω(i(a0)+i(v1),...,i(a0)+i(v k))=kj=1(−1)j+1˜ω(i(a0),i(v1),..., i(v j),...,i(v k))+˜ω(i(v1),...,i(v k))=kj=1(−1)j+1ω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k),which proves condition3.Conversely,assume we are given a k-formωon the aﬃne space A with its associatedω0andω.Fixing a0∈A,we know that every point z∈˜A can be written in the form z=λi(a0)+i(v)forλ∈I R and v∈V.We deﬁne the map ˜ωby˜ω(z1,...,z k)=˜ω(λ1i(a0)+i(v1),...,λk i(a0)+i(v k))=kj=1(−1)j+1λjω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k).By virtue of1and3,it follows that˜ωis multi-linear and skew-symmetric, i.e.it is a k-form on˜A.Moreover,˜ω(z1,...,z k)is independent of the choice of the point a0.Indeed,if we choose a diﬀerent point a′0=a0+w,then z j=λj i(a′0)+v′j with v′j=v j−λj w,and applying the deﬁnition above we get ˜ω(z1,...,z k)=˜ω(λ1i(a′0)+i(v′1),...,λk i(a′0)+i(v′k))=kj=1(−1)j+1λjω0(a′0,v′1,..., v′j,...,v′k)+ω(v′1,...,v′k)=kj=1(−1)j+1λjω0(a0+w,v1−λ1w,..., v j,...,v k−λk w) +ω(v1−λ1w,...,v k−λk w)=kj=1(−1)j+1λjω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k)where we have used the properties ofω0andω.The form˜ωis unique,since,if˜θis a k-form on˜A such that i∗˜θ=0,then it follows that the associatedθ0andθvanish from where we deduce that˜θ=0.Once a reference frame has beenﬁxed on A,a1-form˜ωon˜A is of the form ˜ω=ω0e0+ωi e i,and then the local representation ofω=i∗˜ωlooks exactly the same.8More generally,a p-form˜ωon˜A is of the form˜ω=1(p−1)!ω0i1···i p−1e0∧e i1∧···∧e i p−1+1Deﬁnition3.A Lie algebroid structure on E consists of a Lie algebra structure on the(real)aﬃne space of sections of E together with an aﬃne mapρ:E→T M(the anchor),satisfying the following compatibility conditionDσ(fζ)=ρ(σ)(f)ζ+fDσζ,for everyσ∈Sec(E),ζ∈Sec(V)and f∈C∞(M),and where Dσis the action σ→Dσof Sec(E)on Sec(V).The compatibility condition ensures that the associationσ→Dσ,which acts by derivations on the real Lie algebra Sec(V),also acts by derivations on the C∞(M)-module Sec(V).The anchor mapρextends to a linear map˜ρ:˜E→T M,which we will describe in more detail below.It is of interest,however,to observe now already that the map i:V→˜E is a morphism of Lie algebroids,since we have[i(η1),i(η2)]=i([η1,η2])and˜ρ◦i=ρ,whereρis the linear part ofρ.On the contrary,the map j:˜E→M×I R is not a morphism of Lie algebroids,since we have that j([ζ1,ζ2])=˜ρ(ζ1)f2−˜ρ(ζ2)f1, while[j(ζ1),j(ζ2)]=0since theﬁbres of M×I R are1-dimensional.The aﬃne Lie algebroid structure we studied in[20]is the case that M is furtherﬁbred over the real lineπ:M→I R and the anchor mapρtakes values in J1M.Notice that such an extraﬁbration is not generally available,not even locally.For instance,if we take any aﬃne bundleτ:E→M with the trivial Lie algebroid structure(null bracket and anchor)then there is noﬁbration over I R such that the image ofρis in J1M,since the vectors in i(J1M)are non-zero.The following result shows that one can alternatively deﬁne an aﬃne Lie algebroid structure on E as a vector Lie algebroid structure([,],˜ρ)on˜E such that the bracket of two sections in the image of i belongs to the image of i. Theorem2.A Lie algebroid structure on the vector bundle˜τ:˜E→M which is such that the bracket of sections in the image of i lies in the image of i deﬁnes a Lie algebroid structure on the aﬃne bundleτ:E→M,whereby the brackets and maps are determined by the following relations:i([η1,η2])=[i(η1),i(η2)]i(Dση)=[i(σ),i(η)]ρ(σ)=˜ρ(i(σ)).Conversely,a Lie algebroid structure on the aﬃne bundleτ:E→M extends to a Lie algebroid structure on the vector bundle˜τ:˜E→M such that the bracket of sections in the image of i is in the image of i.If weﬁx a sectionσof E and write sectionsζof˜E(locally)in the formζ=fi(σ)+i(η)then the anchor and the bracket are given by˜ρ(ζ)=fρ(σ)+ρ(η)[ζ1,ζ2]= ˜ρ(ζ1)(f2)−˜ρ(ζ2)(f1) i(σ)+i [η1,η2]+f1Dση2−f2Dση1 .10Proof.The veriﬁcation of the above statements is straightforward but rather lengthy.We limit ourselves to checking that the compatibility conditions be-tween brackets and anchors are satisﬁed.For theﬁrst part,weﬁndi Dσ(fη) =[i(σ),i(fη)]=[i(σ),f i(η)]=˜ρ(i(σ))(f)i(η)+f[i(σ),i(η)]=i ρ(σ)(f)η+fDσ(η) , form which it follows that Dσ(fη)=ρ(σ)(f)η+fDσ(η).For the converse,observe that[ζ1,fζ2]−f[ζ1,ζ2]=f2˜ρ(ζ1)(f)i(σ)+i f1ρ(σ)(f)η2+ρ(η1)(f)η2=˜ρ(ζ1)(f)f2i(σ)+(f1ρ(σ)+ρ(η1))(f)i(η2)=˜ρ(ζ1)(f)ζ2,which is the required compatibility condition.In coordinates,if x i denote coordinates on M and yαﬁbre coordinates on E with respect to some local frame(e0,{eα})of sections of E,then we haveρ(e0+yαeα)=(ρi0+ρiαyα)∂∂x i,and the bracket is determined by[e0,e0]=0[e0,eβ]=Cγ0βeγ[eα,eβ]=Cγαβeγ.As aﬁnal remark we mention that the orbits of the Lie algebroidτ:V→M are subsets of the orbits of the algebroid˜τ:˜E→M and they are equal if and only if there exists a sectionσof E such thatρ(σ)is in the image ofρ.6Exterior diﬀerential and Poisson structure Now that we have proved that a Lie algebroid structure on an aﬃne bundle is equivalent to a Lie algebroid structure on˜E,we can deﬁne the exterior diﬀeren-tial operator on E by pulling back the exterior diﬀerential on˜E.More precisely, given a k-formωon the aﬃne bundle E we know that there exists a unique˜ωon˜E such thatω=i∗(˜ω).Then we deﬁne dωas the(k+1)-form given bydω=i∗(d˜ω).11It is easy to see that this deﬁnition is equivalent to the one given in[20](at least when M isﬁbred over I R).The deﬁnition given here has some clear advantages.For instance,the prop-erty d2=0which was rather diﬃcult to prove in[20],becomes evident now:d2ω=d(d(i∗˜ω))=d(i∗d˜ω)=i∗(d2˜ω)=0.The diﬀerential d on the Lie algebroid V is also related to the diﬀerential on˜E;for every k-form˜ωon E we have thatdi∗˜ω=i∗(d˜ω),which in fact simply expresses that i is a morphism of Lie algebroids.Hence, one canﬁnd the diﬀerential of a formωon V by choosing a k-form˜ωon˜E such that i∗˜ω=ωand then obtain dωas i∗(d˜ω).That this does not depend on the choice of˜ωis expressed in equivalent terms by the following result.Proposition4.The ideal I={˜ω|i∗˜ω=0}is a diﬀerential ideal,i.e.d I⊂I.Proof.If i∗˜ω=0,then the same is true for d˜ωsince i∗(d˜ω)=di∗(˜ω)=0.The ideal I is generated by the1-form e0,I={e0∧θ|θis a form on˜E} so that de0belongs to I.The following result shows that e0in fact is d-closed and that this property characterises aﬃne structures.Theorem3.A Lie algebroid structure on˜E restricts to a Lie algebroid struc-ture on the aﬃne bundle E if and only if the exterior diﬀerential satisﬁes de0=0.Proof.Indeed,taking two sectionsσ1andσ2of E we havede0(i(σ1),i(σ2))=˜ρ(σ1) e0,i(σ2) −˜ρ(σ2) e0,i(σ1) − e0,[i(σ1),i(σ2)]=− e0,[i(σ1),i(σ2)]It follows that[i(σ1),i(σ2)]is in Im(i)=Ker(e0)if and only if de0vanishes on the image of i,which spans˜E.In coordinates,the exterior diﬀerential operator is determined byd f=ρi0∂f∂x ieα,for f∈C∞(M)andde0=0,deγ=−Cγ0βe0∧eβ−1f cannot simultaneously vanish,hence f deﬁnes a localﬁbration and then for any sectionσof E we have thatρ(σ)f= d f,σ = e0,σ =1,which is the condition for the anchor being1-jet-valued.When we have a Lie algebroid structure on˜E,there is a Poisson bracket on the dual bundle˜E∗=E†.Any sectionζof˜E,and in particular any section of E,determines a linear function ζon E†byζ(ϕ)= ζm,ϕ for everyϕ∈E†m.Then the Poisson bracket is determined by the condition{ ζ1, ζ2}= [ζ1,ζ2],which for consistency(using linearity and the Leibnitz rule)requires that we put{ ζ,g}=˜ρ(ζ)(g),and{f,g}=0,for f and g functions on M.It is of some interest to mention yet another characterisation of the result described in Theorem2.The above Poisson bracket in fact is determined by the bracket of linear functions coming from sections of E,since these span the set of all linear functions on E†.But the bracket of sections of E is a section of V;it follows that the corresponding Poisson brackets are independent of the coordinateµ0,and therefore,∂∂µ0∧X0,with X0=ρ(e0).(Noticethatρ(e0)=ρ(σ)for any sectionσof E.)The other brackets vanish,so there are no further conditions.In coordinates,we have{x i,x j}=0{µ0,x i}=ρi0{µα,x i}=ρiα{µ0,µβ}=Cγ0βµγ{µα,µβ}=Cγαβµγ13and therefore the Poisson tensor isΛE†=ρiα∂∂x i+1∂µα∧∂∂µ0∧ ρi0∂∂µβ .7ExamplesThe canonical aﬃne Lie algebroid The canonical example of a Lie alge-broid over an aﬃne bundle is theﬁrst jet bundle J1M→M to a manifold M ﬁbered over the real lineπ:M→I R.The elements of the manifold J1M are equivalence classes j1tγof sectionsγof the bundleπ:M→I R,where two sec-tions are equivalent if they haveﬁrst order contact at the point t.It is an aﬃne bundle whose associated vector bundle is Ver(π)the set of vectors tangent to M which are vertical over I R.In this case it is well-known that J1M†=T∗M, and thereforeJ1M=T M.The canonical inmersion is given byi(j1tγ)=˙γ(t),i.e.it maps the1-jet of the sectionγat the point t to the vector tangent toγat the point t.In coordinates,if j1tγhas coordinates(t,x,v)theni(t,x,v)=∂∂x i (t,x)An element w of the associated vector bundle Ver(π)is of the formw=w i∂∂t +X i(t,x)∂∂x iwhich is obviously a section of the vector bundle.Aﬃne distributions An aﬃne E sub-bundle of J1M is involutive if the bracket of sections of the sub-bundle is a section of the associated vector bundle. Therefore,taking as anchor the natural inclusion into T M and as bracket the restriction of the bracket in J1M to E we have an aﬃne Lie algebroid structure on E.14Lie algebra structures on aﬃne spaces We consider the case in which the manifold M reduces to one point M={m}.Thus our aﬃne bundle is E={m}×A and the associated vector bundle is W≡{m}×V for some aﬃne space A over the vector space V.Then,a Lie algebroid structure over the aﬃne bundle E is just an aﬃne Lie algebra structure over A.Indeed,every section of E and of W is determined by a point in A and V,respectively.The anchor must vanishes since T M={0m},so it does not carry any additional information. Trivial aﬃne algebroids By a trivial aﬃne space we mean just a point A={O},and the associated vector space is the trivial one V={0}.The extended aﬃne dual of A is A†=I R since the only aﬃne maps deﬁned on a space of just a point are the constant maps.It follows that the extended bi-dual is˜A=I RGiven a manifold M,we consider the aﬃne bundle E=M×{O}with associated vector bundle V=M×{0}.On V we consider the trivial bracket [,]=0and the anchorρ=0,and as derivation D O we also take D O=0.Now, to construct a Lie algebroid structure on E,we take an arbitrary vectorﬁeld X0on M as given and deﬁne the mapρ:E→T M byρ(m,O)=X0(m).Then it follows thatρis compatible with D O.The extended dual of E is E†=M×I R and the extended bi-dual is˜E= M×I R.We therefore have one section e0spanning the set of sections of E†, and the dual element e0(which is just the image under the canonical immersion of the constant section of value0.)We want to study the associated exterior diﬀerential operator and Poisson bracket.For the exterior diﬀerential operator,since we have1-dimensionalﬁbre on E†it follows that de0=0.On functions f∈C∞(M)we have d f=ρ(e0)(f)e0= X0(f)e0.For the Poisson structure,since theﬁbre of E†is1-dimensional,it is deter-mined by the equation{ˆe0,f}=ρ(e0)(f)andˆe0=µ0.We have that the only non-trivial brackets are{µ0,f}=X0(f).Therefore,the Poisson tensor is∂Λ=sectionsσi(m)=(m,ξi),is the constant section corresponding to the bracket of the values[σ1,σ2](m)=(m,[ξ1,ξ2]A).If we consider the Lie algebra˜A then˜A acts also on the manifold M.The extension˜E is the Lie algebroid associated to the action of˜A.Poisson manifolds with symmetry Consider a Poisson manifold(M,Λ) and an inﬁnitesimal symmetry Y∈X(M)ofΛ,that is L YΛ=0.Take E to be T∗M with its natural aﬃne structure,where the associated vector bundle is V=T∗M itself.On V we consider the Lie algebroid structure deﬁned by the canonical Poisson structure.For a sectionαof E(i.e.a1-form on M)we deﬁne the map Dα:Sec(V)→Sec(V)byDαβ=L Yβ+[α,β].Since Y is a symmetry ofΛ,Dαis a derivation and clearly satisﬁes the required compatibility condition.If we further consider the aﬃne anchorρ:E→T M, determined byρ(αm)=Λ(αm)+Y m,then we have a Lie algebroid structure on the aﬃne bundle E.In this case,since there is a distinguished section of E(the zero section),we have that E†=T M×I R and˜E=T∗M×I R.Jets of sections in a groupoid Let G be a Lie groupoid over a manifold Mwith sourceαand targetβ(the notation is as in[1]).Let TαG(0)G=ker Tα|G(0)be the associated Lie algebroid,that is,the set ofα-vertical vectors at points in G(0)(the set of identities).The anchor is the mapρ=Tβ.Assume that M is furtherﬁbred over the real line,π:M→I R and consider the bundleE=JαG(0)G of1-jets of sections ofπ◦βwhich areα-vertical,at points in G(0).This is an aﬃne bundle whose associated vector bundle is(TαG(0)G)ver the set of(π◦β)-vertical vectors on TαG(0)G.If i is the natural inclusion of(TαG(0)G)ver intoTαG(0)G and we deﬁne the map j:TαG(0)G→M×I R by j(v)=(α(τG(v)),t(v))(where t=π◦β◦τG),then we have the exact sequence of vector bundles over M0−→(TαG(0)G)ver i−→TαG(0)G j−→M×I R−→0and j−1(M×{1})=JαG(0)G.Moreover,the bracket of two sections of JαG(0)G isvertical over I R from where it follows that the Lie algebroid structure of TαG(0)Grestricts to a Lie algebroid structure on the aﬃne bundle JαG(0)G.8ProlongationIn this section we deﬁne the prolongation of aﬁbre bundle with respect to a (vector)Lie algebroid.We are primarily interested in the prolongation of the bundle E→M,but we will describe explicitly a more general constructionﬁrst, since this does not introduce extra complications(see also[8]for generalities).16。

Small degree representations of finite Chevalley groups in defining characteristic

Small degree representations ofﬁnite Chevalley groups in deﬁning characteristicFrank L¨u beckAugust24,2000AbstractWe determine for all simple simply connected reductive linear algebraic groups deﬁned over aﬁniteﬁeld all irreducible representations in their deﬁning character-istic of degree below some bound.These also give the small degree projective rep-resentations in deﬁning characteristic for the correspondingﬁnite simple groups.For large rank our bound is proportional to and for rank much higher.The small rank cases are based on extensive computer calculations.1IntroductionIn this note we give lists of projective representations of simple Chevalley groups in their deﬁning characteristic.There are two types of results.First we determine for groups of rank all such representations of degree smaller or equal some bound depending on the type(e.g.,100000for type).In particular this contains a complement to the tables of representations in non-deﬁning characteristic up to degree250given in[7].These data are produced using a collection of computer programs developed by the author.Then we determine for groups of classical type of rank all representations of degree at most for type,respectively for the other types.For large there is a small list and for small this range is easily covered by our tables mentioned above. This extends results by Kleidman and Liebeck[11,5.4.11].Weﬁx some notation for the whole paper.Let be aﬁnite twisted or non-twisted simple Chevalley group in characteristic.There is an associated connected reductive simple algebraic group over of simply connected type,a Frobenius endomorphism of,a with and an such that:—is deﬁned over via.—For the group of-ﬁxed points with center we have.—is the quotient of the universal covering group of by the-part of its center.So,asking for the projective representations of in characteristic is the same as asking for the representations of in characteristic.These can be constructed by restricting certain representations,called highest weight representations,of the alge-braic group to.This is explained in Section2.In Section3we shortly describe how our computer programs for computing weight multiplicities work.In Section4we describe our main results consisting of lists of small degree representations for groups12Frank L¨u beckof rank at most11.The lists are printed in Appendix6.Finally,in Section5we consider groups of larger rank.Acknowledgements.I wish to thank Kay Magaard,Gunter Malle and Gerhard Hißfor useful discussions about the topic of this note.2Representations in deﬁning characteristicThere are several well readable introductions to this topic,for example Humphreys’survey[9].A detailed reference is Jantzen’s book[10].We recall some of the basic facts.For of simply connected type of rank as in the introduction let be a maximal torus of,its character group and its co-character group.Let be a set of simple roots for this root system andthe coroot corresponding to,.Viewing as Euclidean space we deﬁne the fundamental weights as the dual basis of. This is a-basis of(because is simply connected).There is a partial ordering on deﬁned by if and only if is a non-negative linear combination of simple roots.A weight is called dominant if it is a non-negative linear combination of the fundamental weights.The Weyl group of,generated by the reﬂections along the,acts on.Under this action each -orbit on contains a unique dominant weight.From now on let be aﬁnite dimensional-module over.Considering this as-module there is a direct sum decomposition into weight spaces such that acts by multiplication with on.The set of with is called the set of weights of.The set of weights of is a union of -orbits and for,we have.The following basic results are due to Chevalley.Theorem2.1Let be as above.(a)If is irreducible then the set of weights of contains a(unique)element such that for all weights of we have.This is called the highest weight of ,it is dominant and we have.(b)An irreducible-module is determined up to isomorphism by its highest weight.(c)For each dominant weight there is an irreducible-module with highest weight.A dominant weight is called-restricted iffor.The following result of Steinberg shows how all highest weight modules of can be constructed out of those with-restricted highest weights.Theorem2.2(Steinberg’s tensor product theorem)Let be the Frobenius auto-morphism of,raising elements to their-th power.Twisting the-action on a -module with,,we get a new-module which we denote by.If are-restricted weights thenSmall degree representations in deﬁning characteristic3 Finally we need to recall the relation between the irreducible modules of the alge-braic group and those of theﬁnite group.This is nicely described by Steinberg in[16,13.3,11.6].Theorem2.3Let and be as in the introduction.We deﬁne a subset of dom-inant weights.If is not a Suzuki or Ree group(i.e.,not of type,or )then for.In the case of Suzuki and Ree groups we deﬁneif is a short root(note that is the square root of an odd power of or,respectively,in these cases).Then the restrictions of the-modules with to form a set of pairwise inequivalent representatives of all equivalence classes of irreducible-modules.These results show that the dimensions of the irreducible representations of the groups over are easy to obtain if we know the dimensions of the representa-tions of the algebraic groups for-restricted weights.3Computation of weight multiplicitiesIn this section we sketch how we compute the degree of the representation for given root datum of,highest weight and prime.For almost all it is the same as for the algebraic group over the complex numbers with same root datum,respectively its Lie algebra.In these cases the degree can be computed by a formula of Weyl,see[8, 24.3].In the other cases no formula is known.But in principle there is an algorithm to compute the degree.This is described in[8,Exercise2of26.4]and goes back to Burgoyne[2].This was also used in[6]to handle some cases in exceptional groups. The idea is to construct a so-called Weyl module generically over the integers. By base change this leads to a module for any with the given root datum over any ring,which has as a highest weight.Over or over for almost all this is irreducible and so isomorphic to.In general is a quotient of.To construct one considers the universal enveloping algebra of the com-plex Lie algebra corresponding to the given root datum.It contains a-lattice,the Kostant-form of,which is deﬁned via a Chevalley basis of the Lie algebra.Up to equivalence there is a unique irreducible highest weight representation for with highest weight.Let be a vector of weight(this is unique up to scalar).Then we set.Weﬁx an ordering of the set of positive roots.Then and have -bases labeled by sequences of non-negative integers.Applying such a basis element of to a vector of weight we get a vector of weight(and similarly for basis vectors of).So,decomposingaccording to the weight spaces,we can describe generating sets of by all non-negative linear combinations of positive roots which are equal to.There is a non-degenerate bilinear form on which can be described via this generating system.Different weight spaces are orthogonal with respect to this form.Let be two coefﬁcient vectors as above such that the corresponding linear combination of the positive roots is the same.Then is again of weight and one deﬁnes.4Frank L¨u beckSince the form is nondegenerate we can compute the rank of a weight latticeby computing the rank of the matrix where and are running through all non-negative linear combinations of positive roots for.The dimension of the weight space of is the rank of modulo.The coefﬁcients can be computed by simplifying the element with the help of the commutator relations in,see[8,25.].These involve the structure con-stants for a Chevalley basis of the corresponding Lie algebra which can be computed as described in[3,4.2].In principle this allows the determination of for all,and.But in practice these computations can become very long,already in small examples.There are technical problems like the question how to apply commutator relations most efﬁ-ciently in order to compute the integers.Different strategies change the number of steps in the calculation considerably.But the main problem is that the generating sets for the weight spaces as described above are very ually the number of non-negative linear combinations of positive roots which yield is much larger than the dimension of.By a careful choice of the ordering of the positive roots one can reduce the linear combinations to consider,because for manywith there is a such thatis not a weight of(and hence).But the main improvement we get by using results of Jantzen and Andersen, see[10,II.8.19].The so-called Jantzen sum formula expresses the determinant of the Gram matrix of the bilinear form on the lattice in terms of weight multi-plicities of various with.The weight multiplicities of these can be efﬁciently computed by Freudenthal’s formula,see[8,22.3].In particular the formula gives exactly the set of primes for which the Weyl module is not isomorphic to .In rare cases it happens that a prime divides such a determinant exactly once-then we know without further calculations that.Using these determinants we compute only parts of the matrices correspond-ing to a subset of its rows and columns until the submatrix has the full rankand the product of its elementary divisors is equal to the known determinant.Then clearly the rank of modulo is the same as the rank of this submatrix modulo.With this approach we never need submatrices of of much larger dimension than.(In[6]the consideration of the matrices was substituted by com-puting somewhat smaller matrices using the action of parabolic subalgebras in cases where has a non-trivial stabilizer in,but these matrices were still big compared to the dimension of the weight spaces.)Of course,our method is limited to representa-tions where no single weight space has a dimension of more than a few thousand.Actually we can also compute by our approach the Jantzenﬁltrations of the Weyl modules,see[10,II.8],since we compute the elementary divisors of the matrices and not just their rank.To do this one also needs a sophisticated algorithm to compute the exact elementary divisors of Gram matrices of this size.We developed the algorithm described in see[12]for this purpose.We have a collection of computer programs for doing the calculations described above,which are based on the computer algebra system GAP[14]and the package CHEVIE[5].They are currently in a usable but still experimental state.It is planned to improve their efﬁciency,to extend their functionality and to put this into a package which will be made available to other users.We postpone a much more detailed version of this very sketchy section until this package is ready.Small degree representations in deﬁning characteristic5 4Representations of small degree for groups of small rankFrom Theorem2.2we see how to construct any highest weight representationof from those with-restricted weights by twisting withﬁeld automorphisms and tensoring.Assume now we are given the type of an irreducible root system and a number .We consider the groups over with this root system for all at once.We want toﬁnd for all primes all-restricted dominant weights such that the highest weight representation of the group over has degree smaller or equal.Our main tool to restrict this question to aﬁnite number of to consider is the following result by Premet,see[13].Recall from Section2that the set of weights of is a union of-orbits and that each-orbit contains a unique representative which is dominant.Theorem4.1If the root system of has different root lengths we assume thatand if is of type we also assume.Then the set of weights of is the union of the-orbits of dominant weights with.The length of the-orbit of a dominant weight is easy to compute by the following remark,see[8,10.3B]for a proof.Remark4.2Let be a dominant weight.Then the stabilizer in the Weyl group of is the parabolic subgroup generated by the reﬂections along the simple roots for which.Here is an algorithm forﬁnding a set of candidate highest weights for representa-tions of of rank at most a given bound.Algorithm4.3Input:An irreducible root system and an.Output:A set of dominant weights which contains all such that there is a prime with-restricted and a group over corresponding to the given root system with highest weight representation of degree at most.—To handle the exceptions in Premet’s theorem we put all-,respectively-restricted weights into,if the root system has roots of different lengths.—If not yet considered we compute for all-restricted weights a lower bound for the dimension of for any corresponding by counting the number of its weights (using4.1and4.2).If the bound is at most we put this weight into.—We choose a linear function which takes positive values on the simple roots,(and hence on the fundamental weights).Then we determine recursively all dominant weights with growing value of and compute for them a lower bound for as above.If it is at most we put the weight into. We proceed until weﬁnd an interval such that for all and such that for all dominant with the dimension of is bigger than.Proof.We show that all with have dimension for any:This is clear for which were considered during the algorithm.Thosewhich were not considered are not-restricted and so one coefﬁcient.But then is also a dominant weight(an expression of as linear combination6Frank L¨u beckof the fundamental weights is given by the-th column of the Cartan matrix of theroot system of,this matrix contains’s on the diagonal and non-positive numberselsewhere).We have.Repeating this step recursively for the smaller weight we must eventuallyﬁnd a dominant weight which is-restrictedand not in or for which.Since the orbit lengths of dominant weights already sum up to more than the same holds for.The algorithm stops since for any given bound there are onlyﬁnitely many dominant weights with less than smaller dominant weights.(If a coefﬁcient of as above is bigger than,,then we have seen that are also dominant for.)In Table1weﬁx some bound for each irreducible root system of rank at most,respectively in case.Table1:MMType3007001000200040005000700080001000012000 MType200030004000500010000150001800020000 MFor each irreducible root system with rank at most11we used the deﬁned aboveas input for Algorithm4.3.For each weight in the output set and for all primes for which is-restricted we computed the exact weight multiplicities of using the techniques described in Section3.Actually we stopped such a computation whenever we found that theﬁnal dimension will be larger than.The bounds were chosen such that the results presented below could be com-puted interactively with our programs within about two days using10computers in parallel.The types,,were added upon request of Gunter Malle,who asked to cover in this note all representations of degree for a speciﬁc application.Here is our main result.Theorem4.4For any type of root system and number as given in Table1and all primes the Tables6.5to6.52list the-restricted weights such that the representa-tion of the algebraic group over has degree at most.The exact degree of is also given.Furthermore we describe the centers of the groups and the action of the fundamental weights on the center in6.2.This allows to determine the kernels of the representations.The Frobenius-Schur indicators in case are given by6.3.As mentioned above we have actually computed the exact weight multiplicities for the representations appearing in the tables of Section6.It would take too much space to print this in detail but the results are available upon request from the author.Small degree representations in deﬁning characteristic7 The types considered above do not include,i.e.,.The reason is that this case is easy to describe systematically.This seems to be well known but also follows immediately from4.1since all weight multiplicities are at most in this case, see[8,7.2].Remark4.5If is of type then the representation with has degree.For odd the center of is non-trivial and of order.It is contained in the kernel of if and only if is even.5Representations of small degree for groups of large rankIn this section we consider classical groups of large rank.We prove the following result.Theorem5.1Let be of classical type and be the rank of.Set if is of type and otherwise.If then all-restricted weights such that the highest weight representation of has dimension at most are given in the following table.The table also includes the dimensions of these modules.(Note that the case of type and is included in case and.)The fundamental weights are labeled as explained in6.1.type,all,allallallallallNote that the corresponding result for of rank is included in Theorem4.4.8Frank L¨u beckProof.Weﬁrst prove that the weights which don’t appear in our table correspond to representations of degree larger than.(Case,)Let be a dominant weight.If somethen the-stabilizer of is contained in a reﬂection subgroup of type, see4.2.Hence,the-orbit of and so is at leastFor and we have.If and then the stabilizer of is contained in a reﬂection group of type,whose index is for all.This shows that only weights with at most one non-zero coefﬁcient,either or ,can appear in our list.If then,as explained in the proof of4.3,is also a dominant weight.But this has coefﬁcient ing the estimate as above for this smaller weight and Premet’s theorem4.1(note that the considered now is not-restricted)we see again that.A similar argument shows that for weights in our list.So,the only weights which could(and actually do)lead to degrees are,, and.(Case)Here weﬁnd the relevant with very similar arguments as in case and.(Case)Because of the symmetry of the Dynkin diagram the weights and must describe representations of equal degree(in fact they are dual to each other).We can again use very similar arguments as above toﬁnd the relevant weights for our list.(Here we rule out andby observing.)It remains to determine the exact degrees of the in our table.Probably they are known to the experts but we could notﬁnd references for all cases.Type and and for the other types are contained in[11,5.4.11]and[8,25.5,Ex.8].We include a proof for the degrees of and in types,and following hints of K.Magaard and G.Hiß.The idea is to use explicit modules.For all types we know the-modules ,.These are the natural modules of, ,or,respectively.We determine the constituents offor.Note that for these types is self-dual.Since the caseis covered by the references above(is not-restricted),we assume in the rest of the proof that is odd.We will use the following general remarks.If is an indecomposable-module then has the trivial module as direct summand if and only if(here denotes the dual module).In that case there is exactly one trivial direct summand, see[1,3.1.9]for a proof.If is a basis of then has two submodules with basisand with basis.Since we have.(Case)Let,be a basis of the natural symplectic module such that for the symplectic form we have,and for all.Then the vectoris invariant under the symplectic group.If this vector must span the unique trivial direct summand mentioned above.Small degree representations in deﬁning characteristic9 The group contains a subgroup isomorphic to.An element acts on the subspace of spanned by and by its inverse on the subspace spanned by.Hence restricted to this subgroup is isomorphic to.Restricting the representation to this subgroup we ﬁnd the decomposition, see[4,I,12.].Using the known degrees for case,we see that this contains at most trivial constituents.The similar restriction to a subgroup of type shows by induction that one irreducible constituent of has degree at least .Comparing degrees we see that the three non-trivial constituents in the restriction to must lie in a single constituent of.To summarize,we have at most one non-trivial and two trivial constituents.If there are trivial constituents then one must be in the sockle,i.e.,there must be an invariant vector.This can only be the one we see in the-summand of the restriction to.We can write down such a vector and check that it is not invariant under the whole group ,apply an element which does not leave the space spanned by invariant. We have proved that is irreducible.We can argue very similarly for.If weﬁnd that it is a direct sum of an irreducible and the trivial module found above.If weﬁnd that there is one non-trivial constituent,exactly one trivial constituent in the sockle and at most two trivial constituents.Since in this case the trivial constituent in the sockle is not a direct summand there must be a second trivial constituent in the head,by duality.If has a highest weight vector then is contained in and has weight.Hence.The other constituents of the tensor product must correspond to dominant weights smaller than,there are only two of them,and.We get that the non-trivial constituent of is isomorphic to.(Case)This can be handled by almost exactly the same arguments as the case .Here the trivial submodule is contained in.(Case)In this case we consider the restrictions of to subgroups of type,.This leads to decompositions.Comparing this decomposition for and or,respectively,and using the results for type and in-duction we see as in type that has only one non-trivial and maybe a few trivial constituents.There is an such that and.The decomposition above for this shows that there are at most two trivial constituents.Furthermore,as in type weﬁnd a trivial submodule.Either this is a direct summand(if)then it is the only trivial constituent or otherwise there is a second constituent in the head of .The argument for is again very similar.Thisﬁnishes the proof.It would be interesting if the degrees in our list could be determined more sys-tematically in the framework of highest weight modules.Then one could work out systematically generalizations of Theorem5.1where the bound is substituted by anyﬁxed polynomial in.Checking that the statement in Theorem5.1is also true for of type and ,we get the following Corollary.This completes the list in[7]which gives all representations of in non-deﬁning characteristic of degree at most.Corollary5.2For any simple over weﬁnd all-restricted weights of such that has degree in the tables given in Theorems4.4and5.1.10Frank L¨u beck6Appendix:Tables for groups of rank at most 11In this section we give the detailed lists for Theorem 4.4.We start by introducing some notation and describe the centers of the groups and the action of the fundamental weights on the center.6.1Ordering of fundamental weightsFor the irreducible types of root systems we choose the following ordering for thesimple roots ,the corresponding corootsand the fundamental weights ,.(We show the Dynkin diagrams with the node of labeled by .)14571457214145432126.2Action of fundamental weights on the center ofWe give for each irreducible type of root system the values of for in the center of .To compute this we use that is contained in a maximal torus of .Such a is isomorphic to and consists of those withfor all .So,we have to solve a system of equations given by the Cartan matrix of the root system.We denote an element whose multiplicative order is .For we write for the maximal divisor of prime to .For elements we write.is cyclic of order .For the generator we have(,odd)is cyclic of order for odd and for .There is a generator such that(,even)is elementary abelian of order with if is odd and if.There are generators and such that.Here is the element such that.()is cyclic of order if and if.There is a generator such that.()is trivial.6.3Frobenius-Schur indicatorsIf is odd we can determine the Frobenius-Schur indicators of the representations in our lists using a result of Steinberg,see[15,Lemmas78and79].If is of type or of type with odd or of type then the representations and,where the permutation is given by the auto-morphism of order two of the Dynkin diagram,are dual to each other.For other all are self-dual.If is self-dual(and recall that is odd)then its Frobenius-Schur indicator is ‘+’if has no element of order.Otherwise it is the sign of where is the only element of order in,respectively in case with even.This can be computed by6.2.6.4An exampleAs an example how to read the data in this Appendix,let us determine all irreducible representations in deﬁning characteristic for groups of type Spin, which have degree:We apply Theorems2.2and2.3.Weﬁrst need to compute all factorizations of into factors which appear as degrees in6.22.Although many divisors of appear as degree the only factorization is.Now we have a closer look at6.22.If then there is no irreducible representation of degree.And the listed representations of degree do not correspond to-restricted weights.So,in charac-teristic there are no irreducible representations of this degree.The same conclusion holds for.In this case there is no irreducible represen-tation of degree.If there is of degree and of degree.Ifweﬁnd for any,,the irreducible representationrestricted to of degree.Furthermore we see in6.22 that and have degree.For larger there is no representation of degree in our list.We onlyﬁndand in case also.In all cases where we have found representations is odd and so the center of and of is of ing6.2we see that for the non-trivial element in the center we have and for.Applying this to the representations found above weﬁnd that only in case is not faithful.From6.3we see that this is also the only representation with Frobenius-Schur indicator‘+’.The others have indicator‘-’.In type the representations andare dual to each other,in particular they have the same dimension.In6.5to6.20we save some space by only including one of such a pair of representations.6.5Case,(Recall the remark before6.5.)deg deg deg6.6Case,(Recall the remark before6.5.)deg deg deg6.7Case,(Recall the remark before6.5.)deg deg deg6.8Case,(Recall the remark before6.5.)deg deg deg6.9Case,(Recall the remark before6.5.)deg deg deg6.10Case,(Recall the remark before6.5.)deg deg deg1(00000000)all966(00010001)22079(10000011)2,3,59(00000001)all966(00010001)32304(00001100)336(00000010)all990(00000012)all2352(00001010)245(00000002)all1008(00001001)52385(00100010)379(10000001)31050(00010001)2,32394(00100010)280(10000001)31135(01000010)72520(00000200)384(00000100)all1214(01000010)22691(00100010)7 126(00001000)all1215(01000010)2,72700(00100010)2,3,7 156(00000011)31278(00000013)52772(00000102)5 165(00000003)all1287(00000005)all2844(00000021)3 240(00000011)31359(10000011)32907(00002000)3 306(01000001)21395(10000003)112922(00000022)5 315(01000001)21440(10000003)112970(00000021)3 387(10000002)51461(01000002)33003(00000006)all 396(10000002)51540(01000002)33060(00010010)5 414(00000020)31554(00000200)33139(00011000)3 495(00000004)all1764(00000110)23168(00000013)5 504(00000101)21864(20000002)113318(00001010)5 540(00000020)31890(00000102)53402(00001010)2,5 630(00000101)21890(00000110)2,33414(02000001)3 684(00100001)71943(20000002)53465(00100002)all 720(00100001)71944(20000002)5,113654(00001002)3 882(00000110)32034(10000011)23744(00010010)3 882(00001001)52043(10000011)53780(00010010)3,5。

给别人建议的英语作文

When writing an essay to offer advice to others,its crucial to structure your thoughts clearly and provide practical,actionable suggestions.Heres a detailed guide on how to approach such an essay:1.Introduction:Start with a hook to grab the readers attention.This could be a thoughtprovoking question,a surprising fact,or a relevant quote.Briefly introduce the topic and the purpose of your advice.2.Understanding the Problem:Clearly define the issue or situation for which you are offering advice. Empathize with the reader by acknowledging the challenges they might be facing.3.Body Paragraphs:Each paragraph should focus on a single piece of advice.Begin each paragraph with a topic sentence that clearly states the advice.Provide explanations,examples,or evidence to support your advice.This could include personal experiences,expert opinions,or research findings.Use transition words to connect your ideas and maintain a logical flow.4.Practical Steps:Offer stepbystep guidance on how to implement your advice.Break down complex advice into manageable actions.5.Addressing Objections:Anticipate potential objections or concerns the reader might have and address them. Provide reassurance and additional information to alleviate these concerns.6.Benefits of Following the Advice:Highlight the positive outcomes that can result from following your advice.Use persuasive language to motivate the reader to take action.7.Conclusion:Summarize the key points of your advice.Reiterate the importance of the advice and its potential impact.End with a call to action,encouraging the reader to implement the advice.nguage and Tone:Use a respectful and supportive tone throughout the essay.Avoid using jargon or overly complex language that might confuse the reader.Be positive and encouraging,even when discussing difficult topics.9.Proofreading:Carefully proofread your essay to ensure there are no grammatical or spelling errors. Check that your advice is clear,concise,and easy to understand.10.Citations and References:If youve used any sources to support your advice,make sure to cite them properly to avoid plagiarism.Remember,the goal of an advice essay is not only to inform but also to inspire action.By following these guidelines,you can craft an essay that is both helpful and persuasive.。

Abstract In this paper

Symplectic and contact Lie algebras with an application to Monge-Ampere equations
Boris S. Kruglikov
In this paper we consider symplectic and contact Lie algebras. We de ne contactization and symplectization procedures and describe its main properties. We also give classi cation of such algebras in dimensions 3 and 4. The classi cation in dimension 4 is closely connected with normal forms of nondegenerate elliptic equations of the second order on two-dimensional surfaces with transitive symmetry group in rst jets. We point out this connection and discuss normal forms.
1. Symplectic and contact Lie algebras
It is natural to consider left-invariant structures on Lie groups. For in this case they may be treated as structures on Lie algebras. For example the de Rham's complex for left-invariant di erential forms on a Lie group G coincides with the complex for Lie algebra G cohomologies' complex with coe cients in the trivial module IR. Hence everything may be expressed in the language of algebras (and re-presented for groups).

Baxter Algebras and Differential Algebras

0.3
Outline
After introducing notations and examples in Section 1, we will focus on the construction of free Baxter algebras in Section 2. We will give three constructions of free Baxter algebras. We ﬁrst explain Cartier’s construction using brackets, followed by a similar construction using a generalization of shuﬄe products. These two constructions are “external” in the sense that each is a free Baxter algebra 2
0
Introduction
This is a survey article on Baxter algebras, with emphasis on free Baxter algebras and their applications in probability theory, Hopf algebra and umbral calculus. This article can be read in conjunction with the excellent introductory article of Rota [28]. See also [31, 29]. A Baxter algebra is a commutative algebra R with a linear operator P that satisﬁes the Baxter identity P (x)P (y ) = P (xP (y )) + P (yP (x)) + λP (xy ), ∀x, y ∈ R, (0.1)

On quasicrystal Lie algebras

On quasicrystal Lie algebrasVolodymyr Mazorchuk2000Mathematics Subject Classiﬁcation:17B68,17B10,17B81Key words:aperiodic Virasoro algebra,highest weight module,Shapovalov form,Kac determinantAbstractWe realize the aperiodic Witt and Virasoro algebras as well as other quasicrystal Lie algebras as factoralgebras of some subalgebras of the higher rank Virasoro alge-bras.This realization allows us to generalize the notion of quasicrystal Lie algebras.In the case when the constructed algebra admits a conjugation,we compute the Kacdeterminant for the Shapovalov form on the corresponding Verma modules.In thecase of the aperiodic Virasoro algebra this proves the conjecture of R.Twarock.1IntroductionThis paper has grown up from my attempt to understand the recently introduced notion of quasicrystal Lie algebras and the aperiodic Witt and Virasoro algebras,[PPT,T1]. These algebras form a new family of inﬁnite-dimensional Lie algebras,whose generators are indexed by points of an aperiodic set(which is in fact a one-dimensional cut-and-project quasicrystal,an object,intensively studied by many authors,see e.g.[Ka,K,R] and references therein).Quasicrystal Lie algebras and their representations were studied in[PT,PPT,T1]and in[T2,T3]some applications of these algebras to construction of some integrable models in quantum mechanics were given.However,there are many important questions about the quasicrystal Lie algebras,which are still open.For example,in[T1]the author constructs a triangular decomposition for the aperiodic Virasoro algebra,hence constructing Verma modules,and conjectures a formula for the Kac determinant of the Shapovalov form on these modules.This formula in important both for description of simple highest weight modules and for picking up those of them which can be unitarizable,which is the question of primary interest in physical applications.It was clear from the veryﬁrst deﬁnition of quasicrystal Lie algebras,that this notion should be closely connected with the notion of the higher rank Virasoro algebras,deﬁned in [PZ].The major diﬀerence between these algebras is that the indexing set for quasicrystal Lie algebras is a discreet subset of R while for the higher rank Virasoro algebras the corresponding set is everywhere dense.In the present paper we establish this connection1by realizing quasicrystal Lie algebras as factoralgebras of some subalgebras of the higher rank Virasoro algebras.This realization allows us to generalize quasicrystal algebras in several directions,preserving the property to have a discreet indexing set.Moreover,the notion and construction of triangular decomposition for these algebras appears naturally in this framework.Further,we discuss the existence of conjugation on constructed algebras, which pairs the components of the positive and negative part.In the case,when such pairing exists,the deﬁnition of the Shapovalov form on Verma modules(see[S,MP])is straightforward and we compute the Kac determinant(see[S,KK,MP,KR])of this form. In the case of the aperiodic Virasoro algebras this proves[T1,Conjecture V.7].The paper is organized as follows:in Section2and Section3we remind the deﬁnitions of quasicrystal Lie algebras and higher rank Virasoro algebras.We give a realization of quasicrystal Lie algebras as factoralgebras of certain subalgebras of the higher rank Virasoro algebras in Section4and use it to construct parabolic and triangular decompositions of our algebras in Section5.In Section6we study the Verma modules and,in particular, calculate the determinant of the Shapovalov form on them.Weﬁnish with discussing several generalizations of our construction in Section7and Section8.2Quasicrystal Lie algebras√Denote by(·)#the unique non-trivial automorphism of theﬁeld Q(5.LetΩbe a non-empty,connected and bounded real set, whose set of inner points does not contain0.Setτ=15).Then the quasicrystal Σ(Ω),associated withΩ,is the set of all x∈Z[τ],such that x#∈Ω.The quasicrystal Lie algebra L(Ω),associated withΩ,is deﬁned as follows(see[PPT]):it is generated over F by L x,x∈Σ(Ω),with the Lie bracket deﬁned via[L x,L y]= (y−x)L x+y,x+y∈Σ(Ω)0,otherwise.To deﬁne the aperiodic Witt and Virasoro algebras as it is done in[T1],we introduce the mapϕ:Z[τ]→Z,which sends x=a+bτtoϕ(x)=b.Then the aperiodic Witt algebra AW([0,1],F)is generated over F by L x,x∈Σ([0,1]),with the Lie bracket deﬁned via[L x,L y]= (ϕ(y)−ϕ(x))L x+y,x+y∈Σ([0,1])0,otherwise.By[T1,Theorem III.4],the algebra AW([0,1],F)admits the unique central extension AV([0,1],F),called the aperiodic Virasoro algebra,which is generated over F by L x,x∈Σ([0,1]),and c,with the Lie bracket deﬁned via[L x,L y]= (ϕ(y)−ϕ(x))L x+y+δϕ(x),−ϕ(y)ϕ(x)3−ϕ(x)33The higher rank Virasoro algebrasLet P denote the free abelian group Z k ofﬁnite rank k andψ:P→(F,+)be a group monomorphism.The rank k Virasoro algebra V(ψ,F),associated withψ,is generated over F by elements e x,x∈P,and central c,with the Lie bracket deﬁned viaψ(x)3−ψ(x)3[e x,e y]=(ψ(y)−ψ(x))e x+y+δx,−yadmissible order on P.Indeed,<is obviously antisymmetric,antireﬂexiv and transitive.So,it is a partial order.But from the deﬁnition it also follows immediately,that<is linear. Further,for any a<b in P and c∈P we have(ψ(a+c)−ψ(b+c))#=(ψ(a+c−b−c))#=(ψ(a−b))#=(ψ(a)−ψ(b))#<0and hence a+c<b+c,thus<is compatible withthe addition in P.Finally,if0<a<b then0<(ψ(a))#and hence there always exists k∈N such that(ψ(b)−ψ(ka))#=(ψ(b))#−k(ψ(a))#<0,which shows that the order is admissible.Consider the rank2Witt algebra G=W(ψ,F).Without loss of generality we can assume thatΩ⊂R0,as in other case we can work with the order,opposite to<.AsΩis a connected bounded subset of R,it has one of the following four forms:[a,b],(a,b],[a,b), (a,b)for some non-negative real a,b.We deﬁne I and J as follows:I is generated by all e x such thatψ(x)#>a(resp.ψ(x)# a)if a∈Ω(resp.a∈Ω);and J is generated by all e x such thatψ(x)#>b(resp.ψ(x)# b)if b∈Ω(resp.b∈Ω).From the deﬁnition it follows immediately that both I and J are non-negative ideals of P with respect to <.Hence,the algebras L(G,<,I)and L(G,<,J)are well-deﬁned subalgebras of G and L(G,<,J)⊂L(G,<,I)by deﬁnition.Now we show that L(G,<,J)is actually a ideal of L(G,<,I).Indeed,if x∈I andy∈J we get that x+y∈J as J is an ideal of P and x∈P0+.Hence[e x,e y]∈L(G,<,J) for any e x∈L(G,<,I)and e y∈L(G,<,J).Finally,we consider the map f:L(G,<,I)→L(Ω)deﬁned byf(e x)= L x,x ∈Ω0,otherwise.From the deﬁnition of the Lie brackets in L(Ω)(Section2)and in G(Section3)weimmediately get that f is a Lie algebra homomorphisms.Moreover,it is also clear that its kernel coincides with L(G,<,J).This completes the proof.Theorem1motivates the following deﬁnition:let G=W(P,ψ)be a higher rank Witt algebra(it is important here that k>1,i.e.that G is not the classical Witt algebra), <be an admissible order on P,and I⊃J be two non-negative ideals of P with respect to the order<.Then we deﬁne the Lie algebra A(P,ψ,<,I,J)of quasicrystal type as the quotient algebra L(G,<,I)/L(G,<,J).In particular,all quasicrystal Lie algebras are Lie algebra of quasicrystal type.Now we can formulate some basic properties of Lie algebras of quasicrystal type and we see that these algebras share a lot of properties of classical quasicrystal Lie algebras.We start with the following easy observation.Lemma 1.Let<be an admissible order on Z k.Then there exists a homomorphism,σ:P→R,such thatσ(P±)⊂R±.Proof.Identify P with Z k⊂R ing the description of admissible orders on an abelian group from[Z],weﬁnd a hyperplane,H,of R k,such that P+coincides with the set of points from Z k,which are settled on the same side with respect to H.Thenσcan be taken,e.g.the projection on H⊥with respect to H(here R k is considered as an Euclidean space in a natural way).4For given G=W(P,ψ)and<weﬁx someσ,existing by Lemma1.We deﬁne a= inf x∈I(σ(x))and b=inf x∈J(σ(x))and will use this notation in the following statement. Proposition1.Let A(P,ψ,<,I,J)be a Lie algebra of quasicrystal type.1.A(P,ψ,<,I,J)is abelian if and only if2a b.2.A(P,ψ,<,I,J)has non-trivial center if and only if a=0.3.A(P,ψ,<,I,J)is nilpotent if and only if a>0and J=∅.4.The algebra A(P,ψ,<,I,J)is perfect if an only if a=0and0∈I.5.If J=∅then anyﬁnite set of elements in A(P,ψ,<,I,J)generates aﬁnite-dimensional Lie subalgebra of A(P,ψ,<,I,J).In particular,A(P,ψ,<,I,J)has ﬁnite-dimensional subalgebras of arbitrary non-negative dimension.Proof.All statements are easy corollaries from the additivity of indices of generating ele-ments under the Lie bracket.Indeed,with this remark theﬁrst statement reduces to the fact that x a and y a implies x+y 2a b;the second one reduces to the fact that for any x a and y>b−a holds x+y>b;and the third one reduces to the fact that for ka>b we have kx>b for any x a.If a=0,the algebra A(P,ψ,<,I,J)is nilpotent by statement three and hence not perfect.It is also clear that it is impossible to get0as a result of the Lie operation.But if a=0and0∈I,then L(G,<,I)=G+,σ(P+)is dense in R+and hence for any x>0there are y,z∈P+such that y+z=x.This implies that A(P,ψ,<,I,J)is perfect in this case and hence the property four.Theﬁrst part of the last statement is equivalent to the trivial statement that aﬁnite subset of R+generates an additive semigroup,whose intersection with any bounded set is ﬁnite.To prove the second part it is suﬃcient to consider the span of e ix,i=1,...,n, such that nx<b and(n+1)x>b.This completes the proof.For example,to realize the aperiodic Witt algebra AW([0,1]),deﬁned in[T1],as a Lie algebra of quasicrystal type,one should take P=Z2,ψbeing the projection on the second coordinate;<deﬁned by x<y if and only if the inner product of y−x with(1,15)) is greater than zero;I=P0+;J={x∈P:(1,0)<x}.5Standard and non-standard triangular decomposi-tionsThe realization of Lie algebras of quasicrystal type,obtained in the previous section al-lows us to adopt the technique from[M]to construct various triangular and parabolic decompositions of these algebras.The general procedure will look as follows.Let A=A(P,ψ,<,I,J)be a Lie algebra of quasicrystal type.Abusing notation we will denote by e x,x∈I\J,the generators of A.Choose any linear pre-order, ,on the abelian5group P ,which is diﬀerent from <and its opposite.Deﬁne A ±as the Lie subalgebras of A ,generated by all e x ,0≺±x ,and set A 0to be the Lie subalgebra of A ,generated by all e x ,0 x and x 0.We get the following obvious fact.Lemma 2.A =A −⊕A 0⊕A +.Proof.Clearly,A =A −+A 0+A +.The fact that this is actually a direct sum decomposition follows easily from the the property x y implies x +z y +z .It is natural to call the decomposition A =A −⊕A 0⊕A +parabolic decomposition of A ,associated with .Given a parabolic decomposition and a simple A 0-module,V ,one can extend V to an A 0⊕A +-module with the trivial action of A +and construct the associated generalized Verma module M (V )as follows:M (V )=U (A )⊗U (0⊕+)V .If A 0happens to be rather special,it is natural to rename the corresponding parabolic decomposition into triangular decomposition .However this is a subtle question and the hierarchy I give here represent only my point of view and is inspired by the corresponding notions for the higher rank Virasoro algebra ([M]).We will say that the decomposition A =A −⊕A 0⊕A +is a standard triangular de-composition provided A 0=F e x for some x ∈P ,which is not maximal in I \J .We call A =A −⊕A 0⊕A +the non-standard triangular decomposition provided A 0is a commutative Lie algebra and the parabolic decomposition fails to be a standard triangular.In the case of triangular decomposition generalized Verma modules become classical Verma modules as in this case dim(V )=1.The ﬁrst case is natural and corresponds to triangular decompositions of the higher rank Virasoro algebras,[M].Actually,here one has to be careful because,depending on whether ≺satisﬁes the Archimed law or not,one can further distinguish two cases of standard triangular decomposition.We will not do this,as we will not study the diﬀerence between the corresponding situations.But the second case has a striking diﬀerence from the ﬁrst one and comes from the deﬁnition of triangular decomposition for the aperiodic Virasoro algebra in [T1].This means that the triangular decomposition for the aperiodic Virasoro algebra,constructed in [T1]is an example of a non-standard triangular decomposition.We now will study analogous situations in more detail.We retain the notation for σ,a,b from the previous section and further assume that a =0and that that there is an element,e u ∈A ,such that σ(u )=b and ψ(u )=0.Since <is an admissible order,such element is unique and we retain the notation e u for it.Lemma 3.Under the above assumptions we consider the vectorspace A =A ⊕F c .Then the formula [e x +a c ,e y +b c ]=[e x ,e y ]+ψ(x )3−ψ(x )The algebra A,constructed in Lemma3is a natural generalization of the aperiodic Virasoro algebra from[T1].In particular,the aperiodic Virasoro algebra coincides with A for A constructed in the end of the previous section.However,if e.g.the rank of P is bigger than two,we get an example of A ,which diﬀers from the aperiodic Virasoro algebra.We will call algebras A the Virasoro-like algebras of quasicrystal type.Assigning the element c index u we easily transfer the notions of parabolic and both standard and non-standard triangular decompositions on algebra A .If P has rank two, then,up to taking the opposite order,the non-standard triangular decomposition of A, such that A 0contains e u,is unique,and in the case of the aperiodic Virasoro algebra this coincides with the triangular decomposition,constructed in[T1,Section V].For the rank two case once can easily construct example of A such that with respect to the unique natural non-standard triangular decomposition,mentioned above,dim(A 0)is an arbitrary positive integer.Hence even in rank two case one gets a lot of examples of A ,diﬀerent from the aperiodic Virasoro algebras.All these algebras will have discrete aperiodic root systems,and,if considered as graded by the action of e0,all roots will be multiple with multiplicity dim(A 0)−2.In the case of the aperiodic Virasoro algebra we have dim(A 0)=3 and hence all roots(with non-zero action of e0)are multiplicity free.We will discuss this situation in more details in the next section,when we will deﬁne the Shapovalov Form on the Verma modules and compute its determinant.6Shapovalov form and Kac determinantIn this section we present several results on the structure of Verma modules over Lie and Virasoro-like algebras of quasicrystal type.As in the case of the Witt and the Virasoro algebras,the representation theory the last one is more complicated,which,in particular, gives a bigger variety of simple highest weight modules.Our main tool in the case of the Virasoro-like algebras of quasicrystal type and the corresponding Lie algebras of quasicrys-tal type will be the Shapovalov form on Verma modules,ﬁrst deﬁned in[S]for simple ﬁnite-dimensional Lie algebras.However,we start with more elementary general case of Lie algebras of quasicrystal type,which happens to be really trivial.Before starting we just note that in this section we always assume that F is an algebraically closedﬁeld of characteristic zero.We recall that,given a triangular decomposition,A=A−⊕A0⊕A+,an A-module,M, is called a highest weight module,if there exists a generator,v∈M,such that A+v=0. Proposition2.All simple highest weight modules over a Lie algebras of quasicrystal type, which correspond to a standard triangular decompositions with A0=F e x⊂[A,A]are one-dimensional.In particular,corresponding Verma modules are always reducible. Proof.This is a direct corollary of A0=F e x and e x∈[A,A].Proposition3.All Verma modules over a Lie algebras of quasicrystal type,which corre-spond to a standard triangular decompositions with A0=F e x⊂[A,A],are reducible.The7corresponding unique simple quotients are one-dimensional if and only if the eigenvalue ofe x on the primitive generator of the module is zero.Otherwise they are inﬁnite-dimensional. Proof.Let v be the canonical generator of the Verma module in question.The reducibilityfollows from the fact that x is not maximal in I\J,and hence there are inﬁnitely manyelements y∈P−satisfying e x∈[e y,A],which implies that U(A)e y v is a proper submodule of the Verma module.The second statement follows considering the set of elements e y,y∈P−,satisfying e x∈[e y,A],which is obviously inﬁnite.So,we can now move on to the case of non-standard triangular decomposition.Firstwe reduce our consideration to the natural case of weight modules withﬁnite-dimensionalweight spaces,which corresponds to the situation,when the root system of A is discrete. This is only possible in the case when P Z2.Here our main tool will be the Shapovalov form and to be able to work with it we will also need the following assumptions from the previous section:e0∈A;and there is e u∈A,such thatσ(u)=b andψ(u)=0.As it was mentioned above,this situation covers,for example,the case of the aperiodic Witt algebra.Since in the case of the algebra A the arguments will be absolutely the same,we consider both cases simultaneously with all the notation for the algebra A .The case of A is then easily obtained by factoring c=0out.We deﬁne the conjugation on P viaω(x)=u−x and it follows immediately from our assumptions that e x∈A implies eω(x)∈A .However,is easily to see thatωdoes not extend to an(anti)involution on A .We note thatσ(ω(x))=b−σ(x).We recall that the algebra A is graded by the adjoint action of e0(or,more general,A 0)and for C α=0the dimension of A αis either0or dim(A 0)−2(dim(A0)−1inthe case of algebra A).We denote by∆the set of all(non-zero)roots of A with respectto this action and by∆±the sets of all positive and negative roots corresponding to our triangular decomposition.Obviously,ωextends to a linear bijection A α→A −αfor any α∈∆∪{0}.As A 0is commutative,simple A 0-modules are one-dimensional and have the form Vλ,λ∈(A 0)∗,where the action is deﬁned via g(v)=λ(g)v for v∈Vλand g∈A 0.Let vλdenote a canonical generator of M(Vλ).Let∆ (resp.∆ ±)denote the semigroup,generated by∆(resp.∆±).Then the module M(Vλ)is a weight module with respect to A 0with the supportλ∪λ−∆ +.All weight spaces of M(Vλ)areﬁnite dimensional.Moreover,M(Vλ)is isomorphic to U(A −)vλas a vectorspace.The∆±-gradation of A ±extends to the∆ ±-gradation of U(A ±)and,in the aniinvolutive way,ωextends to a linear componentwise isomorphism from U(A +)to U(A −)and back, which matches U(A +)αwith U(A −)−α.Forµ∈Supp(M(Vλ)),µ=λ−ν,ν∈∆ +,we deﬁne the Shapovalov form Fλ,νon M(Vλ)µby setting that Fλ,ν(fvλ,gvλ),f,g∈U(A −)−ν,equals the coeﬃcient ofω(f)gvλ∈M(Vλ)λ,written in the basis{vλ}.The following properties of Fλ=⊕ν∈∆Fλ,νare standard and the reader can consult[KK,MP]for the arguments.8Lemma4. 1.M(Vλ)is simple if and only if Fλis non-degenerate.2.The kernel of Fλcoincides with the unique maximal submodule of M(Vλ).Hence in order to study the reducibility of M(Vλ)it is suﬃcient to compute the determinant of Fλ,νfor allλandν.To be able to do this we consider the followingmonomial generators of U(A −)−ν:G=G(ν)={g(x1,...,x k)=e x1...e xk:x i∈∆−;i x i=−ν;σ(x i) σ(x i+1)}.We deﬁne the linear order on this set of generators as the lexicographical order with respect to the values ofσ(x i).The key property of this construction is the following.Lemma 5.If g(x1,...,x k)∈G and g(y1,...,y m)∈G are such that g(x1,...,x k) g(y1,...,y m).Then Fλ,ν(g(x1,...,x k)vλ,g(y1,...,y m)vλ)=0.Proof.Let i be minimal such thatσ(x i)<σ(y i).Thenσ(ω(x i))>b−σ(y i)and henceeω(xi)commutes with e yiand thus with all e yj,j i,since for such j we haveσ(y j) σ(y i)form the deﬁnition of G.For j<i we have x j=u j and thus[eω(xj),e yj]∈A 0.We canwriteeω(xi)(ω(e x1...e xi−1)e y1...e yi−1)e yi...e ymvλ==ε(ω(e x1...e xi−1)e y1...e yi−1)eω(xi)e yi...e ymvλ+other termsfor someε∈F,where in other terms some eω(xj),j<i,occurs already after the corre-sponding e yj .As eω(xi)commutes with all e yj,j i,we get that theﬁrst summand equalszero.Now consider one of the other terms and let eω(xj)be the factor occurring most to theright in the monomial.This means,in particular,that for s<j this monomial contains[eω(xs),e ys],which are the elements of A 0and thus,up to a scalar factor,can be movedto the left.In particular,σ(ω(x s))is the biggest value among all others occurring in thismonomial.If the element e y,standing next to eω(xj)satisﬁes y=x j,this means that eω(xj)commutes with e y and hence the monomial contributes0to the global sum.Otherwise thenumber of factors,standing to the right from eω(xj),which equal x j,is less than the samenumber before the last commutation.Hence induction in this number reduces the problem to the case y=x j thus proving that all monomials occurring in other terms contribute0 to the global sum.From this it follows directly that Fλ,ν(g(x1,...,x k)vλ,g(y1,...,y m)vλ)=0,which com-pletes the proof.From Lemma5we immediately get the following statement,which,in particular,proves [T1,Conjecture V.7].Corollary 1.The determinant of Fλ,νcoincides with the product of diagonal elements Fλ,ν(g(x1,...,x k)vλ,g(x1,...,x k)vλ).9Now we can formulate the computation results for the determinant of the Shapovalov form and the corresponding corollaries for the structure of M(Vλ).Denote by P the set of all non-zero x∈P such that e x∈A is non-zero.Then the decomposition∆=∆−∪∆+ induces a decomposition P =P −∪P +.Forν∈∆ +and x∈P −we denote by pν(x)thenumber of occurrences of e x as factors in the canonical decomposition of all monomials in G(ν).Theorem2.Up to a non-zero constant the determinant of Fλ,νequalsx∈P − λ(e u)−ψ(x)2−112c,asψ(u)=0.Moreover,[e u−x,e u]is in fact central in A .Hence,we can movethe non-zero factor2ψ(x)out and get that,up to a non-zero constant factor,we haveFλ,µ(g(x1,...,x k)vλ,g(x1,...,x k)vλ)=ki=1 λ(e u)−ψ(x i)2−1Proof.Under these conditions all factors of the diagonal elements of the matrix of the Shapovalov form are non-negative and hence all leading minors are non-negative as well. This implies the statement.Using these results we also get some information about highest weight modules,asso-ciated with standard triangular decompositions.Corollary4.The dimensions of the weight spaces of inﬁnite-dimensional highest weight modules over Lie algebras of quasicrystal type,associated with standard triangular decom-positions,are not uniformly bounded.Proof.Let A0=F e x be the zero component of the given standard triangular decomposition. Then we can factor our an ideal of A such that the factoralgebra is still of quasicrystal type,but the element x became maximal in the corresponding I\J,and hence the induced triangular decomposition became non-standard.Now we haveλ(e x)=0and hence the corresponding Verma module over this algebra is simple and the dimensions of its weight spaces are obviously unbounded.Buy this module naturally embeds(as a vector subspace) into the simple highest weight module,which we started with.7Further generalizations of quasicrystal Lie algebras Geometrical realization of the algebra A,obtained in Section4,motivates the following generalization of the class of Lie algebras of quasicrystal type.We consider arbitrary rank n Witt algebra G=G(P,ψ)with P Z n being realized in R n in a natural way.LetΩbe a convex subset of R n,containing at least one non-zero point of P,and satisfying the following0-star condition:v∈Ωimpliesλv∈Ωfor all λ>1.In this case we will callΩa0-star sets.Denote by L(Ω)the vectorsubspace in G, spanned by e x,x∈Ω.Lemma6.L(Ω)is a Lie subalgebra of G.Proof.If x,y∈P∩Ωthen x+y=2(12y).12y belongs toΩbecause of theconvexity and thus x+y∈Ωby the0-star condition.Lemma7.LetΩbe a0-star set,v∈Ωandλ>0.Then,if the setΩλ,v=λv+Ωcontains at least one non-zero point of P,it is a0-star set.Moreover,Ωλ,v⊂Ω.Proof.Clearly,Ωλ,v is convex.Further,if w∈Ωandγ>1,thenγ(w+λv)=γ(λ+1)(1λ+1v)belongs toΩby the same arguments as in Lemma6.This completes theproof.Lemma8.LetΩbe a0-star set,v∈Ωandλ>0.Then L(Ωλ,v)is an ideal of L(Ω). Proof.If w∈Ωand w =w +v∈Ωλ,v for some w ∈Ω,then w+w =w+w +v∈Ωλ,v. This implies the statement.11Hence,for arbitrary G,Ωand v as above we can form the algebra A(P,ψ,Ω,λ,v)= L(Ω)/L(Ωλ,v),which we will call a Lie algebra of convex quasicrystal type.To obtain the usual Lie algebra of quasicrystal type,one should takeΩto be a half-space(open or closed), which does not contain0as an inner point.The basic properties of Lie algebras of convex quasicrystal type are similar to those of Lie algebra of quasicrystal type,however,their formulation is much more complicated because it usually depends on the structure ofΩ. Here we list only some most straightforward ones.Proposition4.Let A=A(P,ψ,Ω,λ,v)be a Lie algebra of convex quasicrystal type and a denote the inﬁnum of distances from points inΩ∩P to0.Assume that dim(A)>1and that for any x∈Ωsome neighborhood(in R n)of2x belongs toΩ.Then1.if a>0then any element of A is nilpotent.2.Anyﬁnite set of elements from A generates aﬁnite-dimensional Lie subalgebra of A. Proof.If x∈S=Ω\Ωλ,v then there alway exists y,such that|x−y| |v|and such that y∈Ω.Let w∈Ω.If some ball of radius r over2w belongs toΩ,then,forλ>1the point 2λw belongs toΩtogether with the ball of radiusλr around it.Makingλr>|v|we get that2λw∈Ωλ,v.This implies theﬁrst statement.If the set{w1,...,w k}⊂Ωisﬁnite,then weﬁnd some r such that2w i belongs toΩtogether with its neighbor ball of radius r.Then the same is true for all linear combinations of these elements with non-negative integer coeﬃcients.By the same arguments as in the previous paragraph,there is N∈N such that any linear combination of{Nw1,...,Nw k} with non-negative integer coeﬃcients belongs toΩλ,v.This implies the second statement.Let us study an example of such algebra,which,as we will show,has some interesting properties.Take P=Z2,ψthe projection on the second component,Ω={w∈R2: (w,(1,1)) 0and(w,(1,−1)) 0},v=(n+ ,0),n∈N, ∈(0,1).The corresponding algebra A=A(P,ψ,Ω,λ,v)is graded with respect to the e0action with graded components corresponding to all integers and having dimension n.In particular,one can deﬁne and study triangular(parabolic)decompositions of this algebra and corresponding(generalized) Verma modules.The set P =P∩(Ω\Ωλ,v)coincides with{(a,b):0 a−|b| n}. Deﬁne the conjugationωon this set viaω(a,b)=(2|b|+n−a,−b).Then we have the natural notions of Verma modules and the Shapovalov form on them.In our situation we have∆+=N.Lemma9.The Verma module M(Vλ)is always reducible.However,the unique sim-ple quotient of M(Vλ)is inﬁnite dimensional if and only if at least one of the numbers λ(e(2,0)),...,λ(e(n,0))is non-zero.Otherwise it is one-dimensional.Proof.We note that the intersection of[A,A]with A0coincides with the linear span˜A0 of elements{e(2,0),...,e(n,0)}.Hence,if the restriction ofλon˜A0is zero,the Shapovalov form in identically zero on all M(Vλ)λ−k,k∈N.Otherwise,assume thatλ(e(i,o))=0and take e x∈A1and e y∈A−1such that[e x,e y]= e(i,0).We get Fλ,k(e k,e k x)=0and the statement is proved.ω(y)12。

Quantum Reduction for Affine Superalgebras

Abstract We extend the homological method of quantization of generalized Drinfeld–Sokolov reductions to aﬃne superalgebras. This leads, in particular, to a uniﬁed representation theory of superconformal algebras. 1991 MSC: 17B65, 17B67, 81R10 1990 PACS: 02.20, 03.65.F, 89
simplest case of g = sℓ2 one recovers thereby the minimal series modules over the Virasoro algebra = Wk (sℓ2 ).) It is straightforward to generalize this construction to the case when f is an even nilpotent element, that is for the sℓ2 -triple e, x, f , such that [e, f ] = x, [x, e] = e, [x, f ] = −f , all eigenvalues 1 of ad x are integers (for general f they lie in 2 Z). One just takes instead of n the subalgebra g+ of g spanned by eigenspaces with positive eigenvalues for ad x. Unfortunately, most nilpotent elements are not even, but often one can replace x by x′ such that ad x′ has integer eigenvalues, so that the construction gives the same homology (see e.g. [BT]). However, it remained unclear how to make it work for a general simple Lie algebra g and a general nilpotent element f . The situation gets worse if one tries to go to the Lie superalgebra case since already the simplest Lie superalgebra spo(2|1) has no good Z- gradations. In the present paper we show how to resolve this problem. It turns out that one needs only to add neutral fermionic ghosts associated to the currents of the eigenspace g1/2 of ad x. This is done in Section 2, where to each quadruple (g, x, f, k), where g is a simple ﬁnitedimensional Lie superalgebra with a ﬁxed even invariant bilinear form (.|.), x is an ad-diagonizable 1 Z, f is a nilpotent even element of g such that [x, f ] = −f , and element of g with eigenvalues in 2 k ∈ C, we associate a BRST complex (C (g, x, f, k) = Vk (g) ⊗ F ch ⊗ F ne , d0 ) . Here Vk (g) is the universal aﬃne vertex algebra of level k associated to g, F ch is the vertex algebra ne is the vertex algebra of free of free charged fermions based on g+ + g∗ + with reversed parity, F neutral fermions based on g1/2 with the form a, b = (f |[a, b]), and d0 is an explicitly constructed odd derivation of the vertex algebra C (g, x, f, k) whose square is 0 (see Section 2.1). The main object of our study is the 0th homology of this complex, which is a vertex algebra, denoted by Wk (g, x, f ). In the case when the pair (x, f ) can be included in an sℓ2 -triple (e, x, f ) (then x is determined by f up to conjugation), we denote this vertex algebra by Wk (g, f ). In this case the map adf : g1/2 → g−1/2 is an isomorphism, which suﬃces for the construction of the energy-momentum ﬁeld L(z ) of Wk (g, x, f ) (see Section 2.2); under the same assumption, we construct ﬁelds J {v} in Wk (g, x, f ) of conformal weight 1, corresponding to each element v ∈ gx,f , the centralizer of x and f (see Section 2.4). As in [FF2, FKW], given a restricted g-module M of level k, hence a Vk (g)-module, we extend it to a C (g, x, f, k)-module C (M ) = M ⊗ F ch ⊗ F ne , which gives rise to a complex (C (M ) , d0 ) of C (g, x, f, k)-modules. Its homology H (M ) is a Wk (g, x, f )-module. In Section 3.1 we compute the Euler–Poincar´ e character of this module: chH (M ) (h) = (−1)j trHj (M ) q L0 e2πiJ0 j ∈Z

1101.4484v1 Some general results on conformal embeddings of affine vertex operator algebras

a r X i v :1101.4484v1[mat h.Q A]24J a n211SOME GENERAL RESULTS ON CONFORMAL EMBEDDINGS OF AFFINE VERTEX OPERATOR ALGEBRAS DRA ˇZEN ADAMOVI ´C AND OZREN PER ˇSE Abstract.We give a general criterion for conformal embeddings of vertex operator algebras associated to aﬃne Lie algebras at arbitrary ing that criterion,we construct new conformal embeddings at admissible rational and negative integer levels.In particular,we con-struct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras.The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.1.Introduction Let U and V be vertex operator algebras of aﬃne type.We say that U is conformally embedded into V if U can be realized as a vertex subalge-bra of V with the same Sugawara Virasoro vector.The most interesting case of conformal embedding is when V is a ﬁnite direct sum of irreducible U –modules.These conformal embeddings are studied in various aspects of conformal ﬁeld theory (cf.[6],[8],[9],[15],[22],[36]),the theory of tensor cat-egories (cf.[21],[28])and in the representation theory of aﬃne Kac-Moody Lie algebras (cf.[12]).The construction and classiﬁcation of conformal embeddings have mostly been studied for simple aﬃne vertex operator alge-bras of positive levels.Some examples of conformal embeddings at rational admissible levels were constructed in [33]and [35].In the present paper we give both necessary and suﬃcient conditions forconformal embeddings at general levels,within the framework of vertex op-erator algebra theory.In this way we will be able to construct new examples of conformal embeddings at rational and negative integer levels.Let us ex-plain our results in more details.Let g be a simple ﬁnite-dimensional Lie algebra and g 0its subalgebra which is a reductive Lie algebra.Let N g (k,0)be the universal aﬃne vertex operator algebra of level k =−h ∨associated to g ,and L g (k,0)its simple quotient.Then g 0generates a subalgebra U g0(resp. U g 0)of N g (k,0)(resp.L g (k,0)).Let ωg 0be the Virasoro vector in U g 0which is a sum of usualVirasoro vectors obtained by using Sugawara construction.Let 2000Mathematics Subject Classiﬁcation.Primary 17B69,Secondary 17B67,81R10.12DRAˇZEN ADAMOVI´C AND OZREN PERˇSEn∈ZL(0)x(−1)1=x(−1)1(x∈W).(1.1)Moreover,the condition(1.1)implies that N g(k,0)contains a singular vector of conformal weight2.If k is a positive integer,then N g(k,0)has a singular vector of conformal weight2if and only if k=1.So we only have conformal embeddings into L g(1,0).But in general we have such singular vectors for other values of k (cf.Remark2.4).Next we apply Theorem1.1in the case when g0is a simple Lie algebra, andg=g0⊕V g0(µ1)⊕···⊕V g(µs),where V g0(µi)is irreducibleﬁnite-dimensional g0-module with highest weightµi,and V g(µi)(i=1,...,s)is orthogonal to g0with respect to the invariant bilinear form on g.Denote by L g0(k′,0)the subalgebra of L g(k,0)generated by g0,where k′=ak and a∈N is the Dynkin index of the embedding g0<g.Note that in the case of general levels,vertex algebra L g0(k′,0)is not necessarily simple.Theorem1.1then implies that the key condition for the conformal em-bedding is that the eigenvalue of the Casimir operator of g0is the same for all V g(µi),i=1,...,s.More precisely,if(µi,µi+2ρ0)0=(µj,µj+2ρ0)0for all i,j=1,...,s, where(·,·)0denotes the suitably normalized invariant bilinear form on g0, then for k,k′such that(µi,µi+2ρ0)03 Our methods enable us to construct all conformal embeddings associated to automorphisms of Dynkin diagrams.Let us illustrate this result by the following table:Table A,g,g0are simple Lie algebras,g0<g.g decomposition of L g(k,0)k′Cℓ−1A2ℓL g0(k′,0)⊕L g(k′,2ω1)2A11DℓL g0(k′,0)⊕L g(k′,ω1)−ℓ+2F4−3D4L g0(k′,0)⊕L g(k′,ω2)⊕−2L(1)g0(k′,ω1)⊕L(2)g(k′,ω1)In order to describe all conformal embeddings for such automorphisms,we include into this table some well-known conformal embeddings at positive integer levels in the cases(A2ℓ,Bℓ),(A2,A1)(cf.[22];see also[4],[37])and the conformal embedding for pair(A2ℓ−1,Cℓ)from[5].But other conformal embeddings at negative integer levels are new.We also apply our methods to aﬃne vertex operator algebras at admissible levels(cf.[1],[3],[26],[27],[32],[33],[34],[37])and get new conformalembedding of L A2(−5/3,0)into L G2(−5/3,0)with decompositionL G2(−5/3,0)=L A2(−5/3,0)⊕L A2(−5/3,ω1)⊕L A2(−5/3,ω2).We also determine the decomposition for the conformal embedding of L Dℓ(−ℓ+3/2,0)into L Bℓ(−ℓ+3/2,0)from[35].We should also mention that our methods give an uniform proof for a family of conformal embeddings which does not require explicit realizations of aﬃne Lie algebras or explicit formulas for singular vectors.In most cases one can check all conditions for conformal embeddings only by using tensor product decompositions ofﬁnite-dimensional modules of simple Lie algebras. The notion of conformal embedding is also closely related to the notion of extension of vertex operator algebra.We note that simple current extensions of aﬃne vertex operator algebras at positive integer levels were studied in [13],[31].We assume that the reader is familiar with the notion of vertex operator algebra(cf.[10],[18],[19]).2.A general criterion for conformal embeddingsIn this section we shall derive a general criterion for the conformal embed-ding of aﬃne vertex operator algebras associated to a pair of Lie algebras (g,g0),where g is a simple Lie algebra and g0its reductive subalgebra.Let g be the simple complexﬁnite-dimensional Lie algebra with a non-degenerate invariant symmetric bilinear form(·,·).We can normalize the form on g such that(θ,θ)=2,whereθis the highest root of g.Letρbe the4DRAˇZEN ADAMOVI´C AND OZREN PERˇSEsum of all fundamental weights for g and let h∨be the dual Coxeter number for g.Assume that g0is a Lie subalgebra of g such that(1)bilinear form(·,·)on g0is non-degenerate and g=g0⊕W is anorthogonal sum of g0and a g0–module W;(2)g0=⊕n i=0g0,i is an orthogonal sum of commutative subalgebra g0,0and simple Lie algebras g0,1,...,g0,n(i.e.,g0is a reductive Lie alge-bra).Let(·,·)g0,i be the non-degenerate bilinear form on g0,i normalized suchthat(θ0,i,θ0,i)g0,i =2,whereθ0,i is the highest root of g0,i.One can showthata i(·,·)g0,i=(·,·)|g0,i×g0,i for certain a i∈N.(For i=0,we can take a0=1).The(n+1)–tuple(a0,a1,···,a n)is called the Dynkin multi-index of embedding g0into g.Let{u i,j}j,{v i,j}j be a pair of dual bases for g0,i such that(u i,j,v i,k)g0,i=δj,k.Let h∨0,i be the dual Coxeter number for g0,i.(For i=0we take h∨0,0=0).Let N g(k,0)be the universal aﬃne vertex operator algebra of level k=−h∨associated to g(cf.[17],[20],[25],[29],[30]).Let N1g(k,0)be the max-imal ideal of N g(k,0),and L g(k,0)=N g(k,0)/N1g(k,0)the corresponding simple vertex operator algebra.Then g0,i generates a vertex subalgebra of N g(k,0)which is isomorphicto N g0,i (k i,0)for k i=a i k.Moreover,g0generates a subalgebra of N g(k,0)which is isomorphic toU g0=⊗n i=0N g0,i(k i,0).Let U g0be the subalgebra of L g(k,0)generated by g0.Then U g0is iso-morphic toU g0=⊗n i=0 L g0,i(k i,0), where L g0,i(k i,0)is a certain quotient of N g0,i(k i,0).Deﬁnition2.1.We say that the vertex operator algebra U g0is conformally embedded into L g(k,0),if U g0is a vertex subalgebra of L g(k,0)with the same Virasoro vector.Letωg be the usual Sugawara Virasoro vector in N g(k,0)and letωg0,i =15 As usual we identify x∈g with x(−1)1∈N g(k,0).Therefore,W can be considered as a subspace of N g(k,0).LetL(n)z−n−2=Y(ωg,z).Theorem2.2.Assume that2(k i+h∨0,i)[[x,u i,j],v i,j](−1)1=k1+h∨0,1+···+k n dim(g0,1)k+h∨.(2.6)So our condition(2.1)from Theorem 2.2implies numerical criterion (2.6).But,in general,(2.6)is only a necessary condition for conformal embeddings.Remark2.4.Assume that the conditions from Theorem2.2are satisﬁed. Then N g(k,0)must have at least one singular vector at conformal weight2. Some explicit formulas for these singular vectors appeared in[1],[2],[32], [33],[34]and recently in[7].We shall now see how the theory presented hereﬁts into some examples.6DRA ˇZEN ADAMOVI ´CAND OZREN PER ˇSE Example 2.5.Let g be a simple complex Lie algebra of type A ,D or E ,h its Cartan subalgebra and ∆its root system.Then g =h ⊕W,W = α∈∆g α.Take now g 0=h in the above construction.ThenL (0)|W ≡Id iﬀk =1or k =−1/2.This leads to the conformal embeddings L g 0(2,0)<L g (1,0), L g 0(−1,0)<L g (−1/2,0).Example 2.7.By using the same method as above we obtain the following conformal embeddings:(1)Let g =sl (ℓ+1,C ),g 0=gl (ℓ,C ).Then we have conformal embeddings:L g 0(1,0)<L g (1,0),L g 0(−ℓ+12,0).(2)Let g =o (2ℓ,C ),g 0=gl (ℓ,C ).Then we have conformal embeddings:L g 0(1,0)<L g (1,0), L g 0(−2,0)<L g (−2,0).Remark 2.8.One can show that in examples presented above the vertex operator algebra L g (k,0)is not a ﬁnite sum of irreducible L g 0(k ′,0)–modules.But it is still possible that there exists certain twisted L g (k,0)–module which is a ﬁnite sum of irreducible twisted L g 0(k ′,0)–modules.In particular,this holds for Z 2–twisted module for L g (1,0)which gives principal realization of level one modules for aﬃne Lie algebras in Example 2.5.It seems that to get conformal embeddings such that L g (k,0)is a ﬁnite sum of irreducible L g 0(k ′,0)–modules,one needs to consider the case when g 0is semisimple.In this article we shall make the ﬁrst step and consider the important case when g 0is a simple Lie algebra.So we shall now assume that g 0is a simple Lie algebra.In the above settings we take n =1,g 0=g 0,1.Set(·,·)0=(·,·)g 0,1,h ∨0=h ∨0,1,a =a 1.Let ρ0be the sum of all dominant integral weights for g 0.7 Assume thatg=g0⊕g1⊕···⊕g s,such that(1)g i=V g0(µi)is irreducibleﬁnite-dimensional highest weight g0-modulewith highest weightµi,(2)g i⊥g0for i=1,...,s(with respect to(·,·)),(3)(µi,µi+2ρ0)0=(µj,µj+2ρ0)0(i,j>0).Then g0generates a subalgebra of N g(k,0)isomorphic to U g0∼=Ng0(k′,0),where k′=ak.Letωg(resp.ωg0)be the Virasoro vector in N g(k,0)(resp.N g(k′,0))obtained by the Sugawara construction.In what follows we choose k such that(µi,µi+2ρ0)0L(0)x(−1)1=x(−1)1(x∈W).As before,let U g0= L g0(k′,0)be the subalgebra of L g(k,0)generated by g0.Now Theorem2.2implies the following result: Corollary2.9.Assume that the above conditions hold.Then L g0(k′,0)is conformally embedded into L g(k,0).The simplicity of the vertex operator algebra L g0(k′,0)will be investigated in the next section.3.On semisimplicity of L g(k,0)as g0–moduleIn this section we give suﬃcient conditions for complete reducibility of L g(k,0)as g0–module.In particular,we obtain the conformal embedding ofsimple vertex operator algebras L g0(k′,0)<L g(k,0).We assume that g0is a simple Lie algebra and that all conditions on embedding g0<g from Section2hold.Furthermore,we assume that g0is theﬁxed point subalgebra of an automorphismσof g of order s+1,and thatV g0(µi)is the eigenspace associated to the eigenvalueξi(for i=1,...,s),whereξdenotes the corresponding primitive root of unity.Thenσcan be extended to aﬁnite-order automorphism of the simple vertex operator algebra L g(k,0)which admits the following decompositionL g(k,0)=L g(k,0)0⊕L g(k,0)1⊕···⊕L g(k,0)swhereL g(k,0)i={v∈L g(k,0)|σ(v)=ξi v}.Clearly L g(k,0)i is a g0–module.For a dominant weightµfor g0we deﬁne µ=k′Λ0+µ.Denote by L g(k′,µ)the irreducible highest weight g0–module with highest weight µ.8DRAˇZEN ADAMOVI´C AND OZREN PERˇSENote that the lowest conformal weight of any g0–module with highest weight µis given by the formula(µ,µ+2ρ0)09Table1.g,g0are simple Lie algebras,g0<g.g decomposition of g k′Cℓ−1Dℓg0⊕V g0(ω1)−ℓ+2F4−3G2g0⊕V g0(ω1)⊕V g(ω2)−5/3Dℓ−ℓ+3/210DRAˇZEN ADAMOVI´C AND OZREN PERˇSELemma4.3.We have:(1)The lowest conformal weights of A(1)2–modules with highest weights−113Λ0+Λ1+Λ2are54,respectively.(2)The lowest conformal weights of B(1)ℓ−1–modules with highest weights−ℓΛ0+ 2Λ1and−ℓΛ0+Λ2are2+1ℓ−1,respectively.(3)The lowest conformal weights of D(1)ℓ–modules with highest weights(−ℓ−12)Λ0+Λ2are2+22ℓ−1,respectively.Lemma4.4.As an F(1)4–module,L E6(−3,0)does not contain singular vec-tors of weights−7Λ0+2Λ4,−5Λ0+Λ1and−7Λ0+Λ3.Proof.Weﬁrst note that F(1)4–modules with highest weights−7Λ0+2Λ4and−5Λ0+Λ1do not have integral conformal weights.(The correspondinglowest conformal weights are132,respectively.)The lowest conformal weight of module of highest weight−7Λ0+Λ3forF(1) 4is2,but one can directly check that there is no non-trivial singular vec-tor of that weight in L E6(−3,0).We use the construction of the root systemof type E6from[11],[23].For a subset S={i1,...,i k}⊆{1,2,3,4,5}withodd number of elements(i.e k=1,3or5),denote by e(i1...i k)the(suitablychosen)root vector associated to positive root1115.Conformal embedding of L G 2(−2,0)into L B 3(−2,0)andL D 4(−2,0)In this section we consider the conformal embedding associated to a pair of simple Lie algebras (D 4,G 2).It turns out that this case is more complicated and one can not directly apply general results from Section 3.The main diﬀerence between this case and cases studied in Section 4is that L D 4(−2,0)contains a non-trivial singular vector for G (1)2of conformal weight 2.In order to prove that L D 4(−2,0)is completely reducible L G 2(−2,0)–module we shall need more precise analysis,which uses the conformal embedding of L B 3(−2,0)in L D 4(−2,0)from Section 4.Let g be the simple complex Lie algebra of type D 4.Then g has an order three automorphism θ,induced from the Dynkin diagram automorphism,such that the ﬁxed point subalgebra g 0is isomorphic to the simple Lie algebra of type G 2.We have the decomposition of g into eigenspaces of θ:g =g 0⊕V (1)g 0(ω1)⊕V (2)g 0(ω1),(5.1)where V (1)g 0(ω1)is generated by highest weight vector e ǫ1−ǫ4+ξe ǫ1+ǫ4+ξ2e ǫ2+ǫ3and V (2)g 0(ω1)by e ǫ1−ǫ4+ξe ǫ2+ǫ3+ξ2e ǫ1+ǫ4,for suitably chosen root vectors.(Here ξdenotes the primitive third root of unity.)Lie algebra g 0is also a subalgebra of Lie algebra of type B 3considered in Section 4,which we now denote by g ′.We have g ′=g 0⊕V (3)g 0(ω1),(5.2)where V (3)g 0(ω1)is generated by highest weight vector e ǫ1−ǫ4+e ǫ1+ǫ4−2e ǫ2+ǫ3.By using (5.2),the decomposition of g from Section 4and the fact that V B 3(ω1)∼=V G 2(ω1)as g 0–module,one obtains another decomposition of g :g =g 0⊕V (3)g 0(ω1)⊕V (4)g 0(ω1),(5.3)where V (4)g 0(ω1)is generated by e ǫ1−ǫ4−e ǫ1+ǫ4.By using (5.2)and results from Section 2we see that L G 2(−2,0)is conformally embedded into L B 3(−2,0),which is conformally embedded in L D 4(−2,0).Recall that Theorem 4.1gives that the vertex operator algebra L D 4(−2,0)has an order two automorphism σwhich deﬁnes the following decomposition:L D 4(−2,0)=L B 3(−2,0)⊕L B 3(−2,ω1).(5.4)Denote by L (i )G 2(−2,ω1)the G (1)2–submodule of L D 4(−2,0)generated by top component V (i )g 0(ω1),for i =1,2,3,4.Furthermore,one can directly check that vectorv =(e ǫ1−ǫ3(−1)e ǫ1−ǫ4(−1)+e ǫ2−ǫ4(−1)e ǫ2+ǫ3(−1)+e ǫ2+ǫ4(−1)e ǫ1+ǫ4(−1)−e ǫ1−ǫ3(−1)e ǫ1+ǫ4(−1)−e ǫ2−ǫ4(−1)e ǫ1−ǫ4(−1)−e ǫ2+ǫ4(−1)e ǫ2+ǫ3(−1))112DRA ˇZEN ADAMOVI ´CAND OZREN PER ˇSE is the unique (up to a scalar)non-trivial singular vector of weight −5Λ0+Λ2for G (1)2in L D 4(−2,0).Denote by L G 2(−2,ω2)the G (1)2–submodule gener-ated by v .We also have that θ(v )=v and σ(v )=−v .This implies that v ∈L B 3(−2,ω1).Now we consider L B 3(−2,0)and L B 3(−2,ω1)as G (1)2–modules.We use the following lemmas:Lemma 5.1.We have the following tensor product decompositions:(1)V G 2(ω1)⊗V G 2(ω1)=V G 2(2ω1)⊕V G 2(ω2)⊕V G 2(ω1)⊕V G 2(0),(2)V G 2(ω1)⊗V G 2(ω2)=V G 2(ω1+ω2)⊕V G 2(2ω1)⊕V G 2(ω1).Lemma 5.2.As a G (1)2–module,L D 4(−2,0)does not contain singular vec-tors of weights −6Λ0+2Λ1and −7Λ0+Λ1+Λ2.Proof.The proof follows from the observation that G (1)2–modules of highest weights −6Λ0+2Λ1and −7Λ0+Λ1+Λ2do not have integral conformal weights.(The corresponding lowest conformal weights are 72,respec-tively.) Proposition 5.3.We have:(1)L B 3(−2,0)= L G 2(−2,0)+ L (3)G 2(−2,ω1).(2)L B 3(−2,ω1)= L(4)G 2(−2,ω1)+ L G 2(−2,ω2).Proof.(1)Since v is the unique (up to a scalar)singular vector of weight −5Λ0+Λ2for G (1)2in L D 4(−2,0),it follows that there is no singular vec-tor of that weight in L B 3(−2,0).The tensor product decomposition from Lemma 5.1(1),Lemma 5.2and standard fusion rules arguments now im-ply that L G 2(−2,0)+ L (3)G 2(−2,ω1)is a vertex subalgebra of L B 3(−2,0)which contains generators of L B 3(−2,0).The claim (1)follows.Claim (1),Lemmas 5.1and 5.2and standard fusion rules arguments now implythat L (4)G 2(−2,ω1)+ L G 2(−2,ω2)is L B 3(−2,0)–submodule of the irreducible module L B 3(−2,ω1).This implies claim (2). Theorem 5.4.We have:L D 4(−2,0)=L G 2(−2,0)⊕L G 2(−2,ω2)⊕L (1)G 2(−2,ω1)⊕L (2)G 2(−2,ω1).Proof.First we notice that relation (5.4)and Proposition 5.3give L D 4(−2,0)= L G 2(−2,0)+ L G 2(−2,ω2)+ L (3)G 2(−2,ω1)+ L (4)G 2(−2,ω1),which also impliesL D 4(−2,0)= L G 2(−2,0)⊕ L G 2(−2,ω2)⊕ L (1)G 2(−2,ω1)⊕ L (2)G 2(−2,ω1).By using the fact that Z 3–orbifold components of simple vertex operator algebra are simple (cf.[14]),we get thatV = L G 2(−2,0)⊕ L G 2(−2,ω2)13 is a simple vertex operator algebra,and that L(1)G2(−2,ω1)and L(2)G2(−2,ω1) are its irreducible modules.Since the automorphismσacts as−1on L G2(−2,ω2),it follows that V is Z2–graded,so its graded components are also simple.Theorem3from[14]now implies that L(1)G2(−2,ω1)and L(2)G2(−2,ω1)are inequivalent V–modules,and Theorem6.1from the same paper then implies that L(1)G2(−2,ω1)and L(2)G2(−2,ω1)are irreducible asL G2(−2,0)–modules.The proof follows. 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Geometric Quantization of Real Minimal Nilpotent Orbits

a r X i v :m a t h /9811033v 1 [m a t h .S G ] 6 N o v 1998Geometric Quantization of Real Minimal Nilpotent Orbits.Ranee Brylinski *Abstract:In this paper,we begin a quantization program for nilpotent orbits O R of a real semisimple Lie group G R .These orbits arise naturally as the coadjoint orbits of G R which are stable under scaling,and thus they have a canonical symplectic structure ωwhere the G R -action is Hamiltonian.These orbits and their covers generalize the oscillator phase space T ∗R n ,which occurs here when G R =Sp (2n,R )and O R is minimal.A complex structure J polarizing O R and invariant under a maximal compact subgroup K R of G R is provided by the Kronheimer-Vergne Kaehler structure (J ,ω).We argue that the Kaehler potential serves as the ing this setup,we realize the Lie algebra g R of G R as a Lie algebra of rational functions on the holomorphic cotangent bundle T ∗Y where Y =(O R ,J ).Thus we transform the quantization problem on O R into a quantization problem on T ∗Y .We explain this in detail and solve the new quantization problem on T ∗Y in a uniform manner for minimal nilpotent orbits in the non-Hermitian case.The Hilbert space of quantization consists of holomorphic half-forms on Y .We construct the reproducing kernel.The Lie algebra g R acts by explicit pseudo-diﬀerential operators on half-forms where the energy operator quantizing the Hamiltonian is inverted.The Lie algebra representation exponentiates to give a minimal unitary ladder representation of a cover of G R .Jordan algebras play a key role in the geometry and the quantization.§1.Introduction.I.Quantization of Phase Space.Quantization of a classical phase space M with symplectic form ωis a process whereby observables φare converted into self-adjoint operators Q (φ)on a Hilbert space H of states.The observables are simply the smooth functions on M .The Hilbert space H should arise,according to the philosophy of Geometric Quantization,as a space of polarized sections of a suitable complex line bundle over M .A real (complex)polarization of M consists of a integrable Lagrangian distribution inside the (complexiﬁed)tangent bundle.A polarized section,of a bundle with connection,is a section annihilated by all vector ﬁelds lying in the polarization;in the real case,this means that the section iscovariantly constant along the leaves of the corresponding Lagrangian foliation.We require that the quantization satisﬁes Dirac’s axioms (see e.g.,[Ki],[A-M])in some form.Dirac’s consistency axiom is that the Poisson bracket of functions on M goes over into the commutator of operators so thatQ ({φ,ψ})=i [Q (φ),Q (ψ)](1.1)(We have set =1.)Additional axioms mandate that the constant function1quantizes to the identity operator,and a complete set of observables quantizes to give a complete set of operators.In Hamiltonian mechanics,the physics of the system in encoded in a single observable F (usually written as E or H)called the Hamiltonian.Often F is the total energy.Any observableφgenerates a Hamiltonianﬂow:this is theﬂow of the Hamiltonian vector ﬁeldξφdeﬁned by the equationξφω+dφ=0(1.2) The Poisson bracket on C∞(M)is given by{φ,ψ}=ξφ(ψ)=ω(ξφ,ξψ).The Hamiltonianﬂow of the F gives the time evolution of the physical system.For any observableφ,the time derivative˙φofφas the system evolves is given by˙φ={F,φ}.This is a concise version of Hamilton’s equations.On physical grounds,in certain circumstances,F should be a positive function on M.In quantization of a Hamiltonian mechanical system,F should be promoted to a self-adjoint operator Q(F)on H with positive spectrum.When F is the total classical energy, the spectrum of Q(F)should be discrete and give the possible quantized energy levels of the quantum system.II.Quantization of the n-dimensional Harmonic Oscillator.The most familiar model situation is the case where M is the cotangent bundle of some(conﬁguration)manifold X andωis the canonical symplectic form so thatω=dθwhereθis the Liouville1-form on T∗X.In this case we have the manifest cotangent polarization where the leaves are the cotangent spaces of X.We expect H to be a space of square integrable half-forms on X (see§2and below starting around(1.5)).A smooth function f on X quantizes to a give a multiplication operator on H.Ifηis a vectorﬁeld on X,then the symbolσηquantizes to the Lie derivative Lηoperator on half-forms.Consistent quantization of additional observables is problematic,as we see already in the oscillator example below.A second model situation is the case where M is a Kaehler manifold andωis the Kaehler form.Then the complex structure J of M gives a complex polarization.Now“polarized”simply means“holomorphic”.Thus H should be a space of holomorphic square-integrable sections of a suitable holomorphic complex line bundle over M.The most familiar example of a Hamiltonian mechanical system,the oscillator phase space, admits both cotangent and Kaehler polarizations.The oscillator phase space is M=T∗R n. The canonical coordinates on T∗R n are the position coordinates q1,...,q n together with the momentum coordinates p1,...,p n.The canonical symplectic form isω= n k=1d p k∧d q k.The Poisson bracket satisﬁes{p j,p k}={q j,q k}=0and{p j,q k}=δjk.For general observableswe have the classical formula{φ,ψ}=nk=1 ∂φ∂q k−∂ψ∂q k (1.3)In physics,T∗R n arises as the phase space of n uncoupled harmonic oscillators with Hamil-2tonian equal to the total energy(kinetic plus potential)F=12are holomorphic coordinates.Now C n is a Kaehler manifold with Kaehler formωand Kaehler metric g= n k=1(d p2k+d q2k).In the z j,z k∧d z k and the Poisson bracket satisﬁes{z j,z k}= {z k}=0and{∂z k +1z k quantize into the creation and annihilation operatorsQ(z k)=z k and Q(∂zk(1.7) Then Q(F)is a grading operator on the quantum space and Q(z k)=z k and Q(2-shift in(1.6)(a quantum correction)is to adopt the sym-metrization procedure of canonical quantization so thatQ(z k2(Q(z k)Q(z k)Q(z k))=1∂zk +∂∂z k+1z k)in(1.7)are mutually adjoint.(The condition that Q(φ)is self-adjoint for realφamounts to the condition that Q(φ)and Q(g(z)e−|z|2|d z dand this expression deﬁnes the inner product on the Hilbert space completion H of H .Thus H consists of all the holomorphic functions f (z 1,...,z n )on C n which are “square integrable”in the sense that ||f ||2= f |f is ﬁnite.The reproducing kernel (see §8)of H is the holomorphic function K (z,C nK (z,w 1+···+z nX denotes the complex conjugate manifold to a complex manifold X ,so that holomorphicfunctions onX is obtained from X by reversing the sign of the complex structure.The Hamiltonian ﬂow of F lies inside a larger symmetry.The Hamiltonian F sits inside the space g of all homogeneous quadratic polynomials z 1,...,z n ,z n .The space g is a ﬁnite-dimensional Lie subalgebra of complex-valued observables under Poisson bracket.The Lie algebra g breaks naturally into 3pieces:g =k ⊕p +⊕p −wherek =span of z j z jz k +z k −z j z jz k )=z j ∂2,Q (z k )=∂2φ)for φ∈g .Moreover this condition by itselfdetermines the inner product f |g uniquely.The No-Go Theorem (see e.g.,[A-M])shows that we cannot extend the quantization to all polynomial observables.A beneﬁt of looking at this large Lie algebra of symmetry is that we can see another sourcefor the 1z j z k )because of the Dirac axiom (1.1).So the term involving 1∂z kand z k do not commute but instead [∂2-shift is to introduce half-forms.This meansthat we replace our Hilbert space H of holomorphic functions on C n by a new Hilbert space4H′of holomorphic half-forms s=f√z k quantize into the operatorsQ′(z j)=z j and Q′(∂z k .We compute L∂k(√ν)=1ν.This givesL∂k(f√∂zk√ν)= z j∂f2δjk f √z k)=L zj ∂k,Q′(z k)=L∂j L∂k(1.18)These operators in(1.16)and(1.18)obey(1.1)and Q′(φ)†=Q′(ν|g√ﬁbers of T∗Z→Z.The good observables correspond to bonaﬁde symbols.See§2,3and[B3] for a way to work this out based on the Hamiltonian F.The result of this is easy to describe directly for the oscillator.We put Z=C n.Letζ1,...,ζn be the holomorphic momentum functions on T∗Z so that z1,...,z n,ζ1,...,ζn are holomorphic coordinates on T∗Z and the canonical holomorphic symplectic form on T∗Z isΩ= n k=1dζk∧d z k.ThenΩdeﬁnes a Poisson bracket{Φ,Ψ}Ωon the algebra of holomorphic functions on T∗Z.We have{z j,z k}Ω={ζj,ζk}Ω=0and{ζj,z k}Ω=δjk.We have an obvious complex Poisson algebra isomorphismα:C[z1,...,z n,z n]→C[z1,...,z n,ζ1,...,ζn](1.21) whereα(z k)=z k andα(w).Thenα(z j z k)=z j z k,α(z j z j∂z k .The quantization of the oscillator has manifold applications in physics–in quantum me-chanics,quantumﬁeld theory,supersymmetry,etc.It also of course occupies a central place in mathematics.III.Quantization of Hamiltonian Symmetry.To formulate a mathematical quan-tization problem generalizing the oscillator case,we suppress(for the time being)the Hamil-tonian F and focus instead on the largeﬁnite-dimensional symmetry algebra g.This brings us to the notion of Hamiltonian symmetry.Suppose we have an action of a connected Lie group G on a symplectic manifold(M,ω). We regard M as a phase space.Assume the action is symplectic,i.e.,G preservesω.Let g be the Lie algebra of G.For each x∈g,we have the1-psg(1-parameter subgroup)γx:R→G,γx(t)=exp(tx),generated by x.By Noether’s Theorem,there is a smooth functionµx (deﬁned at least locally about every point of M),unique up addition of a constant,such that the Hamiltonianﬂow ofµx is the action ofγx.Thenµx is conserved under the action ofγx. Ifµx exists globally on M,thenµx is called aﬁrst integral or momentum function forγx.The symplectic G-action is called Hamiltonian if there exists a mapµ∗:g→C∞(M)(1.23) x→µx,such thatµx is aﬁrst integral forγx and{µx,µy}=µ[x,y]for all x,y∈g,i.e.,µ∗is a Lie algebra homomorphism.Then the functionsµx deﬁne a moment mapµ:M→g∗(1.24) byµx(m)= µ(m),x .If g is semisimple then we often identify g with its dual by means of the Killing form so that moment maps take values in g.6The moment mapµobtained in this way is G-equivariant and Poisson.Consequently theimage ofµin g∗is a union of coadjoint orbits.The image of the moment map is an importantinvariant of the action.It is easy to prove thatµis a covering onto a single coadjoint orbitif and only if the Hamiltonian action of G on M is transitive;thenµis symplectic.Such anaction is called elementary.Thus,symplectically and equivariantly,the elementary Hamiltonian G-spaces are,up tocovering,just the coadjoint orbits of G.Going back to our oscillator phase space,we see that the action of Sp(2n,R)on ourmanifold M=T∗R n=C n,with the origin of C n deleted,is an elementary Hamiltonianaction.The moment map C n−{0}→sp(2n,R)is a2-fold covering on the smallest(non-zero) adjoint orbit O R of Sp(2n,R).This orbit O R is stable under scaling and so consists of nilpotentelements.The quantization problem on the Hamiltonian G-space(M,ω)is to quantize the momen-tum functionsµx into operators in a manner agreeable with Dirac’s axioms.It is natural tostudy the elementary caseﬁrst,as here the symmetry is largest.Thus one seeks a quantizationof the functionsµx,x∈g,for coadjoint orbits and their covers.In analogy with the oscillator,we consider the case where the symmetry group G is areal semisimple Lie group G R(withﬁnite center)and M is an adjoint orbit O R stable underscaling.Then O R is a“nilpotent orbit”of G R–see§2.Quantization of coadjoint orbits has traditionally been considered as part of the OrbitMethod in representation theory.In the Orbit method,one uses polarizations invariant underthe whole symmetry group and obtains unitary representations by induction.The theoryincorporates metaplectic covers and the Mackey machine.Much more can be said about theOrbit Method.We note that unitary representations attached to nilpotent orbits are calledunipotent in representation theory.On the other hand,coming into this problem from geometry,we have found diﬀerentmethods which apply(at least)to nilpotent orbits.The main idea is to transform the quanti-zation problem on O R into a quantization problem on a cotangent bundle,and then solve thatproblem.IV.Outline of this Paper.In this paper,we quantize the nilpotent orbit O R of G R in the case where O R is stronglyminimal(see§3).The oscillator phase space is the double cover of the strongly minimalnilpotent orbit of G R=Sp(2n,R).We assume that g R is simple,the maximal compact subgroup K R of G R hasﬁnite center,and G R is simply-connected.(Thus we exclude the oscillator case as there K R=U(n).)We obtain the analogs of the Fock space model of the quantum mechanical oscillator.Weﬁnd analogs of all the features of the oscillator quantization described above in II.This isworked out in detail in this paper,with the exception of the integral formula(1.10)for theinner product which will be written up elsewhere.We work from scratch and assume no priorknowledge on existence of unitary representations.This completes the work from[B-K4].In[B-K4]we worked out with Kostant the resultscovered in§4-7of this paper for the three cases where G R is a a split group of type E6,E7,E8.7We start from the fact,a product of the work of Kronheimer([Kr])and Vergne([Ve]),thatO R admits a K R-invariant complex structure J which together with the KKS symplectic formσgives a(positive)Kaehler structure on O R.The Vergne diﬀeomorphism V:O R→Y identiﬁes the complex manifold(O R,J)with a complex homogeneous space Y of the complexiﬁcationK of K R.This is a general theory that applies to every nilpotent orbit for G R semisimple.For the oscillator,this recovers the U(n)-invariant Kaehler structure and the identiﬁcationT∗R n=C n used in II.We outline this theory in§2and we explain how it gives rise to an embedding of O Rinto T∗Y as a totally real symplectic submanifold([B1]).This enables us to transform thequantization problem on O R into a quantization problem on T∗Y,as long as the Hamiltonianfunctionsφw,w∈g R extend from O R to T∗Y.An important aspect is that the Kaehler structure on O R possesses a global Kaehlerpotentialρwhich we argue plays the role of the Hamiltonian F.The Hamiltonianﬂow ofρisthe action of the center of K R in the oscillator case.In our cases,the Hamiltonianﬂow ofρlies outside the G R-action.In§3,we specialize to the case where O R is strongly minimal and K R hasﬁnite center.Weexplain how to convert the Hamiltonian functionsφw,w∈g R,on O R into rational meromorphicfunctionsΦw on the cotangent bundle of Y.We interpret theΦw as“pseudo-diﬀerentialsymbols”.To describe the symbols,we consider the Cartan decomposition g R=k R⊕p R(cf.(1.13)). For x∈k R,Φx is just the usual symbol of the holomorphic vectorﬁeldηx on Y deﬁned by diﬀerentiating the K-action.But for v∈p R,Φv is a sum of two terms,each homogeneous under theﬁberwise scaling action of C∗on the leaves of the cotangent polarization of T∗Y. The passage from the observable functionφw to the symbolΦw preserves Poisson brackets.The middle part§4-§7of the paper is devoted to quantizing the symbolsΦw,w∈g R,into skew-adjoint operators on a holomorphic half-form line bundle N12is a multiplicity free ladder representation of K.We get a simple geometric description of the sections which are the highest weight vectors.In§4,we set up the Jordan structure that is used throughout the paper(explicity in§5and§7).A main point is that the polynomial function P constructed in§3is realized in termsof Jordan norms.We construct,in Corollary6.2and Theorem6.3the pseudo-diﬀerential operators Q(Φw)on half-forms which quantize the symbolsΦw,or equivalently,the functionsφw.Theorem6.3says that these operators satisfy(1.1),i.e.,the operatorsπw=i Q(Φw)give a representationof g.In Theorem6.6we construct the g R-invariant inner product B on H.In Theorem6.8we compute B by giving the analog(6.30)of(1.9).Our operators are pseudo-diﬀerential(not purely diﬀerential)in that they involve invertingthe positive-spectrum“energy”operator E′which is the quantization ofρ.In fact,instead ofthe order two operators L∂j L∂k from(1.18)we obtain order4diﬀerential operators divided by E′(E′+1);these are“formally”of order2.The action of the maximal compact group K Ron H is just the natural one deﬁned by the action of K R on Y and N1algebra generated by the operatorsπw on H.It follows in Theorem6.6thatπintegrates to give an irreducible minimal unitary representation of G R on the Hilbert space completion H of H.Next§7is devoted to proving the results of§6.We show that our pseudo-diﬀerential operators satisfy the bracket relations of g R by reformulating the problem and applying the generalized Capelli Identity of Kostant and Sahi([K-S]).An important aspect of their work is that Jordan algebras provide a natural setting for generalizing the classical Capelli identity involving square matrices.The complex Jordan algebra k−1occurring here is semisimple (while in[B-K4]it was simple).It turns out that the simple components of k−1become coupled together in our calculations in a subtle way reﬂected by Proposition7.8.In§8,we compute the reproducing kernel K of the Hilbert space completion H of H.We ﬁnd that K is a holomorphic function on Y×2.Finally,in§9we give some examples.Diﬀerent models,or proofs of existence,for most of the unitary representations we con-struct have been obtained by other authors.These include Binegar,Gross,Howe,Kazhdan, Kostant,Li,Oersted,Rawnsley,Savin,Sijacki,Sternberg,Sabourin,Torasso,Vogan,Wallach, Wolf,and Zierau.Moreover in[T],Torasso constructs in a uniform manner by the Orbit Method Schroedinger type models of all minimal unitary representations.Precisely,Torasso constructs unitary irreducible representations attached to all minimal admissible nilpotent or-bits of simple groups of relative rank at least three over a localﬁeld of zero characteristic.It would be very interesting to construct intertwining operators between our models.There is a rich literature on geometric models of unitary highest weight representations, and there are many interesting ties here with our work.This paper builds on several years of joint work with Bert Kostant on the algebraic holo-morphic symplectic geometry of nilpotent orbits of a complex semisimple Lie group.This work includes[B-K1-5].In addition§4of this paper is joint work.I thank Alex Astashkevich,Olivier Biquard,Murat Gunaydin,Bert Kostant,Michele Vergne,and Francois Ziegler for useful conversations relating to this work.Parts of this work were carried out during visits to Harvard(1993-94,summers of1995and1996),the Institute for Advanced Study(Spring1995)and Brown University(summer of1997).I thank all these departments for their hospitality.I thank Mark Gotay for putting together this volume and for his comments on my paper.I am delighted to dedicate this paper to Victor Guillemin and to be able to contribute it to this volume in his honor.In my graduate student days at MIT I was ensconced in algebraic geometry and algebraic group actions.I was symplectically agnostic.But since my symplectic conversion in the end of the last decade,I have had the opportunity to talk to Victor a lot and learn from him and his many books and papers.I thank him for warmly welcoming me as a visitor into his symplectic group.§2.The Quantization Problem for Real Nilpotent Orbits.The phase spaces we wish to quantize are the so-called“nilpotent orbits”of G R where G R9is a connected non-compact real semisimple Lie group withﬁnite center.Then G R is aﬁnite cover of the adjoint group of its Lie algebra g R,and g R is semisimple.To deﬁne the nilpotent orbits we consider the coadjoint action of G R on the dual g∗R of g R.Each coadjoint orbit O R carries a natural G R-invariant symplectic formσ,often called the KKS or Lie-Poisson form.The formσis uniquely characterized by the following property:letφ:g R→C∞(O R),w→φw(2.1) be the pullback map on functions deﬁned by the embedding O R⊂g∗R.Thenφis a Lie algebra homomorphism with respect to Poisson bracket on C∞(O R)deﬁned byσ.In analogy with the cotangent bundle,we wish to single out those coadjoint orbits which are conical in the sense that they are stable under the Euler scaling action of R+(positive reals).There is a nice Lie theoretic characterization of these orbits.To get this,weﬁrst use to identify g∗R with g R;we do this throughout the paper routinely.Then the Killing form(,)gR(conical)coadjoint orbits get identiﬁed with(conical)adjoint orbits.An adjoint orbit is conical if and only if it consists of nilpotent elements in g R.Such orbits are called“nilpotent orbits”.It is well-known in Lie theory that there are onlyﬁnitely many nilpotent orbits in g R.From now on,we take O R to be a nilpotent orbit in g R.The quantization problem on O R is to quantize into operators the functionsφw,w∈g R.This is a reasonable goal.Ideally quantization would convert all smooth functions on O R into operators in a manner satisfying Dirac’s axioms.See,e.g.,[Ki,§2.1]for a complete axiom list.But full quantization is impossible even for polynomial functions on R2(the infamous No-Go Theorem).We are left hoping that, except for anomalies,ﬁnite-dimensional Hamiltonian symmetry will quantize.In analogy with the Fock space quantization of the oscillator,we look for a Kaehler polar-ization of our phase space(O R,σ)which is invariant under aﬁxed maximal compact subgroup K R of G R.This means that we look for a K R-invariant integrable complex structure J on O R such that J andσtogether give a(positive)Kaehler structure on O R.Fortunately,such a complex structure J on O R arises from the works of Kronheimer([Kr]) and Vergne([Ve])on instantons and nilpotent orbits.This gives the K R-invariant instanton Kaehler structure(J,σ)on O R.This structure is discussed and studied in detail in[B1].We recall two main points.Theﬁrst point is the Vergne diﬀeomorphism([Ve]).To set this up,we introduce the Cartan decompositiong R=k R⊕p R(2.2) where k R⊂g R is the Lie algebra of K R and p R is its orthogonal complement with respect to the Killing form.The natural action of K R on p R complexiﬁes to a complex algebraic action of K on p where K is the complexiﬁcation of K R(so that K is a complex reductive algebraic group)and p=p R⊕i p R.Now the Vergne diﬀeomorphismV:O R→Y(2.3)10is a(K R×R+)-equivariant diﬀeomorphism of real manifolds which maps O R onto a K-orbit Y in p.Y,being a K-orbit,is manifestly a complex submanifold of p.Moreover J is the pullback through V of the complex structure on Y.An important feature is that Y is stable under the Euler scaling action of C∗on p.This follows since O R is R+-stable and V is R+-equivariant.Let E be the inﬁnitesimal generator of the Euler C∗-action so that E is the algebraic holomorphic Euler vectorﬁeld on Y.In general,the target Y of the Vergne diﬀeomorphism is known(by the Kostant-Sekiguchi correspondence[Sek])but not the actual map giving V.A little insight into V comes from Lie theory.To explain this,we introduce the complexiﬁed Lie algebra g=g R⊕i g R.g is a complex semisimple Lie algebra and carries the complex conjugation map x+iy→e)adapted to(g R,k R)such that e+ih+e)=eThe second point is that the Kaehler structure(J,σ)on O R admits a global Kaehler potentialρ.This means thatρis a smooth real valued function on O R such that i∂2over Y in such a way that the Dirac axiomQ(φ[w,w′])=i[Q(φw),Q(φw′)](2.4) is satisﬁed.In the course of doing this,we will end up quantizing one additional function on O R.There are additional axioms which should also be satisﬁed,but these are somewhat hidden as we are only dealing with the functionsφw. E.g.,the axiom that the constant function1 quantizes to the identity operator is“hidden”.These“hidden axioms”are basically incorpo-rated by our methodology developed below using symbols.If the Hamiltonianﬂow ofφpreserves J andφis homogeneous of degree1,then we mandate that the quantized operator is simplyQ(φ)=−i L ξφ(2.5)11Here ξφis the J-Hamiltonian vectorﬁeld on Y deﬁned by the condition that ξφis holomorphic and coincides withξφon holomorphic functions.We write Lηfor the Lie derivative operator (acting on holomorphic half-forms)with respect to a holomorphic vectorﬁeldη.Diﬀerentiating the K-action on Y we get an inﬁnitesimal holomorphic vectorﬁeld actionk→V ect hol Y,x→ηx(2.6) Thenηx= ξφx for x∈k R and soQ(φx)=−i Lηx,for x∈k R(2.7)The problem,since our polarization J is only K R-invariant,is to quantize the remaining functionsφv,v∈p R corresponding to the second piece in the Cartan decomposition(2.2).A key aspect of our program for quantization of real nilpotent orbits(see[B1-3])is that we regardρas the Hamiltonian function on O R.This generalizes the case of the harmonic oscillator discussed in§1where the Hamiltonian is the total energy.It may seem strange that the oscillator energy Hamiltonian is homogeneous quadratic while our functionρis homogeneous linear.However the oscillator phase space R2n−{0}arises as the double cover of a real nilpotent orbit.In that case,our linear potential functionρdoes indeed pull back to a quadratic function on R2n−{0},and it is easy to check that we recover the classical energy p21+q21+···+p2n+q2n (see[B3]).In physical terms,the Hamiltonian governs the time evolution of the classical system. The quantum mechanical problem is toﬁnd the eigenvalues and eigenstates of the operator quantizing the Hamiltonian.Thus we now demand that quantization should not only promote the symmetry functions φw to operators,but should also promoteρto an operator.In fact the Hamiltonianﬂow ofρpreserves J and is periodic;we call this the KV(Kronheimer-Vergne)S1-action on O R([B1]). Under V,the KV S1-action corresponds to the circle part of the Euler C∗-action on Y.It follows that the J-Hamiltonian vectorﬁeld ofρis iE.HenceQ(ρ)=−i L iE=L E(2.8)LetΩbe the canonical holomorphic symplectic form on T∗Y.ThenΩdeﬁnes a Poisson bracket on the algebra of holomorphic functions on T∗Y,and also on theﬁeld of meromorphic functions.A main result of[B1]is to realize the holomorphic cotangent bundle(T∗Y,Ω)as a sym-plectic complexiﬁcation of O R.To do this,we push forwardρto a smooth functionρY on Y so thatρ=ρY◦V.Next we construct the following real1-formβon Yβ=−i∂)ρY(2.9)Thenβdeﬁnes a smooth section of the cotangent bundle T∗Y→Y.12Theorem2.1[B1].The compositionb:O R V−→Yβ−→T∗Y(2.10) embeds O R as a totally real symplectic submanifold of T∗Y.In particular,b∗(ReΩ)=σand b∗(ImΩ)=0.Now,given a functionφon O R which we wish to quantize,we can ask ifφextends to a holomorphic functionΦon T∗Y.(Such an extension,if it exists,is necessarily unique.)If so, thenΦis our candidate for the symbol of Q(φ).This philosophy is consistent with what we already found in(2.7)and(2.8).Indeed we can deﬁne the holomorphic symbols,where x∈k,Φx=symbolηx andλ=symbol E(2.11) Our convention for symbols is speciﬁed by the following formula in holomorphic Darboux coordinates:∂k0+···+k msymbol f(z0,...,z m)。

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a rXiv:alg-ge o m/96414v124Apr1996A LIE ALGEBRA ATTACHED TO A PROJECTIVE V ARIETY Eduard Looijenga and V alery A.Lunts ∗Abstract.Each choice of a K¨a hler class on a compact complex manifold deﬁnes an action of the Lie algebra sl (2)on its total complex cohomology.If a nonempty set of such K¨a hler classes is given,then we prove that the corresponding sl (2)-copies generate a semisimple Lie algebra.We investigate the formal properties of the re-sulting representation and we work things out explicitly in the case of complex tori,hyperk¨a hler manifolds and ﬂag varieties.We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra.Table of Contents Introduction 11.Lefschetz modules 42.Jordan–Lefschetz modules 123.Geometric examples of Jordan type I:complex tori 184.Geometric examples of Jordan type II:hyperk¨a hlerian manifolds 245.Filtered Lefschetz modules 306.Frobenius–Lefschetz modules 337.Appendix:a property of the orthogonal and symplectic Lie algebra’s 41Introduction Let X be a projective manifold of complex dimension n .If κ∈H 2(X )is an ample class,then cupping with it deﬁnes an operator e κin the total complex cohomology(denoted here by H(X ))of degree 2and the hard Lefschetz theorem asserts that for s =0,...,n ,e s κmaps Hn −s (X )isomorphically onto H n +s (X ).As is well-known,this is equivalent to the exististence of a (unique)operator f κin H(X )of degree −2such that the commutator [e κ,f κ]is the operator h which on H k (X )is multiplication by k −n .The elements e κ,h,f κmake up a Lie subalgebra g κof gl (H(X ))isomorphic to sl (2)and the decomposition of H(X )as a g κ-module into isotypical summands is just the primitive decomposition:the primitive cohomology in degree n −s generates the isotypical summand associated to the irreducible representation of dimension s +1.As these operators respect the Hodge decomposition (in the sense that e κresp.f κhas bidegree (1,1)resp.(−1,−1)),the Hodge structure on H(X )is entirely determined by the Hodge structure on the primitive cohomology.However,the primitive decomposition usually depends in a nontrivial way on the choice of2EDUARD LOOIJENGA AND VALERY LUNTSκ.This we regard as a fortunate fact,as it often leads toﬁnding an even smaller Hodge substructure of H(X)that determines the one on H(X).To be explicit,let us deﬁne the N´e ron–Severi Lie algebra g NS(X)as the Lie subalgebra of gl(H(X)) generated by the gκ’s withκan ample class.This Lie algebra is deﬁned over Q and is evenly graded by the adjoint action of its semisimple element h(with its degree2k summand acting as transformations of bidegree(k,k)).We prove in this paper that it is also semisimple.So if we regard H(X)as a representation of this Lie algebra,then the subspace of H(X)annihilated by the negative degree part of g NS(X)is a Hodge substructure that determines the one on H(X).Notice that this Hodge substructure is itself still invariant under the degree zero part of g NS(X) (which is a reductive Lie subalgebra).Despite its naturality,this idea appears to be new(although a note by[Verbitsky1990],of which we were not aware of when we started this research,is suggestive in this repect.)Whereas the eκ’s commute,the corresponding fκ’s don’t in general.This makes it diﬃcult to compute the N´e ron–Severi Lie algebra in practice.It is often helpful when we know of a morphism from X to another projective manifold Y whose base andﬁbers are well-understood:for example,the fact that the associated Leray spectral sequence degenerates yields(among other things)the existence of a copy of g NS(Y)in g NS(X).This is an ingredient of our proof that the N´e ron–Severi Lie algebra ofﬂag variety of a simple complex Lie group is“as big as possible”(reﬂected by the fact that its Hodge structure is as simple as possible):it is the Lie algebra of inﬁnitesimal automorphisms of a naturally deﬁned bilinear form(which is either symmetric or skew)on its cohomology.But if the fκ’s happen to commute,then we are in a very interesting situation: the N´e ron–Severi Lie algebra has degrees−2,0and2only and the(complexiﬁed) N´e ron–Severi group acquires the structure of a Jordan algebra without preferred unit element.For abelian varieties this is a classical fact,although,as far as we know,it had not been seen from this point of view.The N´e ron–Severi Lie algebra appears here as a natural companion of the Mumford–Tate group:the latter helps us toﬁnd the Hodge ring as a ring of invariants,whereas the decomposition of the Hodge ring into Q-irreducible representations of the N´e ron–Severi Lie algebra helps us to say more about its structure.(For example,the subring generated by the divisor classes is one such irreducible summand.)Here are some variants of this construction:instead of working with complex projective manifolds,we could do this for compact complex manifolds that admit a K¨a hler metric and replace the ample classes by K¨a hler classes.Or we could even take all cohomology classes of degree2that have the Lefschetz property;clearly,the complex structure has now become irrelevant.The resulting Lie algebra’s are again semisimple and we call them the K¨a hler Lie algebra and the total Lie algebra of the manifold respectively.Examples of interest here are complex tori and hyperk¨a hler manifolds;in both cases we get Jordan algebra structures.In a diﬀerent direction, we can take for X a projective variety and take instead its complex intersection homology,even with values in a variation of polarized Hodge structure.These examples lead us to formalize the situation by means of what we have called a Lefschetz module.This is essentially a graded vector space equipped with a set of commuting degree two operators that have the Lefschetz property,such that the Lie algebra generated by the corresponding sl(2)-triples is semisimple.So this vector space becomes a representation of a semisimple Lie algebra,and it was oneLIE ALGEBRA ATTACHED TO A PROJECTIVE VARIETY3rather restrictive properties,we did not succeed in this.We now brieﬂy decribe the contents of the separate sections.In section1we introduce the notion that is central to this paper,that of a Lefschetz module,and discuss its basic properties.If a Lefschetz module has a compatible Hodge structure,as is the case for the cohomology of a projective man-ifold,then there is also deﬁned its Mumford–Tate group and we compare the two notions.We next deﬁne and discuss the closely related notion of a Lefschetz pair. This is followed by a partial classiﬁcation of such pairs in case the associated Lie algebra is of classical type.In section2we concentrate on the case when the fκ’s commute.We show that the resulting structure is essentially that of a Jordan algebra and that is why a complete classiﬁcation is available.We are also led to a remarkable class of Frobenius algebra’s associated to each Jordan algebra,some of which we describe explicitly.The next two sections are devoted to examples of K¨a hler manifolds that give rise to Lefschetz modules of Jordan type.First we compute the total Lie algebra and the K¨a hler Lie algebra of a complex torus.Then we turn our attention to the N´e ron–Severi Lie algebra of an abelian variety A and express it in terms of the endomorphism algebra of A.Weﬁnd that that this N´e ron–Severi Lie algebra inter-sects End(A)⊗C in a Lie ideal of End(A)⊗C and we describe the complementary ideal.Our treatment of hyperk¨a hler manifolds(in section4)follows essentially[Ver-bitsky1995],a preprint that in turn is partly based on a preliminary version of the present paper.As an application we show how the Hodge structure on the cohomology algebra of a compact hyperk¨a hlerian manifold is expressed in terms of the Hodge structure on its degree two part.We also give an alternative description of the Beauville-Bogomolov quadratic form on the N´e ron–Severi group.Thus the abelian varieties and the hyperk¨a hler manifolds produce the classical Jordan algebra’s.The exceptional Jordan algebra can be realized topologically and we ask whether it is realizable by a Calabi-Yau threefold.Section5is aboutﬁltered Lefschetz modules.The example to keep in mind here is the Lerayﬁltration on H(X)deﬁned by a surjective morphism f:X→Y of projective manifolds.We apply this to the case where f is a projective space bundle.In combination with a theorem proved in the appendix we are then able to determine the N´e ron–Severi Lie algebra of aﬂag variety.It would be interesting to do the same for the intersection homology of Schubert varieties.In section6we investigate another interesting class of Lefschetz modules,which we have called Frobenius–Lefschetz modules.These arise as the Lefschetz submod-ule of the cohomology of a projective manifold generated by its unit element.The Jordan–Lefschetz modules are among them and we suspect that the remaining sim-ple Frobenius–Lefschetz modules are“tautological representations”of orthogonal or symplectic Lie algebra’s.The main result(6.8)of this section supports this belief: it says that any other simple Frobenius–Lefschetz module must be a representation of an exceptional Lie algebra.We began this work in the Fall of1990,when both of us were at the University of Michigan in Ann Arbor.We would like to thank its Mathematics Department for providing so stimulating working conditions.One of us(Looijenga)thanks in4EDUARD LOOIJENGA AND VALERY LUNTSthe subject matter of this paper as well as closely related questions.Our work continued during the Spring of1991,while Looijenga was at the University of Utah. He gratefully remembers the friendly atmosphere he encountered there.After an interruption we resumed work on this paper in the academic year1994-95.We thank Tonny Springer for some helpful references and Bertram Kostant,Misha Verbitsky and Yuri Zahrin for useful conversations.1.Lefschetz modules(1.1)Weﬁx aﬁeld K of characteristic zero.Let M•be a Z-graded K-vector space ofﬁnite dimension and denote by h:M→M the transformation that is multiplication by k in degree k.So a linear transformation u:M→M has degree k if and only if[h,u]=ku.We say that a linear transformation e:M→M of degree2has the Lefschetz property if for all integers k≥0,e k maps M−k isomorphically onto M k.According to the Jacobson–Morozov lemma this is equivalent to the existence of K-linear transformation f in M of degree−2such that[e,f]=h.This f is then unique and(e,h,f)is a sl(2)-triple:the assignment0100 →e, 100−1 →h, 0010 →fdeﬁnes a representation of sl(2).If h and e happen to be contained in a semisimple Lie subalgebra g⊂gl(M),then so is f.Now let a be aﬁnite dimensional K-vector space.We regard a as a graded abelian Lie algebra which is homogeneous of degree two.We say that a graded Lie homomorphism e:a→gl(M)has the Lefschetz property if for some a∈a, e a has that property.Notice that the set of a∈a with the Lefschetz property is always Zariski open in a.For a in this open set,we have deﬁned the operator f a such that(e a,h,f a)is sl(2)-triple.This deﬁnes a rational map f:a→gl(M)in the sense of algebraic geometry.We let g(a,M)denote the Lie subalgebra of gl(M) generated by the transformations e a,f a.If a is merely an abelian group that acts on M by operators of degree2,then the linear extension a⊗K→gl(M)is a Lie homomorphism and we then often write g(a,M)for g(a⊗K,M).The following example shows that this Lie algebra need not act reductively in M.Example.Consider the graded sl(2)-representation M=sl(2)⊕K2,where sl(2)=Ke+Kh+Kf has the adjoint representation(with its usual grading)and K2is the trivial representation in degree zero.Deﬁne an operator e′of degree 2in M by e′(xf+yh+zf,u,v)=(ve,z,0).Then ee′=e′e=0,so that e and e′span an abelian Lie algebra a.Since e has the Lefschetz property,the Lie algebra g=g(a,M)is deﬁned.Now a kills(0,1,0),but the line spanned by this vector has no g-invariant complement.This example was chosen as to make g inﬁnitesimally preserve a nondegenerate quadratic form on M(namely (xf+yh+zf,u,v)→−2xz+y2+2uv).A smaller example without that property is the submodule sl(2)⊕K⊕0.Notice that g(a,M)is evenly graded and that the grading is induced from the action of ad h.We say that(a,M)is a Lefschetz module if g(a,M)is semisimple. In case M=0,we call greatest integer n with M n=0(or equivalently,M−n=0)LIE ALGEBRA ATTACHED TO A PROJECTIVE VARIETY5 The collection of Lefschetz modules is closed under direct sums,tensor products and taking duals.Also,a Lefschetz module M has always the decomposition M= M ev⊕M odd,where M ev(resp.M odd)is the direct sum of the M k’s with k even(resp.odd).Since any representation of a semisimple Lie algebra is reductive, the category of Lefschetz modules of a is semisimple.A Lefschetz a-module M is irreducible as a Lefschetz module if and only if it is irreducible as a g(a,M)-module.There is also an exterior direct sum and tensor product:if(a′,M′)and(a′′,M′′) are Lefschetz modules,then we have deﬁned Lefschetz modules(a′×a′′,M′⊞M′′),e(a′,a′′)(m′,m′′)=(e a′m′,e a′′m′′)(a′×a′′,M′⊠M′′),e(a′,a′′)(m′⊗m′′)=e a′m′⊗m′′+m′⊗e a′′m′′.The associated Lie algebra is in theﬁrst case equal to g(a′,M′)×g(a′′,M′′).This is also true in the second case if both factors are nonzero.The preceding discussion showed that when studying Lefschetz modules we may restrict ourselves to irreducible ones.The following lemma allows the further re-duction of having the associated Lie algebra simple.(1.2)Lemma.Let M be an irreducible Lefschetz a-module and let g(a,M)= g′×g′′be a decomposition of Lie algebra’s.Then this decomposition is graded and there exist irreducible Lefschetz a-modules M′and M′′such that M∼=M′⊗M′′as Lefschetz a-modules with g′resp.g′′corresponding to g(a,M′)resp.g(a,M′′). Proof.Since the grading of g is the eigen space decomposition of ad h it is immediate that upon writing h=(h′,h′′)∈g′×g′′,g(i)gets a grading from ad h(i)making the decomposition a graded one.Since M is an irreducible module of the semisimple Lie algebra g(a,M),it must have the form M′⊗M′′with M(i)a g(i)-module.This is compatible with the gradings.If the rational map f:a→g−2=g′−2⊕g′′−2is written(f′,f′′),then [h,f]=−2f implies[h′,f′]=−2f′and[h′′,f′′]=−2f′′.So M(i)a Lefschetz module of a with the stated property.(1.3)Given a Lefschetz module M of a,then an invariant bilinear form on M is a bilinear mapφ:M×M→K that deﬁnes a morphism of Lefschetz modules M⊗M→K(where a acts trivially on K):soφis zero on M k×M l unless k+l=0and a preserves the formφinﬁnitesimally:φ(e a m,m′)+φ(m,e a m′)=0 for all m,m′∈M and a∈a.If a is a Lefschetz element,then the Jacobson–Morozov lemma implies that f a also preservesφinﬁnitesimally.So g(a,M)is then a subalgebra of aut(M,φ).Ifφis nondegenerate and symmetric(resp.skew-symmetric),then we call(M,φ)an orthogonal(resp.symplectic)representation. Since a nonzero invariant bilinear form on an irreducible representation is either orthogonal or symplectic,any Lefschetz module with nondegenerate bilinear form is the perpendicular direct sum of Lefschetz modules that are irreducible orthogonal, irreducible symplectic,or the direct sum of an irreducible Lefschetz module with its dual.(1.4)Many Lefschetz modules have the additional structure of an algebra.Let A=⊕2n i=0A i be a graded-commutative algebra with A0=K.We say that A is a Lefschetz algebra of depth n if A[n]is a Lefschetz module of depth n over A2.Such a Lefschetz module can be endowed with an invariant(−)n-symmetric bilinear form:6EDUARD LOOIJENGA AND VALERY LUNTSall other degrees and deﬁneφ(a,b):=(−1)q (ab)if a is homogeneous of degree n+2q or n+2q+1.If this form is nondegenerate(which is for instance the case when A[n]is irreducible as a Lefschetz module),then the form(a,b)→ (ab)is also nondegenerate and so A becomes a Frobenius algebra(in the graded sense).(1.5)Let M be a graded real vector space.A Hodge structure of total weight d on M consists of a bigrading on its complexiﬁcation:M⊗C=⊕p,q∈Z M p,q such that(i)M k⊗C=⊕p+q=k+d M p,q for all k and(ii)complex conjugation interchanges M p,q and M q,p.These data are conveniently described in terms of an action of the Deligne torus on M.We recall[Deligne1979]that this is two-dimensional torus S deﬁned over Q that is obtained from GL(1)by restricting scalars from C to R.It comes with two characters z,¯z that are each others complex conjugate and generate the character group.Their product is for obvious reasons called the norm character and is denoted Nm.There is also a natural homomorphism w:GL(1)→S which on the real points is given by the inclusion R×⊂C×=S(R).We follow Deligne’s convention by letting S act on M so that M p,q becomes the eigen space of z−p¯z−q (this action is deﬁned over R).A positive number t>0,viewed as an element of C×=S(R),acts on M k as multiplication by t−k−d.So w(t)acts on M ast−d−h(=t−d exp(−log(t)h)).√−1)q−p;The action ofit is a real operator,called the Weil operator;we denote it by J.Suppose we are also given a nondegenerate(−)d-symmetric formφ:M×M→R that is zero on M p,q×M p′,q′unless(p+p′,q+q′)=(d,d)(this is equivalent to φ(gm,gm′)=Nm(g)−dφ(m,m′)for all g∈S(C)).Let e:M→M be a real operator of bidegree(1,1)which preservesφinﬁnitesimally.Clearly,e commutes with J and it is easily checked that for k≥0,the map H k e:M−k⊗C×M−k⊗C→C deﬁned byH k e(m,m′):=φ(e k m,JLIE ALGEBRA ATTACHED TO A PROJECTIVE VARIETY7 respect toφ:then itsφ-perpendicular space will be an invariant complement.Con-sider the primitive decomposition of N with respect to e a:N=⊕k≥0C[e k]P−k(N), where P−k(N):=Ker(e k+1a|N−k).This decomposition isφ-perpendicular and so we need to show thatφis nondegenerate on each summand C[e k]P−k(N).For this we observe that P−k(N)is J-invariant.Since H k a is deﬁnite on P−k(N),it follows from the deﬁnition of H k a thatφis nondegenerate on P−k(N)+e k P−k(N).The fact that e k leavesφinﬁnitesimally invariant then implies thatφis nondegenerate on C[e k]P−k(N).(1.7)We brieﬂy explain the relation between g(a,M)and the Mumford–Tate group.For this we have to assume that M,its grading,a,and the action of a on M are all deﬁned over Q.Then g(a,M)is as a Lie subalgebra of gl(M)also deﬁned over Q.We further assume that a acts by transformations of bidegree(1,1).Then g(a,M)2k acts by transformations of bidegree(k,k),in other words,for all g∈S and x∈g(a,M)we have gxg−1=Nm(g)−h.Consider the image of S in GL(M)×GL(1),where the second map is given by the norm.One deﬁnes the Mumford–Tate group of M,MT(M),as the smallest Q-subgroup of GL(M)×GL(1)containing this image.It is clear that this is actually a subgroup of(×k GL(M k))×GL(1).The projection of MT(M)onto the last factor is still called the norm character and denoted likewise.The identitygxg−1=Nm(g)−his now valid for all g∈MT(M)and x∈g(a,M).This shows in particular that the adjoint action of MT(M)on gl(M)leaves g(a,M)invariant.(1.8)This suggests to combine the Mumford–Tate group and the group associ-ated to the Lie algebra g(a,M)into a single group:if G(a,M)denotes the closed subgroup of GL(M)with Lie algebra g(a,M),then MT(M)G(a,M)is a reductive algebraic group deﬁned over Q.Its Lie algebra has a natural Hodge structure;it is obtained by composing the homomorphism S→MT(M)G(a,M)with the adjoint action.In this set up the the rˆo le of the Deligne torus is played by the semidirect product S⋉SL(2),where s∈S acts on SL(2)as conjugation by the diagonal ma-trix diag(z(s)−1,¯z(s)).So the Q-homomorphism w:GL(1)→S⋉SL(2),which on the real points is given by t∈R×→(t,diag(t,t−1)),maps onto a central subgroup.A a polarized Hodge structure of weight d on(M,φ)can now be thought of as a certain representation of this group on M with w(t)acting as multiplication by t−d.The corresponding action of S on sl(2)is given by s→(−z(s)−¯z(s))ad h,so that for the resulting Hodge structure on sl(2),the bidegrees of e,h,f are(1,1), (0,0),(−1,−1)respectively.In this spirit one can also enhance the notion of a set of Shimura data,as deﬁned by[Deligne].(1.9)It is high time to give the examples that motivated the preceding deﬁni-tions.Let X be a compact K¨a hlerian manifold of dimension n.We take for M its shifted total complex cohomology H(X)[n]and we letφbe deﬁned byφ(α,β):=(−1)q Xα∪βifαis homogeneous of degree n+2q or n+2q+1.(We shall always suppose8EDUARD LOOIJENGA AND VALERY LUNTS endomorphisms of H(X)that preserve this form inﬁnitesimally.)The fundamental theorems of Hodge theory tell us that M comes with a Hodge structure of total weight n and thatφtogether with cupping with a K¨a hler class deﬁnes a polarization of M.So by proposition(1.6)H(X)[n]is a Lefschetz module over H2(X).The corresponding semisimple Lie subalgebra of aut H(X)will be called the total Lie algebra of M and be denoted g tot(X);it is deﬁned over Q.It is equivalent to say that the cohomology algebra of X is a Lefschetz algebra.Clearly,g tot(X)is independent of the complex structure.For example,if X is a product of an even number of circles,then g tot(X)is deﬁned.If we want to take the complex structure into account,then it is more natural to regard H(X)[n]as module over H1,1(X).This is by(1.6)also a Lefschetz module structure.We shall refer to the associated Lie algebra as the K¨a hler Lie algebra of X and denote it by g K(X).For a complex projective manifold we can restrict further and take for a the N´e ron–Severi group NS(X).We call the corresponding semisimple Lie algebra the N´e ron–Severi Lie algebra of X(denoted g NS(X)).It is deﬁned over Q.Notice that the N´e ron–Severi Lie algebra and the Mumford–Tate group behave in opposite ways under specialization:the former gets bigger,whereas the latter gets smaller.If one of these Lie algebra’s g∗(X)is deﬁned over a subﬁeld K⊂C,then we often write g∗(X;K)for the corresponding Lie algebra of K-points.(1.10)Here is another example.Let V be a complex vector space and W⊂GL(V)aﬁnite complex reﬂection group acting eﬀectively(that is,V W={0}).This group acts naturally in the symmetric algebra of Sym(V).According to a theorem of Chevalley,the subalgebra of invariants,Sym(V)W is a polynomial algebra on dim(V)homogeneous generators.Let I be the ideal generated by the invariants of positive degree.Then the quotient Sym(V)/I is a graded complete intersection algebra.As a W-representation it is isomorphic to the regular representation. In case W is a Weyl group,then after doubling the degrees,Sym(V)/I has the interpretation of the cohomology algebra of aﬂag variety.From this we see that for a suitable regrading,Sym(V)/I is a Lefschetz representation for the obvious action of V.(We do not know whether this is true for an arbitrary reﬂection group.)We shall determine the Lie algebra g(V,Sym(V)/I)in(5.8).(1.11)The N´e ron–Severi Lie algebra can also be deﬁned when X is an irreducible projective variety:take for M the total intersection cohomology IH(X)with the same shift in the grading.There is in general no such thing as a cup product on this graded vector space,but the cohomology ring of X acts on M and according to [Saito]polarizations have the Hodge–Lefschetz property.This even extends to the case where we take intersection cohomology with values in a local system deﬁned on a Zariski open-dense subset that underlies a polarized variation of Hodge structure. There is an invariant formφdeﬁned as in the previous example.The natural setting here is that of polarizable Hodge modules.We return to general properties of Lefschetz modules.(1.12)Proposition.The Lie algebra g(a,M)is a Lefschetz module of a.If a∈a is such that f a is deﬁned,then the Lie subalgebra g(a,M)≥0(resp.g(a,M)≤0)is generated by g(a,M)0and e a(resp.f a).We haveLIE ALGEBRA ATTACHED TO A PROJECTIVE VARIETY9 Proof.Theﬁrst statement is clear and the second follows from this.The third is clear also.We deﬁne the primitive subspace of M as the set of vectors killed by g(a,M)<0. It is a graded g(a,M)0-subrepresentation of M that we denote by Prim(M).Sinceg(a,M)<0is nilpotent,Prim(M)=0.The previous proposition yields:(1.13)Corollary.In the situation of the previous proposition,the primitive sub-space Prim(M)is the maximal g(a,M)0-invariant subspace contained in Ker(f a).Hence M is irreducible as a g(a,M)-representation if and only if Prim(M)is irre-ducible as a g(a,M)0-representation(and if M=0,then Prim(M)is the summandof M of lowest degree).The preceding suggests to shift,in Tannakian spirit,the emphasis from modules to Lie algebra’s.For suppose that conversely,we are given a semisimple Lie algebrag,a simple element h∈g(in the sense of appearing as the middle element of an sl(2)-triple)and an abelian subalgebra a of g such that(i)the adjoint representation of g makes g a Lefschetz module over a,i.e.,thereis a rational map f:a→g−2so that for e in the domain of f,we have ansl(2)-triple(e,h,f e)and(ii)g is as a Lie algebra generated by a and the image of f.If M is aﬁnite dimensional representation of g,then h determines a grading of M and every e∈a in the domain of f has the Lefschetz property in M with respect tothis grading.So M is then a Lefschetz module of a.Since g is generated by a and the image of f,it follows that g(a,M)is just the image of g in gl(M).This reducesthe classiﬁcation of Lefschetz modules to classifying triples(g,h,a)as above.We shall call such a triple a Lefschetz triple and itsﬁrst two items,(g,h),a Lefschetzpair.If we are given a Lefschetz pair(g,h),then we say that an associated Lefschetztriple(g,h,a)is saturated if a is maximal for this property.The stabilizer G h of h in the adjoint group G permutes these,but we do not know whether this action istransitive or even whether it has onlyﬁnitely many orbits.(1.14)Let(g,h)be a Lefschetz pair.Choose a Cartan subalgebra h of g that contains h.It is clear that then h⊂g0.Let R⊂h∗denote the set of rootsof h in g and let R k be the set ofα∈R such thatα(h)=k,or equivalently,gα⊂g k(remember that only even values of k occur).The subset R0is a closed root subsystem of R;it is the set of roots of h in g0.Choose a root basis B⊂Rsuch thatα(h)≥0for allα∈B(we then say that h and B are adapted).Since h is the semisimple element of a sl(2)-triple,the elements of B take on h values in{0,1,2}[Bourbaki],Ch.VIII,§11,Prop.5.Since these are also even,we get a decomposition B=B0⊔B2.Clearly B0will be a root basis of R0.According to [Bourbaki],Ch.VIII,§11,Prop.8all sl(2)-triples with h as semisimple element are conjugate under the stabilizer G h of h in the adjoint group G of g.So if(e,h,f)is such a triple and M is any representation of g,then the isomorphism class of M as a representation of this sl(2)-copy only depends on(g,h).We call it the sl(2)-type of M.The following property narrows down the possible subsets B2⊂B.Let V(k)denote the standard irreducible representation of sl(2)of dimension k+1 (k=1,2,...).(1.15)Proposition.Let(g,h)be a Lefschetz pair and let M be an irreduciblerepresentation of g of depth n.Then the dimensions of the irreducible sl(2)-10EDUARD LOOIJENGA AND VALERY LUNTSwith increment2.In other words,there exists an integer r with0≤r≤⌊1form.So anﬁnite dimensional sl(2)-representation of even parity always admits a nondegenerate invariant symmetric form,whereas it admits a nondegenerate in-variant skew-symmetric form if and only if all multiplicities are even.In the case of odd parity it is just the other way around.The sl(2)-triple(e,h,f)in g determines a primitive decomposition of V.Ac-cording to(1.15)the set of positive integers i≥0for which V(i)appears in the sl(2)-module V is of the form{n,n−2,...,n−2r}(with n−2r≥0).So if we put k:=⌊n/2⌋,then the dimensions d t:=dim V−n+2t(i=0,1,...,k)satisfy(*)1≤d0<d1<···<d r=d r+1=···=d k.By the remark above,the d i’s must all be even in the orthogonal cases with odd parity and in the symplectic cases with even parity.We choose an h-invariant basis of V indexed as in[Bourbaki]:(e1,...,e l+1) in case A l,(e1,...,e l,e0,e−l,...,e−1)in case B l and in the cases C l and D l, (e1,...,e l,e−l...,e−1,).The same shall apply to our labeling of the simple roots (α1,α2,...αl)(as recalled below).Case A l.We letαi(diag(λ1,...,λl+1))=λi−λi+1.Weﬁnd that the elements of B2are the simple roots with index d0,d0+d1,...,d0+···+d k,d0+···+ d k−1+2d k,...,d0+2(d1+···+d k−1+d k)(then l=2(d0+···+d k))or d0,d0+ d1,...,d0+···+d k,d0+···+2d k−1+d k,...,d0+2(d1+···+d k−1)+d k(then l=2(d0+···+d k−1)+d k).So B2is symmetric with respect to the natural involution of B.Notice that no two elements of B2will be adjacent in the Dynkin diagram:if that would be the case,then d i=1for some i>0and(*)shows that this is impossible.Case B l.Since dim V=2l+1,its parity must be even.This means that n is even and d k is odd;we have l=d0···+d k−1+12d k and the elements of B2are simple roots with index d0,d0+d1,...,d0+···+d k−1. In both cases no two elements of B2will be adjacent in the Dynkin diagram. Case D l,l≥4.We letαi(diag(λ1,...,λl,−λ−l,...,−λ−1))=λi−λi+1if i=1,...,l−1。

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