Vertex algebras, Kac-Moody algebras, and the Monster

第十二届全国代数学术会议大会报告摘要Classical Yang-Baxter equation,its extensions and some related algebraicstructures白承铭南开大学Wefirst give a brief introduction to the classical Yang-Baxter equation which emphasizes the relationship between the tensor and operator forms.Then we give two approaches to extend the classical Yang-Baxter equation,motivated by the study of different topics like integrable systems,Rota-Baxter algebras and Lie bialgebras.Moreover,there are some interesting algebraic structures behind these two approaches.Representation type of algebras:Yesterday,today and tomorrow韩阳中国科学院数学与系统科学研究院Email:***********.cnOne task in representation theory of algebras is to classify all representations of an algebra.Before this,one needs to judge whether it is hopeful for an algebra to do so or not,namely,to determine the representation type of an algebra.In this survey,some concepts,results,ideas,methods,and problems on representation type of algebras are introduced.群环的代数K理论唐国平中科院研究生院数学科学学院报告将分为三个部分：1）本质循环群(具有指数有限的循环子群的群)的整群环的K理论。

量子群表示的完全可约性判据量子群表示是量子力学的基本概念之一，在研究物质微观结构和宏观性质方面具有重要意义。

在量子群表示的研究中，我们常常需要判断一个表示是否是完全可约的，即是否可以完全分解为一系列不可约表示的直和。

本文将介绍几种判据来评估量子群表示的完全可约性。

一、迹和产生函数量子群表示的完全可约性判据之一就是利用迹和产生函数的性质。

迹是指线性算子在不变子空间上的迹。

通过计算线性算子在不变子空间上的迹，我们可以得到一个量子群表示是否是完全可约的判据。

另外，产生函数则用于刻画表示的完全可约性。

通过计算产生函数可以确定一个表示是否拥有某种特定的完全可约性。

二、直和条件直和条件也是判断量子群表示的完全可约性的重要判据之一。

直和条件是指将一个表示分解为一系列不可约表示的直和的条件。

通过直和条件的验证，我们可以确定一个表示是否是完全可约的。

在实际应用中，直和条件则常用于量子群表示的模型选择和优化。

三、特征标方程特征标方程也是判断量子群表示的完全可约性的一种方法。

特征标方程是指通过计算不同表示对应的特征标，判断表示是否完全可约的方程。

对于完全可约的表示来说，其特征标方程必然存在解。

四、不变子空间的直和在量子群表示的研究中，我们可以通过计算不变子空间的维数来判断一个表示的完全可约性。

如果一个表示的不变子空间可以完全分解为一系列不可约表示的直和，那么这个表示就是完全可约的。

通过计算不变子空间的维数，我们可以得知一个表示是否满足完全可约性的要求。

综上所述，我们可以通过迹和产生函数的性质、直和条件、特征标方程以及不变子空间的直和来判断量子群表示的完全可约性。

这些判据在量子力学研究中具有重要的理论和实际应用价值，对于揭示物质微观结构和宏观性质具有重要意义。

总结起来，量子群表示的完全可约性判据是研究量子力学中的一个重要问题，通过迹和产生函数、直和条件、特征标方程以及不变子空间的直和等方法可以进行判断。

正确应用这些判据，我们可以准确评估量子群表示的完全可约性，进而深入研究物质微观结构和宏观性质。

a r X i v :m a t h /0411523v 1 [m a t h .Q A ] 23 N o v 2004Twisted representations of vertex operatorsuperalgebrasChongying Dong 1and Zhongping ZhaoDepartment of Mathematics,University of California,Santa Cruz,CA 95064AbstractThis paper gives an analogue of A g (V )theory for a vertex operator superalgebra V and an automorphism g of finite order.The relation between the g -twisted V -modules and A g (V )-modules is established.It is proved that if V is g -rational,then A g (V )is finite dimensional semisimple associative algebra and there are only finitely many irreducible g -twisted V -modules.1IntroductionThe twisted sectors or twisted modules are basic ingredients in orbifold conformal field theory (cf.[FLM1],[FLM2],[FLM3],[Le1],[Le2],[DHVW],[DVVV],[DL2],[DLM2]).The notion of twisted module [FFR],[D]is derived from the properties of twisted vertex operators for finite automorphisms of even lattice vertex operator algebras constructed in [Le1],[Le2]and [FLM2],also see [DL2].In this paper we study the twisted modules for an arbitrary vertex operator superalgebra following [Z],[KW]and [DLM2].An associative algebra A (V )was introduced in [Z]for every vertex operator algebra V to study the representation theory for vertex operator algebra.The main idea is to reduce the study of representation theory for a vertex operator algebra to the study of represen-tation theory for an associative algebra.This approach has been very successful and the irreducible modules for many well-known vertex operator algebras have been classified by using the associative algebras.This theory has been extended to the vertex operator superalgebras in [KW]and has been further generalized to the twisted representations for a vertex operator algebra in [DLM2].This paper is a “super analogue”of [DLM2].We construct an associative algebra A g (V )for any vertex operator superalgebra V together with an automorphism g of finite order.Then the vacuum space of any admissible g -twisted V -module becomes a module for A g (V ).On the other hand one can construct a ‘universal’admissible g -twisted V -module from any A g (V )-module.This leads to a one to one correspondence between the set of inequivalent admissible g -twisted V -modules and the set of simple A g (V )-modules.As in the case of vertex operator algebra,if V is g -rational then A g (V )is a finite dimensional semisimple associative algebra.The ideas of this paper and other related papers are very natural and go back to the theory of highest weight modules for Kac-Moody Lie algebras and other Lie algebras with triangular decompositions.In the classical highest weight module theory,the highestweight or highest weight vector determines the highest weight module structure to some extend(different highest weight modules can have the same highest weight).The role of the vacuum space for an admissible twisted module is similar to the role of the highest weight space in a highest weight module.So from this point of view,the A g(V)theory is a natural extension of highest weight module theory in the representation theory of vertex operator superalgebras.A vertex operator superalgebra has a canonical automorphismσof order2arising from the structure of superspace.Theσ-twisted modules which are called the Ramond sector in the literature play very important roles in the study of geometry.Important topological invariants such as elliptic genus and certain Witten genus can be understood as graded trace functions on the Ramond sectors constructed from the manifolds.It is expected that the theory developed in this paper will have applications in geometry and physics.Since the setting and most results in this paper are similar to those in[DLM2]we only provide the arguments which are either new or need a lot of modifications.We refer the reader to[DLM2]for details.The organization of this paper is similar to that of[DLM2].We review the definition of vertex operator superalgebra and define various notions of g-twisted V-modules in section2.In section3,we introduce the algebra A g(V)for VOSA V.Section4is devoted to the study of Lie superalgebra V[g]which is kind of twisted affinization of V.A weak g-twisted V-module is naturally a V[g]-module.In section5,we construct the functorΩwhich sends a weak g-twisted V-module to an A g(V)-module.We construct another functor L from the category of A g(V)-modules to the category of admissible g-twisted V-modules in Section6.That is,for any A g(V)-module U we can construct a kind of“generalized Verma module”¯M(U)which is the universal admissible g-twisted V-module generated by U.It is proved that there is a1-1correspondence between the irreducible objects in these two categories.Moreover if V is g-rational,then A g(V)is a finite dimensional semisimple associative algebra.We discuss some examples of vertex operator superalgebras constructed from the free fermions and their twisted modules in Section7.2Vertex Operator superalgebra and twisted mod-ulesWe review the definition of vertex operator superalgebra(cf.[B],[FLM3],[DL1])and various notions of twisted modules in this section(cf.[D],[DLM2],[FFR],[FLM3],[Z]).Recall that a super vector space is a Z2-graded vector space V=V¯0⊕V¯1.The elements in V¯0(resp.V¯1)are called even(resp.odd).Let˜v be0if v∈V¯0,and1if v∈V¯1.Definition2.1.A vertex operator superalgebra is a12Z+V n=V¯0⊕V¯1.(2.1)with V¯0= n∈Z V n and V¯1= n∈112(m3−m)δm+n,0c;(2.6)dz0 Y(u,z1)Y(v,z2)−(−1)˜u˜v z−10δ z2−z1z2 Y(Y(u,z0)v,z2).(2.9) whereδ(z)= n∈Z z n and(z i−z j)n is expanded as a formal power series in z j.Throughout the paper,z0,z1,z2,etc.are independent commuting formal variables.Such a vertex operator superalgebra may be denoted by V=(V,Y,1,ω).In the case V¯1=0,this is exactly the definition of vertex operator algebra given in[FLM3].Definition2.2.Let V be a vertex operator superalgebra.An automorphism g of V is a linear automorphism of V preservingωsuch that the actions of g and Y(v,z)on V are compatible in the sense thatgY(v,z)g−1=Y(gv,z)for v∈V.Note that any automorphism of V commutes with L(0)and preserves each homoge-neous space V n.As a result,any automorphism preserves V¯0and V¯1.Let Aut(V)be the group of automorphisms of V.There is a special automorphism σ∈Aut(V)such thatσ|V¯0=1andσ|V¯1=−1.It is clear thatσis a central element of Aut(V).Fix g∈Aut(V)of order T0.Let o(gσ)=T.Denote the decompositions of V into eigenspaces with respect to the actions of gσand g as followsV=⊕r∈Z/T Z V r∗(2.10)V=⊕r∈Z/T0ZV r(2.11) where V r∗={v∈V|gσv=e2πir/T v}and V r={v∈V|gv=e2πir/T0v}Definition2.3.A weak g-twisted V-module M is a vector space equipped with a linear mapV→(End M)[[z1/T0,z−1/T0]v→Y M(v,z)= n∈1T0+Zu n z−n−1;(2.12)u l w=0for l>>0;(2.13)Y M(1,z)=Id M;(2.14) z−10δ z1−z2−z0 Y M(v,z2)Y M(u,z1)=z−12 z1−z0z2 Y M(Y(u,z0)v,z2).(2.15)Following the arguments in[DL1]one can prove that the twisted Jacobi identity is equivalent to the following associativity formula(z0+z2)k+r T0Y M(Y(u,z0)v,z2)w.(2.16) where w∈M and k∈Z+s.t z k+rz2 −r/T0δz1−z0Lemma2.4.The associativity formula(2.16)is equivalent to the following: (z0+z2)m+s T Y M(Y(u,z0)v,z2)wfor u∈V s∗and some m∈1T Y M(u,z)w involves only nonnegative integral powers of z.Proof:Let u∈V r.It is enough to prove that wt u+sT0are congruent modulo Z.It is easy to see that s≡T2˜u+2r modulo Z if T0is odd.Thus wt u+s2˜u+r2˜u and wt u are congruentmodulo Z,the result follows immediately.Equating the coefficients of z−m−11z−n−12in(2.17)yields[u m,v n]=∞i=0 m i (u i v)m+n−i.(2.18)We may also deduce from(2.12)-(2.15)the usual Virasoro algebra axioms,namely that if Y M(ω,z)= n∈Z L(n)z−n−2then[L(m),L(n)]=(m−n)L(m+n)+1dzY M(v,z)=Y M(L(−1)v,z)(2.20) (cf.[DLM1]).The homomorphism and isomorphism of weak twisted modules are defined in an ob-vious way.Definition2.5.An admissible g-twisted V-module is a weak g-twisted V-module M which carries a1T Z+M(n)(2.21)satisfyingv m M(n)⊆M(n+wt v−m−1)(2.22) for homogeneous v∈V.Definition2.6.An ordinary g-twisted V-module is a weak g-twisted V-moduleM= λ∈C Mλ(2.23) such that dim Mλisfinite and forfixedλ,M nThe admissible g-twisted V-modules form a subcategory of the weak g-twisted V-modules.It is easy to prove that an ordinary g-twisted V-module is admissible.Shifting the grading of an admissible g-twisted module gives an isomorphic admissible g-twisted V-module.A simple object in this category is an admissible g-twisted V-module M such that0and M are the only graded submodules.We say that V is g-rational if every admissible g-twisted V-module is completely reducible,i.e.,a direct sum of simple admissible g-twisted modules.V is called rational if V is1-rational.V is called holomorphic if V is rational and V is the only irreducible V-module up to isomorphism.If M=⊕n∈1T Z+M(n)∗(2.24)where M(n)∗=Hom C(M(n),C).The vertex operator Y M′(a,z)is defined for a∈V via Y M′(a,z)f,u = f,Y M(e zL(1)(−z−2)L(0)a,z−1)u (2.25) where · denotes the natural paring between M′and M.Then we have the following [FHL]:Lemma2.7.(M′,Y M′)is an admissible g−1-twisted V-module.Lemma2.7is needed in the proof of several results in Section6although we do not intend to give these proofs(cf.[DLM2]).3The associative algebra A g(V)Let r be an integer between0and T−1(or T0−1).We will also use r to denote its residue class modulo T or T0.For homogeneous u∈V r∗,we setδr=1if r=0andδr=0 if r=0.Let v∈V we defineu◦g v=Res z (1+z)wt u−1+δr+rz1+δrY(u,z)v(3.1)where(1+z)αforα∈C is to be expanded in nonnegative integer powers of z.Let O g(V) be the linear span of all u◦g v and define the linear space A g(V)to be the quotient V/O g(V).We will use A(V),O(V),u◦v,when g=1.The A(V)was constructed in[KW] and if V is a vertex operator,A g(V)was constructed in[DLM2].Lemma3.1.If r=0then V r∗⊆O g(V).Proof:The proof is the same as that of Lemma2.1in[DLM2].Let I=O g(V)∩V0∗.Then A g(V)≃V0∗/I(as linear spaces).Since O(V0∗)⊂I, A g(V)is a quotient of A(V0∗).We now define a product ∗g on V which will induce an associative product in A g (V ).Let r,u and v be as above and setu ∗gv =Res z (Y (u,z )(1+z )wt uT+nzY (v,z )u ∈O (V 0∗)and(iii)u ∗v −(−1)˜u ˜v v ∗u −Res z (1+z )wt u −1Y (u,z )v ∈O (V 0∗).Proof:See the proofs of Lemmas 2.1.2and 2.1.3of[Z]bynoting thatY (u,z )v ≡(−1)˜u ˜v (1+z )−wtu −wtv Y (v,−zz Y (c,z )u(3.4)andu∗c≡Res z(1+z)wt c−1z0 Y(c,z1)Y(a,z2)b−(−1)˜c˜a z−10δ z2−z1z2 Y(Y(c,z0)a,z2)b.(3.6) Forε=0or1,(3.6)implies:xε=Res z1(1+z1)wt c−εTz1Y(c,z1)(1+z2)wt a−1+δr+rz1+δr2Y(a,z2)b=(−1)˜a˜c Res z1Res z2(1+z1)wt c−εTz1(1+z2)wt a−1+δr+rz1+δr2z−12δ z1−z0z1(1+z2)wt a−1+δr+rz1+δr2Y(a,z2)Y(c,z1)b+Res z2Res z(1+z2+z0)wt c−εTTz1Y(c,z1)b+∞i,j=0(−1)j wt c−εi Res z2(1+z2)wt a−1+δr+r z j+2+δr2Y(c i+j a,z2)b=(−1)˜c˜a Res z2(1+z2)wt a−1+δr+rz1+δr2Y(a,z2)Res z1(1+z1)wt c−εT+j+1−εNext we prove that∗g is associative.We need to verify that(a∗b)∗c−a∗(b∗c)∈O g(V0∗)for a,b,c∈V0∗.A straightforward computation using the twisted Jacobi identity gives(a∗b)∗c=wt a i=0(a i−1b)∗c=wt ai=0 wt a i Res w(Y(a i−1b,w)(1+w)wt(a i−1b)wc)=Res w Res z−w(Y(Y(a,z−w)b,w)(1+z)wt a(1+w)wt bw(z−w)c)−(−1)˜a˜b Res w Res z(Y(b,w)Y(a,z)(1+z)wt a(1+w)wt bwc)−(−1)˜a˜b∞i=0Res w Res z(Y(b,w)Y(a,z)(−1)i+1z i w−i−1(1+z)wt a(1+w)wt bzwc)mod O g(V0∗)≡a∗(b∗c)mod O g(V0∗)Thus A g(V)≃V0∗T0,t−1dt f(t) g(t).(4.1) (see[B]).Then the tensor productL(V)=C[t1T0]⊗V.(4.2)is a vertex superalgebra with vertex operatorY (f (t )⊗v,z )(g (t )⊗u )=f (t +z )g (t )⊗Y (v,z )u.(4.3)The L (−1)operator of L (V )is given by D =dT 0)(t m ⊗ga ).(4.4)Let L (V,g )be the g -invariants which is a vertex sub-superalgebra of L (V ).Clearly,L (V,g )=⊕T 0−1r =0tr/T 0C [t,t −1]⊗V r .(4.5)Following [B],we know thatV [g ]=L (V,g )/D L (V,g )(4.6)is a Lie superalgebra with bracket[u +D L (V,g ),v +D L (V,g )]=u 0v +D L (V,g ).(4.7)For short let a (q )be the image of t q ⊗a ∈L (V,g )in V [g ].Then we have Lemma 4.1.Let a ∈V r ,v ∈V s and m,n ∈Z .Then(i)[ω(0),a (m +r T 0a (m −1+rT 0),b (n +s T 0ia ib (m +n +r +sTZ -graded.Since D increases degree by 1,D L (V,g )is a graded subspace ofL (V,g )and V [g ]is naturally 1TZV [g ]n .By Lemma 4.1,V [g ]is a1TZV [g ]±n .Lemma 4.2.V [g ]0is spanned by elements of the form a (wt a −1)for homogeneous a ∈V 0∗.Proof:Let a∈V.Then the degree wt a−n−1of a(n)is0if and only if a∈V0¯andn=wt a−1or a∈V T0/2¯1and n=wt a−1.The bracket of V[g]0is given by[a(wt a−1),b(wt b−1)]=∞j=0 wt a−1j a j b(wt(a j b)−1).(4.10)Set o(a)=a(wt a−1)for homogeneous a∈V0∗and extend linearly to all a∈V0∗. This gives a linear mapV0∗→V[g]0,a→o(a).(4.11) As the kernel of the map is(L(−1)+L(0))V0∗,we obtain an isomorphism of Lie super-algebras V0∗/(L(−1)+L(0))V0∗∼=V[g]0.The bracket on the quotient of V0∗is given by[a,b]= j≥0 wt a−1j a j b.Lemma4.3.Let A g(V)Lie be the Lie superalgebra of the associative algebra A g(V)intro-duced in section3such that[u,v]=u∗g v−(−1)˜u˜v v∗g u.Then the map o(a)→a+O g(V) is an onto Lie superalgebra homomorphism from V[g]0to A g(V)Lie.Proof:Recall that I=O g(V)∩V0∗.So we have a surjective linear mapV[g]0∼=V0∗/(L(−1)+L(0))V0∗→V0∗/I≃A g(V),o(a)→a+(L(−1)+L(0))V0∗→a+I.(4.12) The Lie homomorphism follows from[o(a),o(b)]=∞j=0 wt a−1j o(a j b).and[a+O g(V),b+O g(V)]≡a∗g b−(−1)˜a˜b b∗g a≡∞j=0 wt a−1j a j b≡Res z(1+z)wt a−1Y(a,z)b mod O g(V0∗)≡∞i=0 wt a−1i a i b mod O g(V0∗).5The functorΩThe main purpose in this section is to construct a covariant functorΩfrom the category of weak g-twisted V-modules to the category of A g(V)-modules(cf.Theorem5.1).Let M be a weak g-twisted V-module.We define the space of“lowest weight vectors”to beΩ(M)={w∈M|u wt u+n w=0,u∈V,n≥0}.The main result in this section says thatΩ(M)is an A g(V)-module.Moreover if f:M→N is a morphism in weak g-twisted V-modules,the restrictionΩ(f)of f toΩ(M)is an A g(V)-module morphism.Note that if M is a weak g-twisted V-module then M becomes a V[g]-module such that a(m)acts as a m.Moreover,M is an admissible g-twisted V-module if and only if M is a1TY(u,z)v.z2The argument in the Proof of Theorem2.1.2in[Z]with suitable modification giveso(u∗v)=o(u)o(v).Note that o(L(−1)u+L(0)u)=0and(L(−1)u+L(0)u)∗v=u◦v.We immediately have o(u◦v)=0onΩ(M).Ifa=Res z (1+z)wt c−1+rzY(u,z)v,we can use Lemma2.4.Since z wt u−1+rT Y M(u,z0+z2)Y M(v,z2)w=(z2+z0)wt u−1+rT2to(5.1)yields0=Res z0Res z2z−10zwt v−rT Y M(Y(u,z0)v,z2)w=∞i=0 wt u−1+rTi o(u i−1v)w=o Res z(1+z)wt u−1+r z Y M(u,z)v w=o(a)w(5.2) as required.If M is a nonzero admissible g-twisted V-modules we may and do assume that M(0) is nonzero with suitable degree shift.With these conventions we haveProposition5.2.Let M be a simple admissible g-twisted V-module.Then the following hold(i)Ω(M)=M(0).(ii)Ω(M)is a simple A g(V)-module.Proof:The proof is the same as in[DLM2].6Generalized Verma modules and the functor LIn this section we focus on how to construct admissible g-twisted V-modules from a given A g(V)-module U.We use the same trick which was used in[DLM2]to do this.We will define two g-twisted admissible V-modules¯M(U)and L(U).The¯M(U)is the universal admissible g-twisted V-module such that¯M(U)(0)=U and L(U)is smallest admissible g-twisted V-module whose L(U)(0)=U.Just as in the classical highest weight module theory,L(U)is the unique irreducible quotient of¯M(U)if U is simple.We start with an A g(V)-module U.Then U is automatically a module for A g(V)Lie.By Lemma4.3U is lifted to a module for the Lie superalgebra V[g]0.Let V[g]−act trivially on U and extend U to a P=V[g]−⊕V[g]0-module.Consider the induced moduleM(U)=Ind V[g]P(U)=U(V[g])⊗U(P)U(6.1)T0Zv(m)z−m−1(6.2)Then Y M(U)(v,z)satisfies condition(2.12)-(2.14).By Lemma4.1(ii),the identity(2.18) holds.But this is not good enough to establish the twisted Jacobi identity for the action (6.2)on M(U).Let W be the subspace of M(U)spanned linearly by the coefficients of(z0+z2)wt a−1+δr+r T Y(Y(a,z0)b,z2)u(6.3) for any homogeneous a∈V r∗,b∈V,u∈U.We set¯M(U)=M(U)/U(V[g])W.(6.4) Proposition6.1.Let M be a V[g]-module such that there is a subspace U of M satisfying the following conditions:(i)M=U(V[g])U;(ii)For any a∈V r∗and u∈U there is k∈wt a+Z+such that(z0+z2)k+r T Y(Y(a,z0)b,z2)u(6.5) for any b∈V.Then M is a weak V-module.Proof:We only need to prove the twisted Jacobi identity,which is equivalent to com-mutator relation(2.17)and the associativity(2.16).But the commutator formula is built in already as M is a V[g]-module.By Lemma2.4,the assumption(ii)can be reformulated as follows:(ii’)For any a∈V r and u∈U there is k∈Z+such that(z0+z2)k+r T0Y(Y(a,z0)b,z2)u(6.6) Since M is a V[g]-module generated by U it is enough to prove that if u satisfies(ii’) then c n u also satisfies(ii’)for c∈V and n∈1T0Y(c i a,z0+z2)Y(b,z2)u=(z2+z0)k2+r+sT0Y(a,z0+z2)Y(c i b,z2)u=(z2+z0)k2+r+sT0+n−k1>k2+r+s(z 0+z 2)k +rT 0c n Y (a,z 0+z 2)Y (b,z 2)u −(−1)˜a ˜c (−1)˜b ˜c∞ i =0n i (z 0+z 2)k +r T 0Y (a,z 0+z 2)Y (c i b,z 2)u=(−1)˜a ˜c (−1)˜b ˜c (z 0+z 2)k +rT 0+n −iY (Y (c i a,z 0)b,z 2)u−(−1)˜b ˜c∞ i =0n iz n −i2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c ∞ i =0n i (z 2+z 0)k +r T 0Y (c i Y (a,z 0)b,z 2)u+(−1)˜a ˜c (−1)˜b ˜c ∞ i =0∞ j =0n j j iz n −i2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c ∞ i =0n i (z 2+z 0)k +r T 0Y (c i Y (a,z 0)b,z 2)u+(−1)˜a ˜c (−1)˜b ˜c ∞ j =0∞ i =jn j n −ji −jz n −i2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c∞ i =0n i z n −i 2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c (z 2+z 0)k +rT 0Y (Y (a,z 0)b,z 2)c n u,=(z 2+z 0)k +rThe proof is complete.Applying Proposition6.1to¯M(U)gives the following main result of this section. Theorem6.2.¯M(U)is an admissible g-twisted V-module with¯M(U)(0)=U and with the following universal property:for any weak g-twisted V-module M and any A g(V)-morphismφ:U→Ω(M),there is a unique morphism¯φ:¯M(U)→M of weak g-twisted V-modules which extendsφ.As in[DLM2]we also haveTheorem6.3.M(U)has a unique maximal graded V[g]-submodule J with the property that J∩U=0.Then L(U)=M(U)/J is an admissible g-twisted V-module satisfying Ω(L(U))∼=U.L defines a functor from the category of A g(V)-modules to the category of admissible g-twisted V-modules such thatΩ◦L is naturally equivalent to the identity.We have a pair of functorsΩ,L between the A g(V)-module category and admissible g-twisted V-module category.AlthoughΩ◦L is equivalent to the identity,L◦Ωis not equivalent to the identity in general.The following result is an immediate consequence of Theorem6.3.Lemma6.4.Suppose that U is a simple A g(V)-module.Then L(U)is a simple admissible g-twisted V-module.Using Lemma6.4,Proposition5.2(ii),Theorems6.2and6.3gives:Theorem6.5.L andΩare equivalent when restricted to the full subcategories of com-pletely reducible A g(V)-modules and completely reducible admissible g-twisted V-modules respectively.In particular,L andΩinduces mutually inverse bijections on the isomor-phism classes of simple objects in the category of A g(V)-modules and admissible g-twisted V-modules respectively.We now apply the obtained results to g-rational vertex operator superalgebras to obtain:Theorem6.6.Suppose that V is a g-rational vertex operator superalgebra.Then the following hold:(a)A g(V)is afinite-dimensional,semi-simple associative algebra(possibly0).(b)V has onlyfinitely many isomorphism classes of simple admissible g-twisted mod-ules.(c)Every simple admissible g-twisted V-module is an ordinary g-twisted V-module.(d)V is g−1-rational.(e)The functors L,Ωare mutually inverse categorical equivalences between the cate-gory of A g(V)-modules and the category of admissible g-twisted V-modules.(f)The functors L,Ωinduce mutually inverse categorical equivalences between the category offinite-dimensional A g(V)-modules and the category of ordinary g-twisted V-modules.The proof is the same as that of Theorem8.1in[DLM2].7ExamplesIn this section we discuss the well known vertex operator superalgebras constructed from the free fermions and their twisted modules.In particular we compute the algebra A g (V )and classify the irreducible twisted modules using A g (V ).The classification results have been obtained previously in [Li2]with a different approach.Let H = li =1C a i be a complex vector space equipped with a nondegenerate symmet-ric bilinear form (,)such that {a i |i =1,2,...l }form an orthonormal basis.Let A (H,Z +12}subject to the relation [a (n ),b (m )]+=(a,b )δm +n,0.Let A +(H,Z +12,n >0},andmake C a 1-dimensional A +(H,Z +12)=A (H,Z +12)C∼=Λ[a i(−n )|n >0,n ∈Z +1∂a i (−n )if n is positive and by multiplication by a i (n )if nis negative.The V (H,Z +12Zso thatV (H,Z +12Zwe define a normal ordering:b 1(n 1)···b k (n k ):=(−1)|σ|b i 1(n i 1)···b i k (n i k )such that n i 1≤···≤n i k where σis the permutation of {1,...,k }by sending j to i j .For a ∈H set Y (a (−1/2),z )=n ∈12)···b k (−n k −12)where n i arenonnegative integers.We setY (v,z )=:(∂n 1b 1(z ))···(∂n k b k (z )):where ∂n =1dz)n .Then we have a linear map:V (H,Z +12))[[z,z −1]]v→Y (v,z )=n ∈Z v n z −n −1(v n ∈End V (H,Z +12li =1a i (−32).The following result is well known (cf.[FFR],[KW]and [Li1]).Theorem7.1.(V(H,Z+12)for i=1,...,l.We have already mentioned in Section2that any vertex operator superalgebra has a canonical automorphismσsuch thatσ=1on V¯0andσ=−1on V¯1.Note thatV(H,Z+12)¯1.We next discuss theσ-twisted V(H,Z+1∂b i(−n)∗if n is nonnegative and multiplication by b i(n)if n is nega-tive.Similarly,b i(n)∗acts as∂2)-module such thatY V(H,Z)(u(−12)is isomorphic to thematrix algebra M2k×2k(C)and V(H,Z)is the unique irreducibleσ-twisted V(H,Z+12)is isomorphic to thematrix algebra M2k×2k(C).Since g=σ,the decomposition(2.10)becomes V=V0∗.By lemma3.2(i),Res z (1+z)1z2+ma i(z)v= s≥0c s a i(−m+s−3lies in O σ(V (H,Z +12s.This implies that a i (−m −32+s )vmod O σ(V (H,Z +12))is spanned by b 1(−1/2)s 1···b k (−1/2)s k b ∗1(−1/2)t 1···b ∗k (−1/2)t kwith s i ,t i =0,1.As a result,dim A σ(V (H,Z +12)-module.By Theorem 5.1,Ω(V (H,Z ))is a simple A σ(V (H,Z +12)≥dim Ω(V (H,Z ))=22k .This forces dim A σ(V (H,Z +12))∼=M 2k ×2k (C ).We now deal with the case dim H =2k +1for some nonnegative integer k.Then H can be decomposed into:H =k i =1C b i +k i =1C b ∗i +C ewith (b i ,b j )=(b ∗i ,b ∗j )=0,(b i ,b ∗j )=δi,j ,(e,b i )=(e,b ∗i )=0,(e,e )=2.Let A (H,Z )be the associative algebra generated same as above,and A (H,Z )+be the subalgebra generated by {b i (n ),b ∗i (m ),e (n )|m,n ∈Z ,m >0,n ≥0,i =1,···,k }and make C a 1-dimensional A (H,Z )+-module so that b i (n )1=0for n ≥0and b ∗i (m )1=e (m )1=0for m >0,i =1,···,k.SetV (H,Z )=A (H,Z )⊗A (H,Z )+C∼=Λ[b i (−n ),b ∗i (−m ),e (−m )|n,m ∈Z ,n >0,m ≥0]and letW (H,Z )=Λ[b i (−n ),b ∗i (−m ),e (−n )|n,m ∈Z ,n >0,m ≥0]=W (H,Z )even⊕W (H,Z )odd be the decomposition into the even and old parity subspaces.Also defineV ±(H,Z )=(1±e (0))W (H,Z )even ⊕(1∓e (0))W (H,Z )odd .ThenV (H,Z )=V +(H,Z )⊕V −(H,Z )and V ±(H,Z )are irreducible A (H,Z )-modules.The actions of b i (n ),b ∗i (n )are the same as before.The e (n )acts as 2∂2),z )=u (z )=n ∈Zu (n )z −n −1/2for u ∈H.Proposition7.3.If dim H=2k+1is odd,then Aσ(V(H,Z+12)has exactly two irreducibleσ-twisted modulesV±(H,Z)up to isomorphism.Proof:The proof is similar to that of Proposition7.2.Note that the automorphismσof V(H,Z+12)as follows:For any a1(−n1)···a s(−n s)∈V(H,Z+1/2),τ(a1(−n1)a2(−n2)···a m(−n m))=(τa1)(−n1)(τa2)(−n2)···(τa m)(−n m).Let o(τσ)=N.We decompose H into eigenspaces with respect to theτσandτas follows:H=⊕r∈Z/N Z H r∗(7.4)H=⊕r∈Z/N0ZH r(7.5) where H r∗={v∈H|τσv=e2πir/N v},and H r={v∈H|τv=e2πir/N0v}.Let l0=dim H0∗.As before we need to consider two separate cases:l0is even or odd. If l0=2k0for some nonnegative integer k0,we haveH0∗=k0i=1C h i+k0 i=1C h∗iwith(h i,h j)=(h∗i,h∗j)=0,(h i,h∗j)=δi,j.Let l r=dim H r∗with r=0.If r=N−r,wefix bases b r,1,b r,2,···b r,lr∈H r∗and b∗r,1,b∗r,2,···b∗r,lr∈H(N−r)∗such that(b r,i,b∗r,j)=(b∗r,j,b r,i)=δi,j.If r=N−r,let{c1,c2,···c lN 2∗.Then M= N−1r=1Λ[b(−n)|n∈r2)-module so that for u∈H r∗, Y M(u(−1N +Zu(n)z−n−1/2(see[Li2]).Note that b r,i(n)acts as∂∂c i(−n)if n is positive andacts as multiplication by c i(n)if n is negative.Also,h i(n)acts as∂∂h i(−n)if n is positive,and acts as multiplication by h∗i(n)if nis nonnegative.One can easily calculate thatΩ(M)=Λ[h∗i(0)|h∗i∈H0∗,i=1,2,···k0]. So dimΩ(M)=2k0.Proposition7.4.If dim H0∗=l0=2k0then M= N−1r=1Λ[b(−n)|n∈r2)-module.Proof:As in the proof of Proposition7.2,it is sufficient to show that dim Aτ(V(H,Z+ 1N−1 z1+m a(z)b=∞l=0 r2l a(−m−12)).So using the same calculation done in Proposition7.2,we conclude that Aτ(V)is spanned byh1(−1/2)s1···h k0(−1/2)s k0h∗1(−1/2)t1···h∗k(−1/2)t k0with s i,t i=0,1.Hence dim Aτ(V(H,Z+1N+Z,1≤r≤N−1,n>0] are irreducibleτ-twisted V(H,Z+1∂e(−n)if n>0andas multiplication by e(n)if n≤0.The proof of Proposition7.4gives Proposition7.5.If dim H0∗=2k0+1is odd,V(H,Z+1[DVVV]R.Dijkgraaf,C.Vafa,E.Verlinde and H.Verlinde,The operator algebra of orbifold models,Comm.Math.Phys.123(1989),485-526.[DHVW]L.Dixon,J.Harvey,C.Vafa and E.Witten,Strings on orbifolds,Nucl.Phys.B261(1985),651;II,Nucl.Phys.B274(1986),285.[D] C.Dong,Twisted modules for vertex algebras associated with even lattice,J.of Algebra165(1994),91-112.[DL1] C.Dong and J.Lepowsky,Generalized Vertex Algebras and Relative Vertex Operators,Progress in Math.,Vol.112,Birkh¨a user Boston,1993.[DL2] C.Dong and J.Lepowsky,The algebraic structure of relative twisted vertex operators,J.Pure and Applied Algebra110(1996),259-295.[DLM1] C.Dong,H.Li and G.Mason,Regularity of rational vertex operator algebras, Adv.Math.132(1997),148–166.[DLM2] C.Dong,H.Li and G.Mason,Twisted representations of vertex operator alge-bras,Math.Ann.310(1998),571–600.[FFR]Alex J.Feingold,Igor B.Frenkel and John F.X.Ries,Spinor Construction of Vertex Operator Algebras,Triality,and E(1)8,Contemporary Math.121,1991.[FHL]I.Frenkel,Y.Huang and J.Lepowsky,On axiomatic approaches to vertex oper-ator algebras and modules,Mem.Amer.Math.Soc.1041993.[FLM1]I.B.Frenkel,J.Lepowsky and A.Meurman,A natural representation of the Fischer-Griess Monster with the modular function J as character,Proc.Natl.A81(1984),3256-3260.[FLM2]I.B.Frenkel,J.Lepowsky and A.Meurman,Vertex operator calculus,in:Math-ematical Aspects of String Theory,Proc.1986Conference,San Diego.ed.byS.-T.Yau,World Scientific,Singapore,1987,150-188.[FLM3]I.B.Frenkel,J.Lepowsky and A.Meurman,Vertex Operator Algebras and the Monster,Pure and Applied Math.,Vol.134,Academic Press,1988.[FZ]I.Frenkel and Y.Zhu,Vertex operator algebras associated to representations of affine and Virasoro algebras,Duke Math.J.66(1992),123-168.[KW]V.Kac and W.Wang,Vertex operator superalgebras and representations,Con-tem.Math.,AMS Vol.175(1994),161-191.[Le1]J.Lepowsky,Calculus of twisted vertex operators,Proc.Natl.Acad A 82(1985),8295-8299.。

个人资料:姓 名: 徐帆 性 别：男出生年月：1972.06.26 民 族：汉籍 贯：广东湛江 政治面貌：群众婚姻状况：已婚电 话：136****8271通讯地址：清华大学数学科学系 邮编：100084E-mail:*******************研究领域：·表示理论：结合到三角范畴的Hall代数，量子群和包罗代数的实现，cluster 代数和cluster范畴学业经历:1989.9-1993.7 在陕西师范大学数学系攻读本科学位；1993.9-1996.7 在北京师范大学数学系攻读硕士学位；2004.9-2007.7 在清华大学数学系攻读博士学位，主要工作经历:1996.10-1999.7 在墨西哥国立自治大学访问学者；2000.1-2002.6 软件公司系统分析师2002.7-2004.7 在广东省东软信息技术职业学院担任数学教师2007.7-2009.12 清华大学数学系 博士后2009.12-至今 清华大学讲师国外学术访问：2008．07．05-2008．08．04，访问日本东京大学IPMU所。

2008．09．01-2008．11．30，德国波恩马普数学所（MPIM）.2009. 02. 01-2010. 01. 31,作为洪堡学者访问德国Bielefeld大学数学系。

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