Baxter Algebras and Hopf Algebras

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。

a rXiv:q -alg/95923v122Se p1995July 1993Published in Advances in Hopf Algebras,Marcel Dekker Lec.Notes Pure and Applied Maths 158(1994)55-105.ALGEBRAS AND HOPF ALGEBRAS IN BRAIDED CATEGORIES 1SHAHN MAJID 2Department of Applied Mathematics &Theoretical Physics University of Cambridge,Cambridge CB39EW,U.K.ABSTRACT This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories.Such objects generalise super-algebras and super-Hopf algebras,as well as colour-Lie algebras.Basic facts about braided categories C are recalled,the modules and comodules of Hopf algebras in such categories are studied,the notion of ‘braided-commutative’or ‘braided-cocommutative’Hopf algebras (braided groups)is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group Aut (C )is given.The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups).It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.One of the main motivations of the theory of Hopf algebras is that they provide a gener-alization of groups.Hopf algebras of functions on groups provide examples of commutativeHopf algebras,but it turns out that many group-theoretical constructions work just as well when the Hopf algebra is allowed to be non-commutative.This is the philosophy associated to some kind of non-commutative (or so-called quantum)algebraic geometry.In a Hopf algebra context one can say the same thing in a dual way:group algebras and enveloping algebras are cocommutative but many constructions are not tied to this.This point of view has been highly successful in recent years,especially in regard to the quasitriangular Hopf algebras of Drinfeld[11].These are non-cocommutative but the non-cocommutativity is controlled by a quasitriangular structure R .Such objects are commonly called quantum ing out of physics,notably associated to solutions of the Quantum Yang-Baxter Equations (QYBE)is a rich supply of quantum groups.Here we want to describe some kind of rival or variant of these quantum groups,which we call braided groups[44]–[52].These are motivated by an earlier revolution that was very popular some decades ago in mathematics and physics,namely the theory of super orZ2-graded algebras and Hopf algebras.Rather than make the algebras non-commutative etc one makes the notion of tensor product⊗non-commutative.The algebras remain commutative with respect to this new tensor product(they are super-commutative).Under this point of view one has super-groups,super-manifolds and super-differential geometry. In many ways this line of development was somewhat easier than the notion of quantum geometry because it is conceptually easier to make an entire shift of category from vector spaces to super-vector spaces.One can study Hopf algebras in such categories also(super-quantum groups).In this second line of development an obvious(and easy)step was to generalise such constructions to the case of symmetric tensor categories[27].These have a tensor product ⊗and a collection of isomorphismsΨgeneralizing the transposition or super-transposition map but retaining its general properties.In particular,one keepsΨ2=id so that these generalized transpositions still generate a representation of the symmetric group.Since only such general properties are used in most algebraic constructions,such as Hopf algebras and Lie algebras,these notions immediately(and obviously)generalise to this setting.See for example Gurevich[19],Pareigis[61],Scheunert[67]and numerous other authors.On the other hand,the theory is not fundamentally different from the super-case.Rather more interesting is the further generalization to relax the condition thatΨ2=id. NowΨandΨ−1must be distinguished and are more conveniently represented by braid-crossings rather than by permutations.They generate an action of the Artin braid group on tensor products.Such quasitensor or braided-tensor categories have been formally in-troduced into category theory in[24]and also arise in the representation theory of quantum groups.The study of algebras and Hopf algebras in such categories is rather more non-trivial than in the symmetric case.It is this theory that we wish to describe here.It has been introduced by the author under the heading‘braided groups’as mentioned above. Introduced were the relevant notions(not all of them obvious),the basic lemmas(such as a braided-tensor product analogous to the super-tensor product of super-algebras)and a construction leading to a rich supply of examples.On the mathematical level this project of‘braiding’all of mathematics is,I believe, a deep one(provided one goes from the symmetric to the truly braided case).Much of mathematics consists of manipulating symbols,making transpositions etc.The situation appears to be that in many constructions the role of permutation group can(with care) be played equally well by the braid group.Not only the algebras and braided groups to be described here,but also braided differential calculus,braided-binomial theorems and braided-exponentials are known[57]as well as braided-Lie algebras[58].Much more can be expected.Ultimately we would like some kind of braided geometry comparable to the high-level of development in the super case.Apart from this long-term philosophical motivation,one can ask what are the more immediate applications of this kind of braided geometry?I would like to mentionfive of them.1.Many algebras of interest in physics such as the degenerate Sklyanin algebra,quantumplanes and exchange algebras are not naturally quantum groups but turn out to be braided ones[52][56].There are braided-matrices B(R)and braided-vectors V(R′) associated to R-matrices.2.The category of Hopf algebras is not closed under quotients in a good sense.Forexample,if H⊂H1is covered by a Hopf algebra projection then H1∼=B>⊳H whereB is a braided-Hopf algebra.This is the right setting for Radford’s theorem as wehave discovered and explained in detail in[52].3.Braided groups are best handled by means of braid diagrams in which algebraicoperations‘flow’along strings.This means deep connections with knot theory and is also useful even for ordinary Hopf algebras.For example,you can dualise theorems geometrically by turning the diagram-proof up-side-down andflip conventions by viewing in a mirror.4.A useful tool in the theory of quasitriangular Hopf algebras(quantum groups)via aprocess of transmutation.By encoding their non-cocommutativity as braiding in a braided category they appear‘cocommutative’.Likewise,dual quasitriangular Hopf algebras are rendered‘commutative’by this process[45][49].5.In particular,properties of the quantum groups O q(G)and U q(g)are most easilyunderstood in terms of their braided versions B q(G)and BU q(g).This includes an Ad-invariant‘Lie algebra-like’subspace L⊂U q(g)and an isomorphism B q(G)∼=U q(g)[50][52].An outline of the paper is the following.In Section1we recall the basic notions of braided tensor categories and how to work in them,and some examples.We recall basic facts about quasitriangular and dual quasitriangular Hopf algebras and the braided cate-gories they generate.In Section2we do diagrammatic Hopf-algebra theory in this setting. In Section3we give a new diagrammatic proof of our generalised Tannaka-Krein-type re-construction theorem.In Section4we explain the results about ordinary quantum groups obtained from this braided theory.In Section5we end with basic examples of braided matrices etc associated to an R-matrix.Although subsequently of interest in physics,the braided matrices arose quite literally from the Tannaka-Krein theorem mentioned above. This is an example of pure mathematics feeding back into physics rather than the other way around(for a change).Our work on braided groups(or Hopf algebras in braided categories)was presented to the Hopf algebra community at the Euler Institute in Leningrad,October1990and at the Biannual Meeting of the American Maths Society in San Francisco,January1991and published in[48][49].The result presented at these meetings was the introduction of Hopf algebras living in the braided category of comodules of a dual quasitriangular Hopf algebra. The connection between crossed modules(also called Drinfeld-Yetter categories)and the quantum double as well as the connection with Radford’s theorem were introduced in[38] in early1990.The braided interpretation of Radford’s theorem was introduced in detail in[52]and circulated at the start of1992.Dual quasitriangular(or coquasitriangular) Hopf algebras themselves were developed in connection with Tannaka-Krein ideas in[36, Sec.4][48][49,Appendix](and earlier in other equivalent forms).A related Tannaka-Krein theorem in the quasi-associative dual quasitriangular setting was obtained in[43]at the Amherst conference and circulated infinal form in the Fall of1990.It is a pleasure to see that some of these ideas have subsequently proven of interest in Hopf algebra circles(directly or indirectly).I would also like to mention some constructions of Lyubashenko[30][31]relating to our joint work[32].Also in joint work with Gurevich[21] the transmutation construction is related to Drinfeld’s process of twisting[13].Several otherpapers can be mentioned here.On the whole I have resisted the temptation to give a full survey of all results obtained so far.Instead,the aim here is a more pedagogical exposition of the more elementary results,with proofs.Throughout this paper we assume familiarity with usual techniques of Hopf algebras such as in the book of Sweedler[69].In this sense the style(and also the motivation)is somewhat different from our braided-groups review article for physicists[42].We work over afield k.With more care one can work here with a ring just as well.When working with matrix or tensor components we will use the convention of summing over repeated indices.Some of the elementary quantum groups material should appear in more detail in my forthcoming book.1Braided CategoriesHere we develop the braided categories within which we intend to work,namely those coming from(co)modules of quantum groups.In fact,the theory in Sections2,3is not tied to quantum groups and works in any braided category.The material in the present section is perfectly standard by now.1.1Definition and General ConstructionsSymmetric monoidal(=tensor)categories have been known for some time and we refer to [27]for details.The model is the category of k-modules.The notion of braided monoidal (=braided tensor=quasitensor)category is a small generalization if this.Briefly,a monoidal category means(C,⊗,ΦV,W,Z,1for the tensor product and associated functorial isomorphisms l V:V→1for all objects V,which we likewise suppress.A monoidal category C is rigid(=has left duals)if for each object V,there is an object V∗and morphisms ev V:V∗⊗V→1→V⊗V∗such that→(V⊗V∗)⊗V∼=V⊗(V∗⊗V)ev→V(1)V coev→V∗⊗(V⊗V∗)∼=(V∗⊗V)⊗V∗ev→V∗(2)V∗coevcompose to id V and id V∗respectively.A single object has a left dual if V∗,ev V,coev V exist. The model is that of afinite-dimensional vector space(orfinitely generated projective module when k is a ring).Finally,the monoidal category is braided if it has a quasisymmetry or‘braiding’Ψgiven as a natural transformation between the two functors⊗and⊗op(with opposite product) from C×C→C.This is a collection of functorial isomorphismsΨV,W:V⊗W→W⊗V obeying two‘hexagon’coherence identities.In our suppressed notation these areΨV⊗W,Z=ΨV,Z◦ΨW,Z,ΨV,W⊗Z=ΨV,Z◦ΨV,W(3)while identities such asΨV,1,V (4)can be deduced.If Ψ2=id then one of the hexagons is superfluous and we have an ordinary symmetric monoidal category.Let us recall that the functoriality of maps such as those above means that they commute in a certain sense with morphisms in the category.For example,functoriality of ΨmeansΨZ,W (φ⊗id)=(id ⊗φ)ΨV,W ∀φV ↓Z ,ΨV,Z (id ⊗φ)=(φ⊗id)ΨV,W ∀φW↓Z .(5)These conditions(3)-(5)are just the obvious properties that we take for granted when transposing ordinary vector spaces or super-vector spaces.In these cases Ψis the twist map ΨV,W (v ⊗w )=w ⊗v or the supertwistΨV,W (v ⊗w )=(−1)|v ||w |w ⊗v (6)on homogeneous elements of degree |v |,|w |.The form of Ψin these familiar cases does not depend directly on the spaces V,W so we often forget this.But in principle there is a different map ΨV,W for each V,W and they all connect together as explained.In particular,note that for any two V,W we have two morphisms ΨV,W ,Ψ−1W,V :V ⊗W →W ⊗V and in the truly braided case these can be distinct.A convenient notation in this case is to write them not as permutations but as braid crossings.Thus we write morphisms pointing downwards (say)and instead of a usual arrow,we use the shorthandV WW V ΨW,V Ψ-1==V W W V V,W .(7)In this notation the hexagons (3)appear as==Z V W Z V W (8)The doubled lines refer to the composite objects V ⊗W and W ⊗Z in a convenient ex-tension of the notation.The coherence theorem for braided categories can be stated very simply in this notation:if two series of morphisms built from Ψ,Φcorrespond to the same braid then they compose to the same morphism.The proof is just the same as Mac Lane’s proof in the symmetric case with the action of the symmetric group replaced by that of the Artin braid group.This notation is a powerful one.We can augment it further by writing any other morphisms as nodes on a string connecting the inputs down to the outputs.Functoriality(5)then says that a morphism φ:V →Z say can be pulled through braid crossings,V W V W=V W V W =φφφφ(9)Similarly for Ψ−1with inverse braid crossings.An easy lemma using this notation is that for any braided category C there is another mirror-reversed braided monoidal category ¯C with the same monoidal structure but with braiding¯ΨV,W =Ψ−1W,V (10)in place of ΨV,W ,i.e with the interpretation of braid crossings and inverse braid crossings interchanged.Finally,because of (4)we can suppress the unit object entirely so the evaluation and co-evaluation appear simply as ev =∪and coev =∩.Then (1)-(2)appear as=V ev = coev =V V* VV V*=V V V V V* V*.(11)There is a similar notion of right duals V ˇand ¯ev V ,¯coev V for which the mirror-reflected double-bend here can be likewise straightened.Example 1.1Let R ∈M n (k )⊗M n (k )be invertible and obey the QYBER 12R 13R 23=R 23R 13R 12then the monoidal category C (V,R )generated by tensor products of V =C n is braided.Proof This is an elementary exercise (and extremely well-known).The notation is R 12=R ⊗id and R 23=id ⊗R in M ⊗3n .The braiding on basis vectors {e i }isΨ(e i ⊗e j )=e b ⊗e a R a i b j (12)extended to tensor products according to (3).The morphisms in the category are linear maps such that Ψis functorial with respect to them in the sense of (5).The associativity Φis the usual one on vector spaces.⊔⊓If R obeys further conditions then C (V,V ∗,R )generated by V,V ∗is rigid.One says that such an R is dualizable .For this there should exist among other things a ‘second-inverse’R =((R t 2)−1)t 2(13)where t 2is transposition in the second M n factor.This defines one of the mixed terms in the braidingΨV ∗,V ∗(f i ⊗f j )=R i a j b f b ⊗f a(14)ΨV,V∗(e i⊗f j)= R a i j b f b⊗e a(15)ΨV∗,V(f i⊗e j)=e a⊗f b R−1i b a j(16) where V∗={f i}is a dual basis.The evaluation and coevaluation are given by the usual morphismsev V(f i⊗e j)=δi j,coev V(1)= i e i⊗f i.(17)One needs also the second-inverse R−1forΨto be invertible.In this way one translates the various axioms into a linear space setting.We see in particular that the QYBE are nothing other than the braid relations in matrix form.We turn now to some general categorical constructions.One construction in[37][40] is based on the idea that a pair of monoidal categories C→V connected by a functor behaves in many ways like a bialgebra with⊗in C something like the product.In some cases this is actually true as we shall see in Section3in the form of a Tannaka-Krein-type reconstruction theorem,but we can keep it in general as motivation.Motivated by this we showed that for every pair C→V of monoidal categories there is a dual one C◦→V where C◦is the Pontryagin dual monoidal category[37].This generalised the usual duality for Abelian groups and bialgebras to the setting of monoidal categories.We also proved such things as a canonical functorC→◦(C◦).(18) Of special interest to us now is the case C→C where the functor is the identity one.So associated to every monoidal category C is another monoidal category C◦of‘representations’of⊗.This special case can also be denoted by C◦=Z(C)the‘center’or‘inner double’of C for reasons that we shall explain shortly.This case was found independently by V.G. Drinfeld who pointed out that it is braided.Proposition1.2[37][10]Let C be a monoidal category.There is a braided monoidal category C◦=Z(C)defined as follows.Objects are pairs(V,λV)where V is an object of C andλV is a natural isomorphism in Nat(V⊗id,id⊗V)such thatλV,1Proof The monoidal structure was found in the author’s paper[37]where full proofs were also given.We refer to this for details.Its preprint was circulated in the Fall of1989. The braiding was pointed out by Drinfeld[10]who had considered the construction from a very different and independent point of view to our duality one,namely in connection with the double of a Hopf algebra as we shall explain below.Another claim to the construction is from the direction of tortile categories[25].See also[40]for further work from the duality point of view.⊔⊓The‘double’point of view for this construction is based on the following example cf[10]. Example1.3Let H be a bialgebra over k and C=H M the monoidal category of H-modules.Then an object of Z(C)is a vector space V which is both a left H-module and an invertible left H-comodule such thath(1)v¯(1)⊗h(2)⊲v¯(2)= (h(1)⊲v)¯(1)h(2)⊗(h(1)⊲v)¯(2),∀h∈H,v∈V.In this form Z(H M)coincides with the category H H M of H-crossed modules[72]with an additional invertibility condition.The braiding isΨV,W(v⊗w)= v¯(1)⊲w⊗v¯(2).The invertibility condition on the comodules ensures thatΨ−1exists,and is automatic if the bialgebra H has a skew-antipode.Proof The proof is standard from the point of view of Tannaka-Krein reconstruction methods(which we shall come to later).From C we can reconstruct H as the representing object for a certain functor.This establishes a bijection Lin(V,H⊗V)∼=Nat(V⊗id,id⊗V) under whichλV corresponds to a map V→H⊗V.ThatλV represents⊗corresponds then to the comodule property of this map.ThatλV is a collection of morphisms corresponds to the stated compatibility condition between this coaction and the action on V as an object in C.To see this in detail let H L denote H as an object in C under the left action.Given λV a natural transformation we definev¯(1)⊗v¯(2)=λV,H L(v⊗1)(19) and check(id⊗λV,HL )(λV,HL⊗id)(v⊗1⊗1)=λV,HL⊗H L(v⊗(1⊗1))=λV,HL⊗H L(v⊗∆(1))=(∆⊗id)◦λV,HL(v⊗1)where thefirst equality is the fact thatλV‘represents’⊗and the last is thatλV is functorialunder the morphism∆:H L→H L⊗H L.The left hand side is the map V→H⊗V in (19)applied twice so we see that this map is a left coaction.Moreover,h(1)v¯(1)⊗h(2)⊲v¯(2)=h⊲λV,H L(v⊗1)=λV,H L(h⊲(v⊗1))= λV,H L(h(1)⊲v⊗R h(2)(1))= (λV,H L(h(1)⊲v⊗1))(h(2)⊗1) where thefirst equality is the definition(19)and the action of H on H L⊗V.The secondequality is thatλV,HLis a morphism in C.Thefinal equality uses functoriality underthe morphism R h(2):H L→H L given by right-multiplication to obtain the right handside of the compatibility condition.The converse directions are easier.Given a coaction V→H⊗V defineλV,W(v⊗w)= v¯(1)⊲w⊗v¯(2).This also implies at once the braiding Ψ=λas stated.Finally we note that in Proposition1.2the definition assumes that theλV are invertible. If we were to relax this then we would have a monoidal category which is just that of crossed modules as in[72],but thenΨwould not necessarily be invertible and hence would not be a true braiding.The invertibleλV correspond to left comodules which are invertible in the following sense:there exists a linear map V→V⊗H sending v to v[2]⊗v[1]say,suchthatv[2]¯(1)v[1]⊗v[2]¯(2)=1⊗v= v¯(2)[1]v¯(1)⊗v¯(2)[2],∀v∈V.(20) One can see that if such an‘inverse’exists,it is unique and a right comodule.Moreover, it is easy to see that the invertible comodules are closed under tensor products.They correspond toλ−1V in a similar way to(19)and withλ−1V,W(w⊗v)= v[2]⊗v[1]⊲w for the converse direction.In thefinite-dimensional case they provide left duals V∗with left coactionβV∗(f)(v)= v[1]f(v[2]).If the bialgebra H has a skew-antipode then every left comodule is invertible by composing with the skew-antipode.So in this case the condition becomes empty.From the categorical point of view in Proposition1.2,if C has right duals then everyλV,W is invertible,cf[37].The inverse is the right-adjoint ofλV,Wˇ,namelyλ−1V,W=(¯ev W⊗id)◦λV,Wˇ◦(id⊗¯coev W).When C=H M then thefinite-dimensional left modules have right duals if the bialgebra H has a skew-antipode,so in this case the invertibility ofλV is automatic.On the other hand,we do not need to make these suppositions here.This completes our computation of Z(H M).Apart from the invertibility restriction we see that it consists of compatible module-comodule structures as stated.⊔⊓Note that the notion of a crossed module is an immediate generalisation of the notion of a crossed G-module[71]with H=kG,the group algebra of afinite group G.In this case the category of crossed G-modules is well-known to be braided[18].Moreover,because the objects can be identified with underlying vector spaces,we know by the Tannaka-Krein reconstruction theorem[66]that there must exist a bialgebra coD(H)such that our braided-category is equivalent to that of right coD(H)-comodulesM coD(H)f.d.=H H M f.d.(21) Here we take the modules to befinite-dimensional as a sufficient(but not necessary)con-dition for a Tannaka-Krein reconstruction theorem to apply and the co-double coD(H)to exist.In the nicest case the category is also D(H)M f.d.for some D(H).This is an abstract definition of Drinfeld’s quantum double and works for a bialgebra.If it happens that H is a Hopf algebra with invertible antipode then one can see from the above that H H M f.d.is rigid and so coD(H)and D(H)will be Hopf algebras.The categorical reason is that H M f.d.is rigid and this duality extends to Z(C)with the dual ofλV defined by the left-adjoint ofλ−1V,W,namelyλV∗,W=(ev V⊗id)◦λ−1V,W◦(id⊗coev V). We will study details about categories of modules and comodules and the reconstruction theorems later in this section and in Section3.The point is that these categorical methods are very powerful.Proposition1.4[33]cf[11]If H is afinite-dimensional Hopf algebra then D(H)(the quan-tum double Hopf algebra of H)is built on H∗⊗H as a coalgebra with the product(a⊗h)(b⊗g)= b(2)a⊗h(2)g<Sh(1),b(1)><h(3),b(3)>,h,g∈H,a,b∈H∗where<,>denotes evaluation.Proof The quantum double D(H)was introduced by Drinfeld[11]as a system of genera-tors and relations built from the structure constants of H.The formula stated on H∗⊗H is easily obtained from this as done in[33].We have used here the conventions introduced in[38]that avoid the use of the inverse of the antipode.Also in[38]we showed that the modules of the double were precisely the crossed modules category as required.To see this simply note that H and H∗op are sub-Hopf algebras and hence a left D(H)-module is a left H-module and a suitably-compatible right H∗-module.The latter is equally well a left H-comodule compatible as in Example1.3.See[38]for details.⊔⊓In[33]we introduced a further characterization of the quantum double as a member of a class of double cross product Hopf algebras H1⊲⊳H2(in which H i are mutually acting on each other).Thus,D(H)=H∗op⊲⊳H where the actions are mutual coadjoint actions. In this form it is clear that the role of H∗can be played by H◦in the infinite dimensional Hopf algebra case.We will not need this further here.1.2Quasitriangular Hopf AlgebrasWe have already described one source of braided categories,namely as modules of the double D(H)(or comodules of the codouble)of a bialgebra.Abstracting from this one has the notion,due to Drinfeld,of a quasitriangular Hopf algebra.These are such that their category of modules is braided.Definition1.5[11]A quasitriangular bialgebra or Hopf algebra is a pair(H,R)where H is a bialgebra or Hopf algebra and R∈H⊗H is invertible and obeys(∆⊗id)R=R13R23,(id⊗∆)R=R13R12.(22)τ◦∆h=R(∆h)R−1,∀h∈H.(23) Here R12=R⊗1and R23=1⊗R etc,andτis the usual twist map.Thus these Hopf algebras are like cocommutative enveloping algebras or group algebras but are cocommutative now only up to an isomorphism implemented by conjugation by an element R.Some elementary(but important)properties areLemma1.6[12]If(H,R)is a quasitriangular bialgebra then R as an element of H⊗H obeys(ǫ⊗id)R=(id⊗ǫ)R=1.(24)R12R13R23=R23R13R12(25) If H is a Hopf algebra then one also has(S⊗id)R=R−1,(id⊗S)R−1=R,(S⊗S)R=R(26)∃S−1,u,v,S2(h)=uhu−1,S−2(h)=vhv−1∀h∈H(27)Proof For(24)applyǫto(22),thus(ǫ⊗id⊗id)(∆⊗id)R=R23=(ǫ⊗id⊗id)R13R23so that(since R23is invertible)we have(ǫ⊗id)R=1.Similarly for the other side.For(25)compute(id⊗τ◦∆)R in two ways:using the second of axioms(22)directly or usingaxiom(23),and then the second of(22).For(26)consider R(1)(1)S R(1)(2)⊗R(2)=1by the property of the antipode and equation(24)already proven,but equals R(S⊗id)R by axiom(22).Similarly for the other side,hence(S⊗id)R=R−1.Similarly for(id⊗S)R−1=Ronce we appreciate that(∆⊗id)(R−1)=(R13R23)−1=R−123R−113etc,since∆is an algebrahomomorphism.For(27)the relevant expressions areu= (S R(2))R(1),u−1= R(2)S2R(1),v=Suwhich one can verify to have the right properties.In addition one can see that∆u=(R21R12)−1(u⊗u)and similarly for v so that uv−1is group-like(and implements S4).Fordetails of the computations see[12]or reviews by the author.⊔⊓Here(25)is the reason that Physicists call R the‘universal R-matrix’(compare Ex-ample1.1).Indeed,in anyfinite-dimensional representation the image of R is such anR-matrix.There are well-known examples such as U q(sl2)and U q(g)[11][23].Here we giveperhaps the simplest known quasitriangular Hopf algebrasExample1.7[47]Let Z n=Z/n Z be thefinite cyclic group of order n and k Z n its groupalgebra with generator g.Let q be a primitive n-th root of unity.Then there is a quasitri-angular Hopf algebra Z′n consisting of this group algebra and∆g=g⊗g,ǫg=1,Sg=g−1,R=n−1n−1a,b=0q−ab g a⊗g b.(28)Proof We assume that k is of suitable characteristic.To verify the non-trivial quasitrian-gular structure we use that n−1 n−1b=0q ab=δa,0.Then R13R23=n−2 q−(ab+cd)g a⊗g c⊗g b+d =n−2 q−b(a−c)q−cb′g a⊗g c⊗g b′=n−1 q−ab′g a⊗g a⊗g b′=(∆⊗id)R where b′=b+d was a change of variables.Similarly for the second of(22).The remaining axiom(23)is automatic because the Hopf algebra is both commutative and cocommutative.⊔⊓Example1.8[2][54]Let G be afinite Abelian group and k(G)its function Hopf algebra. Then a quasitriangular structure on k(G)means a function R∈H⊗H obeying R(gh,f)=R(g,f)R(h,f),R(g,hf)=R(g,h)R(g,f),R(g,e)=1=R(e,g)for all g,h,f in G and e the identity element.I.e.,a quasitriangular structure on k(G) means precisely a bicharacter of G.Proof We identify k(G)⊗k(G)with functions on G×G,with pointwise multiplication. Using the comultiplication given by multiplication in G we have at once that(22)corre-sponds to thefirst two displayed equations.Axiom(23)becomes hg R(g,h)=R(g,h)gh and so is automatic because the group is Abelian.Given thesefirst two of the stated conditions,the latter two hold iffR is invertible.⊔⊓The Z′n example here also has an immediate generalization to the group algebra kG of afinite Abelian group equipped with a bicharacter onˆG.This just coincides with the last example applied to k(ˆG)=kG.Finally,we return to our basic construction,。

"Algebra" 这个词的词根源自阿拉伯数学传统。

它的起源可以追溯到阿拉伯数学家、科学家和学者穆罕默德·本·穆萨·昂瓦尔·赞杜（Muhammad ibn Musa al-Khwarizmi）的名字，他生活在9世纪。

赞杜的一本书"Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala"（关于恢复和平方根的摘要书）对代数学的发展起到了重要作用。

因此，"algebra" 这个词的词根"al-" 实际上是来源于穆罕默德·本·穆萨·昂瓦尔·赞杜的名字"Al-Khwarizmi"。

这个词根表明"algebra" 最初是指阿拉伯数学家所发展的代数学体系。

代数学是数学的一个分支，涉及数字、符号、变量和数学运算，用于解决未知数的方程和问题。

"Algebra" 一词在数学领域广泛使用，用于描述代数学的研究领域和相关的数学概念。

此外，它还被广泛用于描述和命名与代数学有关的数学课程和教材。

a rX iv:mat h /3628v1[mat h.QA ]12J un23BASIC CONCEPTS OF TERNARY HOPF ALGEBRAS ANDRZEJ BOROWIEC,WIESŁAW A.DUDEK,AND STEVEN DUPLIJ A BSTRACT .The theory of ternary semigroups,groups and algebras is reformulated in the abstract arrow language.Then using the reversing arrow ansatz we define ternary comultiplication,bialgebras and Hopf algebras and investigate their properties.The main property ”to be binary derived”is considered in detail.The co-analog of Post theorem is formulated.It is shown that there exist 3types of ternary coassociativity,3types of ternary counits and 2types of ternary antipodes.Some examples are also presented.Ternary and n -ary generalizations of algebraic structures is the most natural way for further development and deeper understanding of their fundamental properties.Firstly ternary algebraic operations were introduced already in the XIX-th century by A.Cayley.As the development of Cayley’s ideas it were considered n -ary generalization of matrices and their determinants [29,15]and general theory of n -ary algebras [19,4]and ternary rings [20](for physical applications in Nambu mechanics,supersymmetry,Yang-Baxter equation,etc.see [18,31]as surveys).The notion of an n -ary group was introduced in 1928by W.D¨o rnte [7](inspired by E.N¨o ther)which is a natural generalization of the notion of a group and a ternary group considered by Certaine [5]and Kasner [16].For many applications of n -ary groups and quasigroups see [25,32]and [3]respectively.From another side,Hopf algebras [1,30]and their generalizations [23,22,13,21]play a basic role in the quantum group theory (see e.g.[6,17,27]).In the first part of this paper we reformulate necessary material on ternary semigroups,groups and algebras [3,25]in the abstract arrow language.Then according to the general scheme [1]using systematic reversing order of arrows,we define ternary bialgebras and Hopf algebras,investigate their properties and present examples.T ERNARY SEMIGROUPS A non-empty set G with one ternary operation []:G ×G ×G →G is called a ternary groupoid and is denoted by (G,[])or G,m (3) .We will present some results using second notation,because it allows to reverse arrows in the most clear way.In proofswe will mostly use the first notation due to convenience and for short.If on G there is a binary operation ⊙(or m (2))such that [xyz ]=(x ⊙y )⊙zor(1)m (3)=m (3)der =m (2)◦ m (2)×id for all x,y,z ∈G ,then we say that []or m (3)der is derived from ⊙or m (2)and denote thisfact by (G,[])=der (G,⊙).If[xyz ]=((x ⊙y )⊙z )⊙b2ANDRZEJ BOROWIEC,WIESŁAW A.DUDEK,AND STEVEN DUPLIJholds for all x,y,z ∈G and some fixed b ∈G ,then a groupoid (G,[]is b -derived from (G,⊙).In this case we write (G,[])=der b (G,⊙)(cf.[9,10]).We say that (G,[]is a ternary semigroup if the operation []is associative ,i.e.if(2)[[xyz ]uv ]=[x [yzu ]v ]=[xy [zuv ]]holds for all x,y,z,u,v ∈G ,or (3)m (3)◦ m (3)×id ×id =m (3)◦ id ×m (3)×id =m (3)◦ id ×id ×m(3) Obviously,a ternary operation m (3)der derived from a binary associative operation m (2)is also associative in the above sense,but a ternary groupoid (G,[])b -derived (b is a cansellative element)from a semigroup (G,⊙)is a ternary semigroup if and only if b lies in the center of (G,⊙).Fixing in a ternary operation m (3)one element a we obtain a binary operation m (2)a .A binary groupoid (G,⊙)or G,m (2)a ,where x ⊙y =[xay ]or(4)m (2)a =m (3)◦(id ×a ×id)for some fixed a ∈G is called a retract of (G,[])and is denoted by ret a (G,[]).In some special cases described in [9,10]we have (G,⊙)=ret a (der b (G,⊙))or (G,⊙)=der c (ret d (G,[])).Lemma 1.If in the ternary semigroup (G,[])or G,m (3) there exists an element e suchthat for all y ∈G we have [eye ]=y ,then this semigroup is derived from the binary semigroup G,m (2)e,where (5)m (2)e =m(3)◦(id ×e ×id)In this case (G,[])=der (ret e (G,[]).Proof.Indeed,if we put x ⊛y =[xey ],then (x ⊛y )⊛z =[[xey ]ez ]=[x [eye ]z ]=[xyz ]and x ⊛(y ⊛z )=[xe [yez ]]=[x [eye ]z ]=[xyz ],which completes the proof.The same ternary semigroup G,m (3) can be derived from two different semigroups(G,⊛)or G,m (2)e and (G,⋄)or G,m (2)a .Indeed,if in G there exists a =e such that[aya ]=y for all y ∈G ,then by the same argumentation we obtain [xyz ]=x ⋄y ⋄z for x ⋄y =[xay ].In this case for ϕ(x )=x ⋄e =[xae ]we havex ⊛y =[xey ]=[x [aea ]y ]=[[xae ]ay ]=(x ⋄e )⋄y =ϕ(x )⋄yandϕ(x ⊛y )=[[xey ]ae ]=[[x [aea ]y ]ae ]=[[xae ]a [yae ]]=ϕ(x )⋄ϕ(y ).Thus ϕis a binary homomorphism such that ϕ(e )=a .Moreover for ψ(x )=[eax ]we haveψ(ϕ(x ))=[ea [xae ]]=[e [axa ]e ]=x,ϕ(ψ(x ))=[[eax ]ae ]=[e [axa ]e ]=xandψ(x ⋄y )=[ea [xay ]]=[ea [x [eae ]y ]]=[[eax ]e [aey ]]=ψ(x )⊛ψ(y ).Hence semigroups (G,⊛)and (G,⋄)are isomorphic.BASIC CONCEPTS OF TERNARY HOPF ALGEBRAS3 Definition2.An element e∈G is called a middle identity or a middle neutral element of (G,[])if for all x∈G we have[exe]=x or(6)m(3)◦(e×id×e)=id.An element e∈G satisfying the identity[eex]=x or(7)m(3)◦(e×e×id)=id.is called a left identity or a left neutral element of(G,[]).By the symmetry we define a right identity.An element which is a left,middle and right identity is called a ternary identity(briefly:identity).There are ternary semigroups without left(middle,right)neutral elements,but there are also ternary semigroups in which all elements are identities[25,24].Example.In ternary semigroups derived from the symmetric group S3all elements of order 2are left and right(but no middle)identities.Example.In ternary semigroup derived from Boolean group all elements are ternary identi-ties,but ternary semigroup1-derived from the additive group Z4has no left(right,middle) identities.Lemma3.For any ternary semigroup(G,[])with a left(right)identity there exists a binary semigroup(G,⊙)and its endomorphismµsuch that[xyz]=x⊙µ(y)⊙zfor all x,y,z∈G.Proof.Let e be a left identity of(G,[]).It is not difficult to see that the operation x⊙y= [xey]is associative.Moreover,forµ(x)=[exe],we haveµ(x)⊙µ(y)=[[exe]e[eye]]=[[exe][eey]e]=[e[xey]e]=µ(x⊙y)and[xyz]=[x[eey][eez]]=[[xe[eye]]ez]=x⊙µ(y)⊙z.The case of right identity the proof is analogous. Definition4.We say that a ternary groupoid(G,[])is:a left cancellative if[abx]=[aby]=⇒x=y,a middle cancellative if[axb]=[ayb]=⇒x=y,a right cancellative if[xab]=[yab]=⇒x=yholds for all a,b∈G.A ternary groupoid which is left,middle and right cancellative is called cancellative. Theorem5.A ternary groupoid is cancellative if and only if it is a middle cancellative,or equivalently,if and only if it is a left and right cancellative.Proof.Assume that a ternary semigroup(G,[])is a middle cancellative and[xab]=[yab]. Then[ab[xab]]=[ab[yab]]and in the consequence[a[bxa]b]=[a[bya]b]which implies x=y.Conversely if(G,[])is a left and right cancellative and[axb]=[ayb]then[a[axb]b]= [a[ayb]b]and[[aax]bb]=[[aay]bb]which gives x=y.The above theorem is a consequence of the general result proved in[12].4ANDRZEJ BOROWIEC,WIESŁAW A.DUDEK,AND STEVEN DUPLIJDefinition6.A ternary groupoid(G,[])is semicommutative if[xyz]=[zyx]for all x,y,z∈G.If the value of[xyz]is independent on the permutation of elements x,y,z, viz.(8)[x1x2x3]= xσ(1)xσ(2)xσ(3)or m(3)=m(3)◦σ,then(G,[])is a commutative ternary groupoid.Ifσisfixed,then a ternary groupoid satisfying(8)is calledσ-commutative.The group S3is generated by two transpositions;(12)and(23).This means that(G,[]) is commutative if and only if[xyz]=[yxz]=[xzy]holds for all x,y,z∈G.As a simple consequence of Theorem5from[11]we obtainCorollary7.If in a ternary semigroup(G,[])satisfying the identity[xyz]=[yxz]there are a,b such that[axb]=x for all x∈G,then(G,[])is commutative.Proof.According to the above remark it is sufficient to prove that[xyz]=[xzy].We have [xyz]=[a[xyz]b]=[ax[yzb]]=[ax[zyb]]=[a[xzy]b]=[xzy].Mediality in the binary case(x⊙y)⊙(z⊙u)=(x⊙z)⊙(y⊙u)can be presentedas a matrix⇓⇓⇒x y⇒z uand for groups coincides with commutativity.Definition8.A ternary groupoid(G,[])is medial if it satisfies the identity [[x11x12x13][x21x22x23][x31x32x33]]=[[x11x21x31][x12x22x32][x13x23x33]]or(9)m(3)◦ m(3)×m(3)×m(3) =m(3)◦ m(3)×m(3)×m(3) ◦σmedial, whereσmedial= 123456789147258369 ∈S9.It is not difficult to see that a semicommutative ternary semigroup is medial.An element x such that[xxx]=x is called an idempotent.A groupoid in which all elements are idempotents is called an idempotent groupoid.A left(right,middle)identity is an idempotent.T ERNARY GROUPS AND ALGEBRASDefinition9.A ternary semigroup(G,[])is a ternary group if for all a,b,c∈G there are x,y,z∈G such that[xab]=[ayb]=[abz]=c.One can prove[24]that elements x,y,z are uniquely determined.Moreover,according to the suggestion of[24]one can prove(cf.[8])that in the above definition,under the assumption of the associativity,it suffices only to postulate the existence of a solution of [ayb]=c,or equivalently,of[xab]=[abz]=c.In a ternary group the equation[xxz]=x has a unique solution which is denoted by z=·)◦D(3)=id,where D(3)(x)=(x,x,x)is a ternary diagonal map.As a consequence of results obtained in[7]we haveBASIC CONCEPTS OF TERNARY HOPF ALGEBRAS5 Theorem10.In any ternary group(G,[])for all x,y,z∈G the following relations take place[xx x x]=[x]=[y x y]=[[xyz]=[yx=x for all x,an idempotent ternary group is semicommutative.From results obtained in[8](see also[11])for n=3we obtain Theorem11.A ternary semigroup(G,[])with a unary operation−:x→x]=[x·×id)◦ D(2)×id =Pr2,m(3)◦(id×id×x]=yor[a is its identity.x−1(in(G,⊛)is[x axa]is an automorphism of(G,⊛).The easy calculation proves that the above formula holds for b=[a6ANDRZEJ BOROWIEC,WIESŁAW A.DUDEK,AND STEVEN DUPLIJ One can prove that the group(G,⊛)is unique up to isomorphism.From the proof of Theorem3in[14]it follows that any medial ternary group satisfies the identityx z],which together with our previous results shows that in such groups we have[xyz].But x=x.Hence,any medial ternary group is semicommutative.Thus any retract of such group is a commutative group.Moreover,forϕfrom the proof of the above theorem we haveϕ(ϕ(x))=[axa]a]=[a a]]=xCorollary15.Any medial ternary group(G,[])has the form[xyz]=x⊙ϕ(y)⊙z⊙b,where(G,⊙)is a commutative group,ϕits automorphism such thatϕ2=id and b∈G is fixed.Corollary16.A ternary group is medial if and only if it is semicommutative. Corollary17.A ternary group is semicommutative(medial)if and only if[xay]=[yax] holds for all x,y∈G and somefixed a∈G.Corollary18.A commutative ternary group is b-derived from some commutative group.Indeed,ϕ(x)=[a]=x.Theorem19(Post).For any ternary group(G,[])there exists a binary group(G∗,⊛) and H⊳G∗,such that G∗ H≃Z2and[xyz]=x⊛y⊛zfor all x,y,z∈G.Proof.Let c be afixed element in G and let G∗=G×Z2.In G∗we define binary operation⊛putting(x,0)⊛(y,0)=([xyc,1)is its neutral element. The inverse element(in G∗)has the form:(x,0)−1=(c c],1)Thus G∗is a group such that H={(x,1):x∈G}⊳G∗.Obviously the set G can be identified with G×{0}and[xyz]=((x,0)⊛(y,0))⊛(z,0)=([xyc]cz],0)=([xy[BASIC CONCEPTS OF TERNARY HOPF ALGEBRAS7Definition21.Autodistributivity in a ternary group is[[xyz]ab]=[[xab][yab][zab]].Let us consider ternary algebras.Take2ternary operations{,,}and[,,],then distributivity is{[xyz]ab}=[{xab}{yab}{zab}],and additivity is[{x+z}ab]=[xab]+[zab].Definition22.Ternary algebra is a pair A,m(3) ,where A is a linear space and m(3)is a linear mapm(3):A⊗A⊗A→Acalled ternary multiplication which is associativem(3)◦ m(3)⊗id⊗id =m(3)◦ id⊗m(3)⊗id =m(3)◦ id⊗id⊗m(3) .T ERNARY COALGEBRASLet C is a linear space over afield K.Definition23.Ternary comultiplication∆(3)is a linear map over afixedfield K∆(3):C→C⊗C⊗C.For convenience we also use the short-cut Sweedler-type notations[30](11)∆(3)(a)=ni=1a′i⊗a′′i⊗a′′′i=a(1)⊗a(2)⊗a(3).Now we discuss various properties of∆(3)which are in sense(dual)analog of the above ternary multiplication m(3).First consider different possible types of ternary coassociativity.(1)Standard ternary coassociativity(12)(∆(3)⊗id⊗id)◦∆(3)=(id⊗∆(3)⊗id)◦∆(3)=(id⊗id⊗∆(3))◦∆(3),In the Sweedler notationsa(1) (1)⊗ a(1) (2)⊗ a(1) (3)⊗a(2)⊗a(3)=a(1)⊗ a(2) (1)⊗ a(2) (2)⊗ a(2) (3)⊗a(3) =a(1)⊗a(2)⊗ a(3) (1)⊗ a(3) (2)⊗ a(3) (3)≡a(1)⊗a(2)⊗⊗a(3)⊗a(4)⊗a(5).(2)Nonstandard ternaryΣ-coassociativity(Gluskin-type—positional operatives)(∆(3)⊗id⊗id)◦∆(3)=(id⊗ σ◦∆(3) ⊗id)◦∆(3),whereσ◦∆(3)(a)=∆(3)σ(a)=a(σ(1))⊗a(σ(2))⊗a(σ(3))andσ∈Σ⊂S3.(3)Permutational ternary coassociativity(∆(3)⊗id⊗id)◦∆(3)=π◦(id⊗∆(3)⊗id)◦∆(3),whereπ∈Π⊂S5.8ANDRZEJ BOROWIEC,WIESŁAW A.DUDEK,AND STEVEN DUPLIJ Ternary comediality is∆(3)⊗∆(3)⊗∆(3) ◦∆(3)=σmedial◦ ∆(3)⊗∆(3)⊗∆(3) ◦∆(3),whereσmedial is defined in(9).Ternary counit is defined as a mapε(3):C→K.In general,ε(3)=ε(2)satisfying one of the conditions below.If∆(3)is derived,then maybeε(3)=ε(2),but another counits may exist.Example.Define[xyz]=(x+y+z)|mod2for x,y,z∈Z2.It is seen that here there are 2ternary counitsε(3)=0,1.There are3types of ternary counits:(1)Standard(strong)ternary counit(13)(ε(3)⊗ε(3)⊗id)◦∆(3)=(ε(3)⊗id⊗ε(3))◦∆(3)=(id⊗ε(3)⊗ε(3))◦∆(3)=id,(2)Two sequensional(polyadic)counitsε(3)1andε(3)2(14)(ε(3)1⊗ε(3)2⊗id)◦∆=(ε(3)1⊗id⊗ε(3)2)◦∆=(id⊗ε(3)1⊗ε(3)2)◦∆=id,(3)Four long ternary counitsε(3)1–ε(3)4satisfying(15) id⊗ε(3)3⊗ε(3)4 ◦∆(3)◦ (id⊗ε(3)1⊗ε(3)2)◦∆(3) =idBy analogy with(8)σ-cocommutativity is defined asσ◦∆(3)=∆(3).Definition24.Ternary coalgebra is a pair C,∆(3) ,where C is a linear space and∆(3) is a ternary comultiplication which is coassociative in one of the above senses.We will consider below onlyfirst standard type of associativity(12).Let A,m(3) is a ternary algebra and C,∆(3) is a ternary coalgebra and f,g,h∈Hom K(C,A).Definition25.Ternary convolution product is=m(3)◦(f⊗g⊗h)◦∆(3)(16)[f,g,h]∗(a)= f a(1) g a(2) h a(3) .or[f,g,h]∗Definition26.Ternary coalgebra is called derived,if there exists a binary(usual,see e.g. [1,30])coalgebra∆(2):C→C⊗C such that(cf.1))= id⊗∆(2) ⊗∆(2).(17)∆(3)derThe derived ternary and n-ary coalgebras were considered e.g.in[26]and[2]respec-tively.T ERNARY H OPF ALGEBRASDefinition27.Ternary bialgebra B is triple B,m(3),∆(3) for which B,m(3) is a ternary algebra and B,∆(3) is a ternary coalgebra and(18)∆(3)◦m(3)=m(3)◦∆(3)One can distinguish four kinds of ternary bialgebrs with respect to a”being derived”property”:BASIC CONCEPTS OF TERNARY HOPF ALGEBRAS9 (1)∆-derived ternary bialgebra∆(3)=∆(3)der= id⊗∆(2) ◦∆(2)(2)m-derived ternary bialgebram(3)der =m(3)der=m(2)◦ m(2)⊗id(3)Derived ternary bialgebra is simultaneously m-derived and∆-derived ternary bial-gebra.(4)Full ternary bialgebra is not derived.Now we define possible types of ternary antipodes using analogy with binary coalge-bras.Definition28.Skew ternary antipod ism(3)◦(S(3)skew ⊗id⊗id)◦∆(3)=m(3)◦(id⊗S(3)skew⊗id)◦∆(3)=m(3)◦(id⊗id⊗S(3)skew)◦∆(3)=idor in terms of the ternary convolution product(16)S(3)skew,id,id∗= id,S(3)skew,id ∗= id,id,S(3)skew ∗=id.Definition29.Strong ternary antipod ism(2)⊗id ◦(id⊗S(3)strong⊗id)◦∆(3)=1⊗id,id⊗m(2) ◦(id⊗id⊗S(3)strong)◦∆(3)=id⊗1,where1is a unit of algebra.Definition30.Ternary coalgebra is derived,if∆(3)is derived.Lemma31.If in a ternary coalgebra C,∆(3) there exists a linear mapε(3):C→K satisfying(19) ε(3)⊗id⊗ε(3) ◦∆(3)=id,then∃∆(2)such that∆(3)=∆(3)der= id⊗∆(2) ⊗∆(2)Definition32.If in ternary coalgebra∆(3)◦S=τ13◦(S⊗S⊗S)◦∆(3),whereτ13= 123321 ,then it is called skew-involutive.Definition33.Ternary Hopf algebra is a ternary bialgebra with a ternary antipod of the corresponding type,i.e. H,m(3),e(3),∆(3),S(3) .REMARK.There are8types of associative ternary Hopf algebras and4types of medial Hopf algebras.Also it can happen that there are several ternary units e(3)i and several ternary counitsε(3)i(see(13)–(15)),which makes number of possible ternary Hopf algebras enormous.10ANDRZEJ BOROWIEC,WIESŁAW A.DUDEK,AND STEVEN DUPLIJTheorem 34.For any a ternary Hopf algebra there exists a binary Hopf algebra,auto-morphism φand a linear map λ,such that (20)∆(3)=(id ⊗φ⊗id)◦ ∆(2)⊗idProof.The binary coproduct is ∆(2)=(id ⊗λ⊗id)◦∆(3)and ∆(3)=(id ⊗id ⊗id ⊗λ)◦ ∆(2)⊗∆(2) ◦∆(2).The co-analog of the Post Theorem 19is Theorem 35.For any ternary Hopf algebra H,∆(3) there exists a binary Hopf algebra H ∗,∆(2) and ∆(3)=∆(3)der |H ,such that H H ∗≃k (Z 2)and (21)(id ⊗id ⊗id)◦∆(3)= id ⊗∆(2) ◦∆(2).E XAMPLESExample.Ternary dual pair k (G )(push-forward)and F (G )(pull-back)which are related by k ∗(G )∼=F (G ).Here k (G )=span (G )is a ternary group (G has a ternary product[]G or m (3)G )algebra over a field k .If u ∈k (G )(u =u i x i ,x i ∈G ),then [uvw ]k =u i v j w l [x i x j x l ]G is associative,and so (k (G ),[]k )becomes a ternary algebra.Define aternary coproduct ∆(3)k :k (G )→k (G )⊗k (G )⊗k (G )by ∆(3)k (u )=u i x i ⊗x i ⊗x i (derive and associative),then ∆(3)k ([uvw ]k )= ∆(3)k (u )∆(3)k (v )∆(3)k (w ) k ,and k (G )is a ternary bialgebra.If we define a ternary antipod by S (3)k =u i ¯x i ,where ¯x i is askew element of x i ,then k (G )becomes a ternary Hopf algebra.In the dual case offunctions F (G ):{ϕ:G →k }a ternary product []F or m (3)F (derive and associative)acts on ψ(x,y,z )as m (3)F ψ (x )=ψ(x,x,x ),and so F (G )is a ternary algebra.Let F (G )⊗F (G )⊗F (G )∼=F (G ×G ×G ),then we define a ternary coproduct ∆(3)F :F (G )→F (G )⊗F (G )⊗F (G )as ∆(3)F ϕ (x,y,z )=ϕ([xyz ]F ),which is derive and associative.Thus we can obtain ∆(3)F ([ϕ1ϕ2ϕ3]F )= ∆(3)F (ϕ1)∆(3)F (ϕ2)∆(3)F (ϕ3) F,and therefore F (G )is a ternary bialgebra.If we define a ternary antipod byS (3)F (ϕ)=ϕ(¯x ),where ¯x is a skew element of x ,then F (G )becomes a ternary Hopf algebra.Example.Matrix representation.Possible non-derived matrix representations of the ternary product can be done only by four-rank tensors:twicely covariant and twicely contravariant and allow only 2possibilities A oi jk B jl oo C ko il and A ij ok B ol io C ko il (where o is any index).Acknowledgments .One of the authors (S.D.)would like to thank Jerzy Lukierski for kind hospitality at the University of Wrocław,where this work was initiated and begun.R EFERENCES[1]E.Abe,Hopf Algebras ,Cambridge Univ.Press,Cambridge,1980.[2]A.Ballesteros and O.Ragnisco,A systematic construction of completely integrable Hamiltonians fromcoalgebras ,J.Phys.A31(1998),3791–3813.[3]V .D.Belousov,n -ary Quasigroups ,Shtintsa,Kishinev,1972.[4]R.Carlsson,Cohomology of associative triple systems ,Proc.Amer.Math.Soc.60(1976),1–7.[5]J.Certaine,The ternary operation (abc )=ab −1c of a group ,Bull.Amer.Math.Soc.49(1943),869–877.[6]E.E.Demidov,Quantum Groups ,Factorial,Moscow,1998.BASIC CONCEPTS OF TERNARY HOPF ALGEBRAS11[7]W.D¨o rnte,Unterschungen¨u ber einen verallgemeinerten Gruppenbegriff,Math.Z.29(1929),1–19.[8]W.A.Dudek,K.Głazek,and B.Gleichgewicht,A note on the axioms of n-groups,in Coll.Math.Soc.J.Bolyai.29.Universal Algebra,Esztergom(Hungary),1977,pp.195–202.[9]W.A.Dudek and J.Michalski,On a generalization of Hossz´u theorem,Demonstratio Math.15(1982),437–441.[10]———,On retract of polyadic groups,Demonstratio Math.17(1984),281–301.[11]W.A.Dudek,Remarks on n-groups,Demonstratio Math.13(1980),165–181.[12]———,Autodistributive n-groups,Annales Sci.Math.Polonae,Commentationes Math.23(1993),1–11.[13]S.Duplij and F.Li,On regular solutions of quantum Yang-Baxter equation and weak Hopf algebras,Journalof Kharkov National University,ser.Nuclei,Particles and Fields521(2001),15–30.[14]K.Głazek and B.Gleichgewicht,Abelian n-groups,in Coll.Math.Soc.J.Bolyai.29.Universal Algebra,Esztergom(Hungary),1977,pp.321–329.[15]M.Kapranov,I.M.Gelfand,and A.Zelevinskii,Discriminants,Resultants and Multidimensional Determi-nants,Birkh¨a user,Berlin,1994.[16]E.Kasner,An extension of the group concept,Bull.Amer.Math.Soc.10(1904),290–291.[17]C.Kassel,Quantum Groups,Springer-Verlag,New York,1995.[18]R.Kerner,Ternary algebraic structures and their applications in physics,Univ.P.&M.Curie preprint,Paris,2000.[19]wrence,Algebras and triangle relations,in Topological Methods in Field Theory,(J.Mickelson andO.Pekonetti,eds.),World Sci.,Singapore,1992,pp.429–447.[20]W.G.Lister,Ternary rings,Trans.Amer.Math.Soc.154(1971),37–55.[21]F.Li and S.Duplij,Weak Hopf algebras and singular solutions of quantum Yang-Baxter equation,Commun.Math.Phys.225(2002),191–217.[22]D.Nikshych and L.Vainerman,Finite quantum groupoids and their applications,Univ.California preprint,math.QA/0006057,Los Angeles,2000.[23]F.Nill,Axioms for weak bialgebras,Inst.Theor.Phys.FU preprint,math.QA/9805104,Berlin,1998.[24]E.L.Post,Polyadic groups,Trans.Amer.Math.Soc.48(1940),208–350.[25]S.A.Rusakov,Some Applications of n-ary Group Theory,Belaruskaya navuka,Minsk,1998.[26]A.E.Santana and R.Muradian,Hopf structures in Nambu-Lie n-algebras,Theor.Math.Phys.114(1998),67–72.[27]S.Shnider and S.Sternberg,Quantum Groups,International Press,Boston,1993.[28]E.I.Sokolov,On the theorem of Gluskin-Hossz´u on D¨o rnte groups,Mat.Issled.39(1976),187–189.[29]N.P.Sokolov,Introduction to the Theory of Multidimensional Matrices,Naukova Dumka,Kiev,1972.[30]M.E.Sweedler,Hopf Algebras,Benjamin,New York,1969.[31]L.Vainerman and R.Kerner,On special classes of n-algebras,J.Math.Phys.37(1996),2553–2565.[32]G.ˇCupona,N.Celakoski,S.Markovski,and D.Dimovski,Vector valued groupoids,semigroups and groups,in Vector Valued Semigroups and Groups,(B.Popov,G.ˇCupona,and N.Celakoski,eds.),Macedonian Acad.Sci.,Skopje,1988,pp.1–79.I NSTITUTE OF T HEORETICAL P HYSICS,U NIVERSITY OF W ROCŁAW,P L.M AXA B ORNA9,50-204 W ROCŁAW,P OLANDE-mail address:borow@ift.univ.wroc.plI NSTITUTE OF M ATHEMATICS,T ECHNICAL U NIVERSITY OF W ROCŁAW,W YBRZEZE W YSPIANSKIEGO 27,50-370W ROCŁAW,P OLANDE-mail address:dudek@im.pwr.wroc.plD EPARTMENT OF P HYSICS AND T ECHNOLOGY,K HARKOV N ATIONAL U NIVERSITY,K HARKOV61001, U KRAINEE-mail address:Steven.A.Duplij@univer.kharkov.uaURL:http://www.math.uni-mannheim.de/˜duplij。

浙江大学学报(理学版)Journal of Zhejiang University (Science Edition )http :///sci第 48 卷第 2 期2021年3月Vol. 48 No. 2Mar. 2021DOI ： 10.3785/j.issn.1008-9497.2021.02.006素特征域上Witt 代数及极大子代数的2-局部导子姚裕丰，王惠(上海海事大学文理学院：上海201306)摘要：李代数的导子代数对李代数结构的研究有重要作用。

特征零的代数闭域上有限维半单李代数的导子都是内导子，该类李代数同构于其导子代数。

作为导子的自然推广，李代数的2-局部导子对李代数局部性质的研究，具有重要作用，研究了素特征域上李代数的2-局部导子。

设F 是特征p >3的代数闭域，g 是域F 上p _维Witt 代 数，g 0是g 的极天子代数，讨论了 g 和g 0的2-局部导子的性质，证明了 g 和g 0的所有2-局部导子均为导子。

关键词：Witt 代数；导子；2-局部导子中图分类号：O 151.26文献标志码：A 文章编号：10()8-9497(2()21)02-174-()6YAO Yufeng, WANG Hui ( College of A rts and Sciences , Shanghai Maritime University Shanghai 201306, China)2-local derivations of the Witt algebra and its maximal subalgebra over a field of prime characteristic . Journal ofZhejiang University (Science Edition), 2021,48(2):174-179Abstract ： The derivation algebra of a Lie algebra plays an important role to study of the structure of the Lie algebra. Allderivations of finite dimensional semisimple Lie algebras over an algebraically closed field are inner. So the Lie algebrasof this kind are isomorphic to their derivation algebras. As a natural generalization of derivation , 2-local derivation of aLie algebra plays an important role in study of local properties of the Lie algebra.This paper is devoted to study 2-localderivations of Lie algebras over fields of prime characteristic. Let g be the p dimensional Witt algebra over analgebraically closed field of characteristic p > 3, g ()be its maximal subalgebra. We investigate the properties of 2-localderivations on g and g 0,and show that all 2-local derivations on g and g 0 are derivations.Key Words ： Witt algebra ； derivation ； 2-local derivation代数的导子指该代数上满足Leibniz 法则的线 性变换。

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Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.Fritzshe GTM039《An Invitation to C*-Algebras》William Arveson （C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, /doc/e96250642.htmlurie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probabi lity TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》WilhelmKlingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《 A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra （黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport （乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru Iitaka GTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters （遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surface s Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway （泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Bre?cker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson （2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL? (R)》Serge Lang（SL? (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael Range GTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen （现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel （线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《T ensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》/doc/e96250642.htmlm（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris （代数几何）GTM134《Coding and Information Theory》Steven Roman GTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic NumberTheory》Henri Cohen（计算代数数论教程）GTM139《T opology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gr?bner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang Walter GTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》/doc/e96250642.htmlm（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to T opological Manifolds》John M.Lee GTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward Cheney GTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek （离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre （矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee （光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjr?ner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《T opics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal （哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠT ools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAna lytic and Modern T ools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki Hibi GTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。