Explicit generators and relations for the centre of the quantum group
aa r X i v : . [ m a t h . QA ] F e b EXPLICIT GENERATORS AND RELATIONS FORTHE CENTRE OF THE QUANTUM GROUP
YANMIN DAI AND YANG ZHANG
Abstract.
For the standard Drinfeld-Jimbo quantum group U q p g q associatedwith a simple Lie algebra g , we construct explicit generators of the centre Z p U q p g qq , and determine the relations satisfied by the generators. For g oftype A n p n ě q , D k ` p k ě q or E , the centre Z p U q p g qq is isomorphic to aquotient of a polynomial algebra in multiple variables, which is described in auniform manner for all cases. For g of any other type, Z p U q p g qq is generatedby n “ rank p g q algebraically independent elements. Introduction
Let g be a finite dimensional simple complex Lie algebra of rank n . In theliterature [Tan92, Jan96], there are two different versions of the Drinfeld-Jimboquantum group [Dri86, Jim85], which are denoted by U q p g q and U q p g q respectivelywith q being an indeterminate. The former contains among generators K λ with λ in the weight lattice P of g , while the latter contains those K α with α in the rootlattice Q of g . We will focus on the latter quantum group U q p g q (see Section 2.1)and study the structure of its centre.Drinfeld [Dri90] and Reshetikhin [Res90] constructed explicitly a natural iso-morphism from the representation ring to the centre of the quantum group U q p g q .Their method exploits the quasi-triangular structure of U q p g q and can be gener-alised to the quantum affine algebras [Eti95]. It turns out that the centre Z p U q p g qq is a polynomial algebra generated by n algebraically independent central elementsassociated to certain representations. Algebraically independent explicit generatorsof Z p U q p g qq have been constructed in [Dai].In contrast to the case of U q p g q , the centre Z p U q p g qq of the quantum group U q p g q is not necessarily a polynomial algebra, and much remains to be understood aboutits algebraic structure.A fundamental problem, analogous to the first and second fundamental theoremsof classical invariant theory, is to describe explicit generators of Z p U q p g qq and therelations which they obey. The generators of Z p U q p g qq are elements of U q p g q whichcommute with all elements of U q p g q . They are usually referred to as quantumCasimir operators, and play important roles in studying symmetries of physicalsystems.We give a complete solution of this problem for all g in Theorem 2.5.Now we briefly describe the key ingredients used in the proof of Theorem 2.5.We point out here that [LXZ16] proved to be very useful for our study, and willmake comments later on results of op. cit. . Mathematics Subject Classification.
Key words and phrases. quantum groups, central elements, Harish-Chandra isomorphism.
Given any finite dimensional U q p g q -module V of type- with some conditions onthe weights, we employ the quasi R -matrix of U q p g q (see e.g., [KS97, § C p k q V for k “ , , . . . in Definition 2.2 by following a method developed in [ZGB91a, ZGB91b].Our main theorem (i.e., Theorem 2.5) states that there exists a finite set Σ ofU q p g q -modules such that C V “ C p q V for V P Σ generate the centre Z p U q p g qq .We determine the set Σ and obtain the relations satisfied by the generatorsby making essential use of the quantised Harish-Chandra isomorphism of U q p g q ,which is an isomorphism from the centre Z p U q p g qq to the Weyl group W invariantsubalgebra p U q W [Jan96], where U is spanned by the even elements K λ for λ P M : “ Q X P with Q : “ t α | α P Q u the half root lattice. In particular,we require a specific representation-theoretical description of the isomorphism. Forthis, we consider the Grothendieck algebra S p U q p g qq of the category of finite di-mensional U q p g q -modules whose weights are contained in M . Then the quantisedHarish-Chandra isomorphism leads to an isomorphism from S p U q p g qq to Z p U q p g qq ,sending each isomorphism class r V s to C V .To gain a conceptual understanding of the Grothendieck algebra S p U q p g qq , webring the monoid algebra C r M ` s into the picture [LXZ16], where M ` : “ Q X P ` denotes the additive monoid consisting of dominant weights in the half root lattice Q . We describe the Hilbert basis Hilb p M ` q , a minimal generating set of M ` ,and then split the simple Lie algebras into two types (see (3.1)). In the caseof type I, the set Hilb p M ` q comprises exactly all fundamental weights of g bystraightforward calculation, and hence the associated monoid algebra C r M ` s is apolynomial algebra. In the case of type II, where g is of A n p n ě q , D k ` p k ě q or E , the automorphism of the corresponding Dynkin diagram (see Figure 1) inducesan involution of the monoid M ` , which permits us to describe generators andrelations of the monoid algebra C r M ` s in a unified way.We prove that there is a natural isomorphism between S p U q p g qq and the monoidalgebra C p q qr M ` s : “ C p q q b C C r M ` s over the field C p q q of rational functions, andtherefore obtain Z p U q p g qq – S p U q p g qq – C p q qr M ` s . By means of these isomor-phisms, for each generator of C p q qr M ` s we construct an explicit generator C T of Z p U q p g qq associated to a certain tensor product T of fundamental representations of g . Using the presentation of C r M ` s , we determine relations among these generators C T .We must point out that the isomorphism Z p U q p g qq – C p q qr M ` s and a presen-tation of the monoid algebra C p q qr M ` s were previously obtained in [LXZ16] bya case by case study. Here we have developed a new method for deriving theseresults, which is conceptual and uniform. Also the presentation of C p q qr M ` s givenin [LXZ16] has different (but equivalent) relations from ours despite the fact thatthe generating set is the same.We should emphasise the difference between the present paper and [LXZ16].While we have given explicit generators and relations of the centre Z p U q p g qq , theauthors of [LXZ16] gave a presentation for the isomorphic algebra C p q qr M ` s in-stead. Their results, while being interesting in their own right, do not help inconstructing explicit generators of Z p U q p g qq , which is one of our main concerns inthis paper.We note that the eigenvalues of higher order central elements C p k q V are computedexplicitly in [LZ93, DGL05] for quantum supergroups U q p gl m | n q and U q p osp m | n q , XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 3 where V is the natural representation. With the eigenvalue formula, it is shown in[Li10] that the centre of U q p gl n q is generated by C p k q V for 1 ď k ď n . In a sequel tothis paper, we will prove an analogue of this result for quantum groups of types B , C and D .This paper is organised as follows. In Section 2 we construct an explicit centralelement C V from any finite dimensional U q p g q -module V whose weights are con-tained in M , and then state our main theorem (Theorem 2.5), which will be provedin later sections. In Section 3 we describe the Hilbert basis Hilb p M ` q , and givea presentation of the monoid algebra C r M ` s in Theorem 3.15 and Theorem 3.16,which correspond to the Lie algebras of type I and II respectively. In Section 4 weprove our main theorem by showing that the centre Z p U q p g qq is isomorphic to themonoid algebra C p q qr M ` s from a representation theoretical point of view. Acknowledgements.
We would like to thank Professor Ruibin Zhang for advicesand help during the course of this work.2.
Construction of central elements
We first recall the definition of quantum groups. Given any finite dimensionalU q p g q -module V whose weights are contained in M , we construct explicitly a centralelement C V by using the quasi R -matrix of U q p g q . Finally, we state the maintheorem of this paper.2.1. Quantum groups.
Let g be a finite dimensional simple Lie algebra of rank n over the complex field C . Let h be the Cartan subalgebra of g , and let Φ Ă h ˚ be theset of roots. Fix a set Φ ` of positive roots, and denote by Π “ t α , . . . , α n u Ď Φ ` the set of corresponding simple roots.Let p´ , ´q be a non-degenerate invariant symmetric bilinear form on h ˚ . TheCartan matrix A “ p a ij q is the n ˆ n matrix with a ij “ p α i , α j q{p α i , α i q . Thefundamental weights ̟ i of g are defined by 2 p ̟ i , α j q{p α j , α j q “ δ ij for 1 ď i, j ď n .We define P “ n à i “ Z ̟ i , Q “ n à i “ Z α i to be the weight lattice and root lattice, respectively. Let P ` Ă P be the set ofdominant weights, i.e., weights that are non-negative integer combinations of ̟ i .Throughout, let q be an indeterminate and C p q q the field of rational functions.The quantum group U q p g q [Jan96] is the unital associative algebra over C p q q gen-erated by E i , F i and K i : “ K α i for 1 ď i ď n , subject to the following relations: K i K ´ i “ “ K ´ i K i , K i K j “ K j K i K i E j K ´ i “ q p α i ,α j q E j ,K i F j K ´ i “ q ´p α i ,α j q F j ,E i F j ´ F j E i “ δ ij K i ´ K ´ i q i ´ q ´ i , ´ a ij ÿ s “ p´ q s „ ´ a ij s q i E ´ a ij ´ si E j E si “ , i ‰ j, Y DAI AND Y ZHANG1 ´ a ij ÿ s “ p´ q s „ ´ a ij s q i F ´ a ij ´ si F j F si “ , i ‰ j, where q i “ q p α i ,α i q{ , and for any m P N r m s q i “ q mi ´ q ´ mi q i ´ q ´ i , r m s q i ! “ r s q i r s q i ¨ ¨ ¨ r m s q i , „ mk q i “ r m s q i ! r m ´ k s q i ! r k s q i ! . It is well known that U q p g q is a Hopf algebra with co-multiplication ∆, co-unit ε and antipode S given by∆ p K i q “ K i b K i , ∆ p E i q “ K i b E i ` E i b , ∆ p F i q “ F i b K ´ i ` b F i ,ε p K i q “ , ε p E i q “ , ε p F i q “ ,S p K i q “ K ´ i , S p E i q “ ´ K ´ i E i , S p F i q “ ´ F i K i . Write U “ U q p g q . The quantum group is graded by the root lattice Q , i.e.,U “ À ν P Q U ν withU ν “ t u P U | K i uK ´ i “ q p ν,α i q u, @ i “ , . . . , n u . Define U ` (resp. U ´ ) to be the subalgebra generated by all E i (resp. F i ), andintroduce U ` ν “ U ν X U ` (resp. U ´´ ν “ U ´ ν X U ´ ).The representation theory of U q p g q is parallel to that of the Lie algebra g [Hum72, Jan96]. Throughout, we are concerned with finite dimensional U q p g q -modules of type 1. Each U q p g q -module V admits the weight space decomposition V “ À µ P Π p V q V µ , where V µ is the weight space of V and Π p V q Ă P is the set ofweights. Define m V p µ q : “ dim V µ . The dominant weights λ P P ` are in bijectionwith the simple U q p g q -modules L p λ q with the highest weight λ . For any two weights λ, µ we have the partial ordering λ ą µ if and only if λ ´ µ is a sum of positiveroots.2.2. Central elements.
Let V be an arbitrary finite dimensional U q p g q -module.Let Tr denote the partial trace on the first tensor factor of End p V q b U q p g q , i.e.,Tr p ξ b x q “ Tr p ξ q x for any ξ P End p V q and x P U q p g q . Note that Tr p ξ b x q is anelement of U q p g q .The following crucial lemma is essentially from [ZGB91a], where the setting isslightly different from ours. We include a proof in Appendix A. Lemma 2.1. [ZGB91a, Proposition 1]
Given an operator Γ V P End p V q b U q p g q satisfying (2.1) r Γ V , ∆ p K ˘ i qs “ r Γ V , ∆ p E i qs “ r Γ V , ∆ p F i qs “ , @ i, the elements C p k q V P U q p g q for k “ , , . . . defined by (2.2) C p k q V : “ Tr pp K ρ b qp Γ V q k q are central in U q p g q , where ρ denotes the half sum of positive roots of g . Using the quasi R -matrix of U q p g q , we shall construct an explicit operator Γ V satisfying (2.1).Recall that the Drinfeld version of quantum group defined over formal powerseries C rr h ss admits a universal R matrix [Dri86, ZGB91a], which is absent for thequantum group U q p g q considered here. But a quasi R -matrix R exists for U q p g q ,which can be described as follows [Lus10]. XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 5
Let U q p g q p b U q p g q be a completion of the tensor product U q p g q b U q p g q . There isan algebra automorphism φ of U q p g q b U q p g q defined by φ p K i b q “ K i b , φ p E i b q “ E i b K ´ i , φ p F i b q “ F i b K i ,φ p b K i q “ b K i , φ p b E i q “ K ´ i b E i , φ p b F i q “ K i b F i , and φ can be extended to U q p g q p b U q p g q . The quasi R -matrix R is an element ofU q p g q p b U q p g q which has the form R “ b ` ÿ ν ą Θ ν P U q p g q p b U q p g q , where Θ ν P U ´´ ν b U ` ν for positive ν P Q . An explicit formula for R can be foundin, e.g., [KS97, § q p sl q ). The quasi R -matrixsatisfies the following relations(2.3) R ∆ p x q “ φ p ∆ p x qq R , R T ∆ p x q “ φ p ∆ p x qq R T , where R T “ T p R q with T being the linear map defined by T p x b y q “ y b x for x, y P U q p g q (see, e.g., [Tan92, § q p g q -module V satisfies(2.4) Π p V q Ă M “ Q X P, where Π p V q is the set of all weights of V . Denote by ζ V : U q p g q Ñ GL p V q thelinear representation. We define the following elements of End p V q b U q p g q :(2.5) R V : “ p ζ V b id qp R q , r R TV : “ p ζ V b id q φ p R T q On the other hand, we define the diagonal part by(2.6) K V : “ ÿ η P Π p V q P η b K η , where P η is the linear projection from V to its weight space V η . Note that thecondition (2.4) guarantees that 2 η P Q for any weight η of V and hence K V P End p V q b U q p g q . Definition 2.2.
Given a finite dimension U q p g q -module V whose weights are con-tained in M “ Q X P , we define the operatorΓ V : “ K V r R TV R V P End p V q b U q p g q , where R V and r R TV are given in (2.5) and K V is defined by (2.6). Let C p k q V : “ Tr pp K ρ b qp Γ V q k q , k “ , , . . . , and write C V “ C p q V .By Lemma 2.1 and Proposition 2.3 below, C p k q V are central elements of U q p g q . Proposition 2.3.
The element Γ V given in Definition 2.2 satisfies the commutativerelations (2.1) , i.e., r Γ V , ∆ p x qs “ for any x P U q p g q . A proof of Proposition 2.3 is given in Appendix A.The following is an example of our construction.
Y DAI AND Y ZHANG
Example 2.4.
Let g “ sl . The quasi R -matrix of U q p sl q is given by R “ ÿ n “ q n p n ` q p ´ q ´ q n r n s q ! F n b E n . Let V be the 2-dimensional simple module, then the weights of V are contained in M “ Q X P “ Z . Denote by ζ : U q p g q Ñ End p V q the representation correspondingto the standard basis t e , e u of V , we have ζ p E q “ ˆ ˙ , ζ p F q “ ˆ ˙ , ζ p K q “ ˆ q q ´ ˙ . Now R V “ b ` p q ´ q ´ q ζ p F q b E, r R TV “ b ` p q ´ q ´ q ζ p EK q b K ´ F, K V “ P b K ` P ´ b K ´ , where P (resp. P ´ ) is the linear projection from V onto the weight space C p q q e (resp. C p q q e ), and we have P “ ˆ ˙ (res. P ´ “ ˆ ˙ ). We haveΓ V “ K V r R TV R V “ P b K ` P ´ b K ´ ` p q ´ q ´ q ζ p F q b K ´ E ` p ´ q ´ q ζ p E q b F ` p q ´ q ´ q q ´ P b F E.
Note that in Γ V the first tensor factors ζ p E q and ζ p F q have no contributions tothe partial trace Tr . Using K ρ P ˘ “ KP ˘ “ q ˘ P ˘ , we obtain the followingcentral element associated to V : C V “ Tr pp K ρ b q Γ V q “ qK ` q ´ K ´ ` p q ´ q ´ q F E.
By similar straightforward calculation, one can also express the higher order centralelements C p k q as C p q q -linear combinations of the powers C kV : C p q V “ q ´ C V ´ q ´ ´ q ´ ,C p q V “ q ´ C V ´ p q ´ ` q ´ q C V ,C p q V “ q ´ C V ´ p q ´ ` q ´ q C V ` q ´ ` q ´ . The main theorem.
We shall state our main theorem which exhibits explicitgenerators and relations of the centre of the quantum group.Recall from Definition 2.2 that the central elements are associated with U q p g q -modules V whose weights are contained M . In particular, the highest weights ofthese modules are contained in M ` “ Q X P ` , which is an additive monoid, i.e.,a commutative semigroup with the identity 0. An element x P M ` is said to beirreducible if x “ y ` z implies either y “ z “
0. The Hilbert basis Hilb p M ` q of M ` is a minimal set of generators given by its irreducible elements.The generators of the centre Z p U q p g qq can be chosen in bijection with the ele-ments of Hilb p M ` q . Given any λ “ ř ni “ a i ̟ i P Hilb p M ` q , we define the tensormodule T p λ q : “ n â i “ L p ̟ i q b a i , XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 7 where L p ̟ i q is the fundamental representation of U q p g q . Note that for any weight µ P Π p T p λ qq , we have λ ´ µ P Q Ď M “ Q X P . Therefore, Π p T p λ qq Ď M and wemay define the associated central element C T p λ q .Particularly, if g is of type A , B n p n ě q , C n p n ě q , D k p k ě q , E , E , F ,and G , we will show that the Hilbert basis Hilb p M ` q consists of all fundamentalweights of g . Hence we obtain n generators C L p ̟ i q of Z p U q p g qq , which will be shownto be algebraically independent.To describe relations among central elements C T p λ q for g of one of the remainingtypes A n p n ě q , D k ` p k ě q and E , we introduce the automorphism σ ofthe corresponding Dynkin diagram. This is depicted as in Figure 1, where eachpair of vertices which are connected by a curved double arrow means they areswapped by the involution σ and the rest vertices are fixed by σ . For instance, σ p i q “ n ` ´ i, ď i ď n for type A n p n ě q . . . . n ´ n . . . σ . . . n ´ n σ σ Figure 1.
The involutions σ of types A , D and E .The automorphism σ induces an involution σ M ` of the monoid M ` . Precisely,if λ “ ř ni “ a i ̟ i P Hilb p M ` q , then we define λ : “ σ M ` p λ q “ ř ni “ a σ p i q ̟ i ; referto Lemma 3.6. The element λ is said to be self-conjugate if λ “ λ ; otherwise,it is called non-self-conjugate. These elements will be characterised explicitly inLemma 3.8 and Lemma 3.10.The following is our main theorem of this paper. Theorem 2.5.
Let g be a complex simple Lie algebra of rank n , and let Hilb p M ` q be the Hilbert basis of the monoid M ` “ Q X P ` , where Q denotes the half rootlattice and P ` is the monoid of dominant weights of g .(1) If g is one of the types A , B n p n ě q , C n p n ě q , D k p k ě q , E , E , F ,and G , then the centre Z p U q p g qq of the quantum group U q p g q is generatedby n algebraically independent elements C L p ̟ q , . . . , C L p ̟ n q , where L p ̟ i q are simple modules corresponding to the fundamental weights ̟ i ;(2) If g is one of the types A n p n ě q , D k ` p k ě q and E , then the centre Z p U q p g qq of the quantum group U q p g q is generated by C T p λ q , λ P Hilb p M ` q ,subject to the following relations: C T p λ q C T p ¯ λ q “ ź i : i ă σ p i q C max t a i ,a σ p i q u T p µ i q ,C ℓ p λ q T p λ q “ n ź i “ C ℓ p λ q a i { s i T p ν i q with λ ‰ ν i , ď i ď n for each non-self-conjugate pair t λ, λ u of Hilb p M ` q with λ “ ř ni “ a i ̟ i and λ “ ř ni “ a σ p i q ̟ i , where σ is the involution of the Dynkin diagram given byFigure 1, µ i P Hilb p M ` q are self-conjugate elements given by Lemma 3.8, Y DAI AND Y ZHANG ν i “ s i ̟ i P Hilb p M ` q are scalar multiples of the fundamental weights ̟ i with s i given by Lemma 3.13, and ℓ p λ q is a positive integer defined by (3.3) . The remainder of the paper is on the proof of Theorem 2.5. The main idea ofthe proof is to show that the centre Z p U q p g qq is isomorphic to the monoid algebra C p q qr M ` s , which is more conceptual and will be studied systematically in Section 3.The actual proof of Theorem 2.5 is given in Section 4.3. The monoid M ` In this section we describe the Hilbert basis Hilb p M ` q for the monoid M ` asso-ciated to a simple Lie algebra g . Using the automorphism of the Dynkin diagram,we give a presentation of the monoid algebra of M ` .3.1. The Hilbert basis of M ` . Some results in this subsection can be found in[LXZ16]. We include proofs for them to make the paper more accessible.In the sequel, for explicit formulae of fundamental weights of g we refer to[Hum72, § §
11, Theo-rem 11.4].
Lemma 3.1. [LXZ16, Lemma 3.4]
For each simple Lie algebra g , the Hilbert basis Hilb p M ` q is finite.Proof. Observe from [Hum72, § ̟ i , there exists a minimal positive integer s i such that s i ̟ i P M ` . Given any λ “ ř ni “ a i ̟ i P Hilb p M ` q , we just need to prove that 0 ď a i ď s i for all i “ , . . . , n .Assuming for contradiction that there exists an index i such that a i ą m i , wehave λ “ ÿ i ‰ i a i ̟ i ` p a i ´ s i q ̟ i ` s i ̟ i . Let µ “ ř ni “ ,,i ‰ i a i ̟ i ` p a i ´ s i q ̟ i . Then we have µ P P ` . As λ P Q and s i ̟ i P Q , we also have µ “ λ ´ s i ̟ i P Q and hence µ P M ` “ Q X P ` . Itfollows that λ is a sum of two nonzero elements µ and s i ̟ i of M ` , contrary tothe irreducibility of λ . (cid:3) Lemma 3.2. [LXZ16, Lemma3.5]
Let ̟ i be the fundamental weights of g .(1) If g is of one of types A , B n p n ě q , C n p n ě q , D k p k ě q , E , E , F ,and G , we have Hilb p M ` q “ t ̟ , ¨ ¨ ¨ , ̟ n u . (2) If g is of type D k ` p k ě q , we have Hilb p M ` q “ t ̟ , ¨ ¨ ¨ , ̟ n ´ , ̟ n ´ , ̟ n , ̟ n ´ ` ̟ n u , where n “ k ` .(3) If g is of type E , we have Hilb p M ` q “t ̟ , ̟ , ̟ , ̟ , ̟ , ̟ , ̟ ` ̟ , ̟ ` ̟ , ̟ ` ̟ ,̟ ` ̟ , ̟ ` ̟ , ̟ ` ̟ , ̟ ` ̟ , ̟ ` ̟ u . Proof.
Part (1) can be checked case by case by using [Hum72, § M ` “ Q X P ` “ P ` in these cases, and hence Hilb p M ` q consists of allfundamental weights. XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 9
For part (2), let s i be the smallest positive integer such that s i ̟ i P Q for1 ď i ď n . Using [Hum72, § s i “ , ď i ď n ´ , and s n ´ “ s n “ n ě ̟ i P Hilb p M ` q for 1 ď i ď n ´ ̟ n ´ , ̟ n P Hilb p M ` q .Next we consider the irreducible element of Hilb p M ` q which can be written asa sum of fundamental weights. Assume that λ “ ř ni “ a i ̟ i P Hilb p M ` q is not amultiple of some fundamental weight. Then by the proof of Lemma 3.1, we have a i ď s i for each i . It is verified readily that ̟ n ´ ` ̟ n P Q , while ̟ i ` ̟ j R Q for any i P t , . . . , n ´ u and j P t n ´ , n u . Therefore, the only irreducible elementwhich is a sum of fundamental weights is ̟ n ´ ` ̟ n . This completes the proof.Part (3) can be done similarly with a suitable computer program (cid:3) Now we consider the case of type A n for n ě
2. The following lemma is useful.
Lemma 3.3. [LXZ16, Lemma 4.3]
Let λ “ ř ni “ a i ̟ i P P ` be a dominant weightof the Lie algebra of type A n . Then λ P M ` if and only if ř ni “ ia i P r n ` Z , where r n ` “ n ` p n ` , q .Proof. We need to show that λ P Q if and only if the given condition holds. Notethat in the case of type A we have ̟ i “ i̟ ´ p α i ´ ` α i ´ ` . . . p i ´ q α i ´ q for2 ď i ď n . Then we have the expression λ “ a ̟ ` n ÿ i “ a i p i̟ ´ p α i ´ ` α i ´ ` . . . p i ´ q α i ´ qq“ p n ÿ i “ ia i q ̟ ´ n ÿ i “ a i p α i ´ ` α i ´ ` . . . p i ´ q α i ´ q . It follows that λ P Q if and only if p ř ni “ ia i q ̟ P Q . Recall that ̟ “ nn ` α ` n ` p α n ` α n ´ ` . . . p n ´ q α q . We have p ř ni “ ia i q ̟ P Q if and only if r n ` “ n ` p n ` , q divides ř ni “ ia i . (cid:3) Corollary 3.4.
In the case of type A n , assume that λ “ ř ni “ a i ̟ i P Hil p M ` q .Then we have ř ni “ a i ď r n ` , where r n ` “ n ` p n ` , q .Proof. For convenience, we denote λ by p a , a , . . . , a n q P N n , where N is the set ofnon-negative integers. Let ă be the lexicographical order on N n . Without loss ofgenerality, we may assume a ‰
0. Then there is a strictly increasing sequence: λ ă λ ă ¨ ¨ ¨ ă λ p “ λ, where p “ ř ni “ a i , λ “ p , , . . . , q , and λ i ` ´ λ i “ ̟ j for some j ě i . For any λ i “ p a i , . . . , a in q , we define b i ” n ÿ j “ ja ij mod r n ` , ď i ď p. Assume for contradiction that p ą r n ` . Then by the pigeonhole principle, there ex-ists a pair i ă i such that b i “ b i , which implies that r n ` divides ř nj “ j p a i j ´ a i j q . It follows from Lemma 3.3 that λ i ´ λ i P M ` , and hence λ “ p λ i ´ λ i q `p λ ´ λ i ` λ i q , contrary to the irreducibility of λ . (cid:3) In accord with the previous lemmas, we split simple Lie algebras into the follow-ing two types:(3.1) Type I: A , B n p n ě q , C n p n ě q , D k p k ě q , E , E , F , and G , Type II: A n p n ě q , D k ` p k ě q and E . For each Lie algebra g of type I, the Hilbert basis Hilb p M ` q comprises exactly allfundamental weights of g . For type II, we will focus on the symmetry propertyof M ` derived from the involution of the corresponding Dynkin diagram. This istreated in the following subsection. Remark . The classification (3.1) is also in accord with the fact that ´ P W ifand only if g is of type I (see, e.g., [Bou68, Chapter V, § ´ W .3.2. The involution of M ` . For the purpose of this paper, we are only concernedwith the involutions corresponding to Lie algebras of type II.
Lemma 3.6.
Let g be a simple Lie algebra of type II , and let σ be the automorphismof the Dynkin diagram given in Figure 1. Then we have the involution of M ` : (3.2) σ M ` : M ` Ñ M ` , λ “ n ÿ i “ a i ̟ i ÞÑ λ “ n ÿ i “ a σ p i q ̟ i such that σ M ` “ . In particular, if λ P Hilb p M ` q , then λ P Hilb p M ` q .Proof. We only give the proof for type A , and the other cases can be treatedsimilarly. By definition σ sends the simple root α i to α σ p i q “ α n ` ´ i , and hence σ gives rise to an involution ψ of the vector space Q Q : “ Q b Z Q over Q . In particular, ψ restricts to an involution of the half root lattice Q . On the other hand, recallthat the fundamental weights are given by ̟ i “ i ÿ j “ p n ` ´ i q jn ` α j ` n ÿ j “ i ` p n ` ´ j q in ` α j P Q Q, ď i ď n. It is straightforward to check that ψ p ̟ i q “ i ÿ j “ p n ` ´ i q jn ` α n ` ´ j ` n ÿ j “ i ` p n ` ´ j q in ` α n ` ´ j “ n ` ´ i ÿ k “ ikn ` α k ` n ÿ k “ n ` ´ i p n ` ´ i qp n ` ´ k q n ` α k , “ ̟ n ` i ´ . It follows that ψ induces an involution of the monoid P ` with ψ p ̟ i q “ ̟ n ` ´ i for1 ď i ď n . Therefore, the restriction σ M ` : “ ψ | M ` is an involution satisfying σ M ` p n ÿ i “ a i ̟ i q “ n ÿ i “ a i ̟ σ p i q “ n ÿ i “ a σ p i q ̟ i , and σ M ` “
1. For the last assertion, it is clear that λ is irreducible if and only ifits image λ is irreducible. (cid:3) Definition 3.7.
The elements λ P M ` satisfying λ “ λ are said to be self-conjugate; the other elements of M ` are called non-self-conjugate. XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 11
Now we can split the finite generating set Hilb p M ` q into two disjoint subsets,depending on whether they are self-conjugate. In the following we figure out theself-conjugate elements of Hilb p M ` q . Lemma 3.8.
Let σ be the involution of the Dynkin diagram given in Figure 1, andlet λ P Hilb p M ` q . Then λ “ λ if and only if λ is of the following form:(1) µ i “ ̟ i ` ̟ σ p i q for all i with i ă σ p i q ;(2) µ i “ ̟ i for all i with σ p i q “ i .Proof. Assume that λ “ ř ni “ a i ̟ i P Hilb p M ` q satisfies λ “ λ . Then we have a i “ a σ p i q for 1 ď i ď n . It follows that λ “ ÿ i : i ă σ p i q a i p ̟ i ` ̟ σ p i q q ` ÿ i : σ p i q“ i a i ̟ i . It suffices to show that ̟ i ` ̟ σ p i q P Hilb p M ` q whenever i ă σ p i q and ̟ i P Hilb p M ` q whenever σ p i q “ i .We only do it for type A , and the other two cases can be treated similarly. For g of type A n , we first claim that ̟ i R M ` and hence ̟ i R Hilb p M ` q whenever i ă σ p i q . By definition (refer to Figure 1) i ă σ p i q if and only if either n is even or n is odd and i ‰ n ` . Using Lemma 3.3, we obtain that ̟ i R M ` for 1 ď i ď n if n is even, and ̟ i R M ` for i ‰ n ` if n is odd. Thus our claim follows. Secondly,by Lemma 3.3 we have ̟ i ` ̟ σ p i q P M ` for all i with i ă σ p i q . Moreover, by ourclaim ̟ i ` ̟ σ p i q is irreducible and hence ̟ i ` ̟ σ p i q P Hilb p M ` q . It remains todeal with the case σ p i q “ i . This happens if and only if n is odd and i “ n ` . Inthis case, ̟ n ` P Hilb p M ` q by Lemma 3.3. (cid:3) Example 3.9.
Using Lemma 3.8, we can write out all self-conjugate elements µ i ofHilb p M ` q explicitly as follows (the indices of fundamental weights ̟ i are in accordwith the labellings of Dynkin diagrams Figure 1).(1) Type A n p n ě q . If n is even, we have µ i : “ ̟ i ` ̟ n ` ´ i , ď i ď n . If n is odd, we have µ i : “ ̟ i ` ̟ n ` ´ i , ď i ď n ` ´ ,µ n ` : “ ̟ n ` . (2) Type D k ` p k ě q . We have µ i : “ ̟ i , ď i ď n ´ , µ n ´ : “ ̟ n ´ ` ̟ n , where n “ k `
1. This can also be verified directly by Lemma 3.2.(3) Type E . We have µ : “ ̟ ` ̟ , µ “ ̟ , µ : “ ̟ ` ̟ , µ “ ̟ . For the non-self-conjugate elements of Hilb p M ` q , we have the following. Lemma 3.10.
Let λ “ ř ni a i ̟ i P Hilb p M ` q . For each i “ , . . . , n , we have(1) If i ‰ σ p i q , then either a i a σ p i q “ or a i “ a σ p i q “ . In the latter case, allother a i are , i.e. λ “ w i ` w σ p i q . (2) If i “ σ p i q , then either a i “ or a i “ . In the latter case, all other a i are , i.e. λ “ ̟ i .Therefore, for any non-self-conjugate element λ P Hilb p M ` q , we have a i a σ p i q “ for i ‰ σ p i q , and a i “ for i “ σ p i q .Proof. For part (1), we assume that i ‰ σ p i q . If a i “ a σ p i q , then we may write λ “ a i p ̟ i ` ̟ σ p i q q ` ÿ j : j ‰ i,σ p i q a j ̟ j . It follows from Lemma 3.8 that ̟ i ` ̟ σ p i q P Hilb p M ` q . By the irreducibility of λ ,we have a i “ a σ p i q “ a i are 0, i.e. λ “ w i ` w σ p i q . If a i ‰ a σ p i q ,we need to prove that one of a i and a σ p i q is 0, that is, a i a σ p i q “
0. Assume forcontradiction that 0 ă a i ă a σ p i q . Then we may write λ “ a i p ̟ i ` ̟ σ p i q q ` p a σ p i q ´ a i q ̟ σ p i q ` ÿ j : j ‰ i,σ p i q a j ̟ j . By the irreducibility of λ we have a i “ a σ p i q “
1, which is a contradiction.Part (2) can be proved similarly, by using the fact from Lemma 3.8 that w i P Hilb p M ` q if i “ σ p i q . (cid:3) Lemma 3.11.
For any non-self-conjugate element λ “ ř ni “ a i ̟ i P Hilb p M ` q , wehave the relation λ ` λ “ ÿ i : i ă σ p i q max t a i , a σ p i q u µ i . Proof. As λ ` λ is fixed by σ M ` , it can be expressed linearly by elements µ i givenin Lemma 3.8. On the other hand, since λ ‰ λ we have a i “ i “ σ p i q and a i a σ p i q “ i ‰ σ p i q by Lemma 3.10. It follows that λ ` λ “ ÿ i : i ă σ p i q p a i ` a σ p i q q µ i “ ÿ i : i ă σ p i q max t a i , a σ p i q u µ i , where the second equation holds since one of a i and a σ p i q is zero. (cid:3) Example 3.12.
For convenience, we denote λ “ ř ni a i ̟ i by p a , a , . . . , a n q . Thenin type A case λ “ p a n , a n ´ , . . . , a q .(1) Type A . Self-conjugate: p , q , and non-self-conjugate: p , q and p , q .(2) Type A . Self-conjugate: p , , q , p , , q , and non-self-conjugate: p , , q , p , , q .(3) Type A . Self-conjugate: p , , , q , p , , , q , and non-self-conjugate: p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q , p , , , q . We proceed to explore other relations among elements of Hilb p M ` q . Note thatfor each fundamental weight ̟ i there exists a minimal positive integer s i such that s i ̟ i P M ` . Since s i is minimal, we have s i ̟ i P Hilb p M ` q . Therefore, we mayform a sequence p s , s , . . . , s n q , which is determined for each Lie algebra of type IIas follows (note that for type I all s i are equal to 1). Lemma 3.13.
Let p s , s , . . . , s n q be a sequence of minimal positive integers suchthat s i ̟ i P Hilb p M ` q for each i . Then XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 13 (1) For type A n p n ě q , we have s i “ s n ` ´ i “ n ` p n ` , i q , ď i ď n. (2) For type D k ` p k ě q , we have s “ ¨ ¨ ¨ “ s n ´ “ , s n ´ “ s n “ . (3) For type E , we have s “ s “ s “ s “ , s “ s “ . Recalling that σ is the involution of the Dynkin diagram given in Figure 1, we have s i “ s σ p i q for all i . In particular, s i “ s σ p i q “ if and only if i “ σ p i q .Proof. In the case of type A n p n ě q , recall from Lemma 3.3 that s i ̟ i P M ` ifand only if r n ` | is i , where r n ` “ p n ` q{ gcd p n ` , q . By the minimality wehave s i “ r n ` { gcd p r n ` , i q “ p n ` q{ gcd p n ` , i q . The other two cases can beverified directly by Lemma 3.2. (cid:3) For any λ “ ř ni “ a i ̟ i P M ` we may define(3.3) ℓ p λ q : “ lcm t s i | ď i ď n and a i ‰ u , i.e., ℓ p λ q is the least common multiple of s i for all i for which a i is nonzero. Thefollowing lemma is trivial. Lemma 3.14.
Let p s , . . . , s n q be the sequence as defined in Lemma 3.13, and let ν i “ s i ̟ i for ď i ď n . For any λ “ ř ni “ a i ̟ i P M ` , we have the relation ℓ p λ q λ “ n ÿ i “ ℓ p λ q a i s i ν i , i.e., ℓ p λ q λ is an integer combination of ν i . We have obtained relations among generators of Hilb p M ` q from Lemma 3.11and Lemma 3.14. Next we will show that these relations are enough in the sensethat they give rise to complete relations among generators of the monoid algebraof M ` .3.3. The monoid algebra.
Given the monoid M ` of any simple Lie algebra,we may form the monoid algebra C r M ` s . As a C -vector space, C r M ` s has abasis consisting of symbols X λ , λ P M ` , with multiplication given by the bilinearextension of X λ X µ “ X λ ` µ . We agree that X “ t X λ | λ P Hilb p M ` qu is a generating set of C r M ` s . In what follows, weshall determine relations among these generators, and hence obtain a presentationof the monoid algebra C r M ` s .Recall that for the simple Lie algebra of type I considered in Lemma 3.2, theHilbert basis Hilb p M ` q consists of all fundamental weights ̟ i of g . Theorem 3.15.
Let g be a simple Lie algebra of type I with rank n , and let M ` be the monoid associated with g . Then the monoid algebra C r M ` s is isomorphic tothe polynomial algebra C r x , . . . , x n s in n variables x i .Proof. Since g is of type I, the monoid algebra C r M ` s is generated by X ̟ i for1 ď i ď n . As the fundamental weights ̟ i are linearly independent, C r M ` s isisomorphic to the polynomial algebra C r x , . . . , x n s in n variables x i , with each X ̟ i assigned to x i . (cid:3) We consider the type II case, i.e., g is one of the types A n p n ě q , D k ` p k ě q and E . In this case, Hilb p M ` q is a disjoint union of self-conjugate elements andnon-self-conjugate elements. The non-self-conjugate elements appear in pairs; weuse t λ, λ u to indicate that λ and λ are conjugate to each other. Theorem 3.16.
Let g be a simple Lie algebra of type II with rank n , and let σ be theinvolution of the corresponding Dynkin diagram given by Figure 1. Let Hilb p M ` q be the Hilbert basis of the monoid M ` associated with g . Then the monoid algebra C r M ` s is isomorphic to A “ P { I , where P is the polynomial algebra over C invariables x λ , λ P Hilb p M ` q , and I is the ideal of P generated by x λ x ¯ λ ´ ź i : i ă σ p i q x max t a i ,a σ p i q u µ i ,x ℓ p λ q λ ´ n ź i “ x ℓ p λ q a i { s i ν i with λ ‰ ν i , ď i ď n for each non-self-conjugate pair t λ, λ u of Hilb p M ` q with λ “ ř ni “ a i ̟ i and λ “ ř ni “ a σ p i q ̟ i , where µ i P Hilb p M ` q are self-conjugate elements given by Lemma 3.8, ν i “ s i ̟ i P Hilb p M ` q are scalar multiples of the fundamental weights ̟ i with s i given by Lemma 3.13, and ℓ p λ q is defined by (3.3) . Before embarking on the proof, let us illustrate this theorem with examples.
Example 3.17. (1) Type A . Hilb p M ` q consists of the following elements: µ “ ̟ ` ̟ , t ν “ ̟ , ν “ ̟ u . The ideal I is generated by x ν x ν ´ x µ . Ideals for type A and A canbe obtained by using Example 3.12.(2) Type D k ` p k ě q . Hilb p M ` q consists of the following elements: µ i “ ν i “ ̟ i , ď i ď n ´ ,µ n ´ “ ̟ n ´ ` ̟ n , t ν n ´ “ ̟ n ´ , ν n “ ̟ n u , where n “ k `
1. The ideal I is generated by x ν n ´ x ν n ´ x µ n ´ .(3) Type E . Hilb p M ` q consists of the following elements: µ “ ̟ ` ̟ , µ “ ν “ ̟ , µ “ ̟ ` ̟ , µ “ ν “ ̟ , t ν “ ̟ , ν “ ̟ u , t ν “ ̟ , ν “ ̟ u , t ̟ ` ̟ , ̟ ` ̟ u , t ̟ ` ̟ , ̟ ` ̟ u , t ̟ ` ̟ , ̟ ` ̟ u . Hence the ideal I is generated by the following binomials: x ν x ν ´ x µ , x ν x ν ´ x µ , x ̟ ` ̟ x ̟ ` ̟ ´ x µ x µ ,x ̟ ` ̟ x ̟ ` ̟ ´ x µ x µ , x ̟ ` ̟ x ̟ ` ̟ ´ x µ x µ ,x ̟ ` ̟ ´ x ν x ν , x ̟ ` ̟ ´ x ν x ν , x ̟ ` ̟ ´ x ν x ν . Proof of Theorem 3.16.
We define the following surjective algebra homomorphism ϕ : P Ñ C r M ` s , x λ ÞÑ X λ , λ P Hilb p M ` q . Then by Lemma 3.11 and Lemma 3.14 we have I Ď Ker ϕ . We need to prove thatKer ϕ Ď I , whence A “ P { I is isomorphic to C r M ` s . XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 15
We start by defining a polynomial subalgebra R Ď P and show that the restric-tion ϕ | R is injective. Let I : “ t i | i ă σ p i q for 1 ď i ď n u andΥ : “ t µ i | i P I u Y t ν i | i P t , , . . . , n uz I u . are linearly independent. Then it can be verified case by case for all A n p n ě q , D k ` p k ě q and E that Υ is a linearly independent set. Denote by R : “ C r x λ , λ P Υ s the polynomial subalgebra of P . By the linear independence of Υ, theimage ϕ p R q is a polynomial subalgebra of C r M ` s generated by the algebraicallyindependent elements X λ , λ P Υ. Therefore, ϕ | R is an isomorphism and R X Ker ϕ “
0. Define the multiplicatively closed sets S : “ R ´ t u and ϕ p S q .Next we consider the rings of fractions S ´ P and ϕ p S q ´ C r M ` s , and the inducedsurjective homomorphism ϕ S : S ´ P Ñ ϕ p S q ´ C r M ` s given by ϕ S p s ´ a q “ ϕ p s q ´ ϕ p a q for a P P and s P S . Clearly, Ker ϕ Ď Ker ϕ S X P .We claim that S ´ I is a maximal ideal of S ´ P . If the claim is done, then Ker ϕ S “ S ´ I since S ´ I Ď Ker ϕ S . Moreover, I is a prime ideal since S ´ I is maximal(hence a prime ideal) and I X S “ R X Ker ϕ “ S ´ I X P “ I . Combing all above together, we obtainKer ϕ Ď Ker ϕ S X P “ S ´ I X P “ I as required.It remains to prove our claim, which is equivalent to showing that S ´ P { S ´ I – S ´ A is a field. Consider the fraction field F “ S ´ R of R . We will adjoinextra elements x λ , λ P Hilb p M ` qz Υ to F , and then S ´ A is equal to the resultingextension field of F . Note that by Lemma 3.10 and Lemma 3.13 we have x µ i “ x ν i “ x ̟ i P F for all i “ σ p i q . For any i ă σ p i q , we define F : “ F r x ν i s – F r t s{p t ´ x ´ ν σ p i q ś i,i ă σ p i q x s i µ i q . Since x ´ ν σ p i q ś i,i ă σ p i q x s i µ i P F , we have F “ F andhence x ν i P F . Therefore, all x µ i and x ν i belong to the fraction field F . Now takingan arbitrary non-self-conjugate pair t λ, λ u of Hilb p M ` q with λ ‰ ν i , we define F “ F r x λ s – F r t s{p t ℓ p λ q ´ ś ni “ x ℓ p λ q a i { s i ν i q . Since x λ is algebraic over F , F r x λ s “ F p x λ q is a field. Similarly, define F “ F r x ¯ λ s – F r t s{p t ´ x ´ λ ś i,i ă σ p i q x max t a i ,a σ p i q u µ i q .Since x ´ λ ś i,i ă σ p i q x max t a i ,a σ p i q u µ i P F , we have F “ F is a field. Repeating theabove step for each non-self-conjugate pair t λ, λ u , we obtain an extension field F “ S ´ A of F . This completes the proof. (cid:3) Proof of the main theorem
This section is devoted to proving Theorem 2.5. We will review the quantisedHarish-Chandra theorem, which allows us to construct explicitly an isomorphismbetween the centre Z p U q p g qq and the Grothendieck algebra S p U q p g qq of the categoryof finite dimensional U q p g q -modules whose weights are contained in M . Then weshow that S p U q p g qq is isomorphic to the monoid algebra C p q qr M ` s over C p q q , andhence Z p U q p g qq – C p q qr M ` s . Combining with the presentation of C r M ` s , weobtain explicit generators and relations of Z p U q p g qq as given in Theorem 2.5. The Harish-Chandra isomorphism.
We will follow [Jan96, Chapter 6] andretain notation from Section 2.1. Write U “ U q p g q . Recall that the quantumgroup U is graded by the root lattice Q , i.e., U “ À ν P Q U ν . In particular, U “ U ‘ ‘ ν ą U ´´ ν U U ` ν , where U denotes the subalgebra generated by all K ˘ i . It isknown that the projection π : U Ñ U is an algebra homomorphism, and the centre Z p U q p g qq is contained in U .The Harish-Chandra isomorphism identifies Z p U q p g qq Ď U with a W -invariantsubalgebra of U . Precisely, we define an algebra automorphism of U by γ ´ ρ : U Ñ U , K α ÞÑ q p´ ρ,α q K α , for any α P Q , where ρ denotes the half sum of positive roots of g . Then thecomposite γ ´ ρ ˝ π is called the Harish-Chandra homomorphism, under which theimage of Z p U q can be described as follows.Recall that Q : “ t α | α P Q u and M “ Q X P . We defineU : “ x K λ | λ P M y to be the subalgebra of U spanned by K λ for all λ P M . Recall that the Weylgroup W of g acts naturally on U via w.K α “ K wα for any w P W and α P Q . Thisaction carries over to U , and we denote by p U q W the W -invariant subalgebra. Theorem 4.1. [Jan96, Theorem 6.25]
The Harish-Chandra homomorphism (4.1) γ ´ ρ ˝ π : Z p U q p g qq Ñ p U q W is an isomorphism. Note that for each λ P M “ Q X P , there exists a unique w P W such that wλ P M ` “ Q X P ` . Lemma 4.2.
The elements av p λ q “ ř w P W K wλ , λ P M ` form a basis of p U q W .Proof. Clearly, av p λ q P p U q W . If the finite sum f “ ř λ c λ K λ is W -invariant,then f “ | W | ÿ λ ÿ w P W c λ w.K λ “ | W | ÿ λ c λ p ÿ w P W K wλ q . Since ř w P W K wλ “ av p λ q for a unique dominant weight λ P M ` , the element f can be expressed uniquely as a linear combination of some av p λ q , λ P M ` . There-fore, av p λ q , λ P M ` form a basis for p U q W . (cid:3) Representation theoretical viewpoint.
Recall from Section 2.2 our con-struction of the central element C V . The image of C V under the Harish-Chandraisomorphism can be calculated as follows. Lemma 4.3.
Let V be a finite dimensional U q p g q -module with Π p V q Ď M , and let C V be the associated central element as defined in Definition 2.2. Then we have γ ´ ρ ˝ π p C V q “ ÿ µ P Π p V q m V p µ q K µ P p U q W , where m V p µ q “ dim L p λ q µ . XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 17
Proof.
Recall that π is an algebra homomorphism from U to U , where the latteris a subalgebra generated by all K ˘ i . Hence we have π p C V q “ π p Tr pp K ρ b q K V r R TV R V qq“ Tr pp K ρ b q K V q“ ÿ µ P Π p V q q p µ, ρ q m V p µ q K µ . By the definition of γ ´ ρ we have γ ´ ρ ˝ π p C V q “ ř µ P Π p V q m V p µ q K µ as desired. (cid:3) Recall that the character of V is defined by χ p V q “ ř µ P Π p V q m V p µ q e µ [Hum72,Jan96]. Hence γ ´ ρ ˝ π p C V q is equal to the character χ p V q with e µ replaced with K µ , and we may study the centre Z p U q p g qq from a representation-theoretic pointof view.We define ¯ R p U q p g qq : “ C p q q b Z R p U q p g qq , where R p U q p g qq is the Grothendieckring of the category of finite dimensional U q p g q -modules of type 1. It is well knownthat the isomorphism classes r L p λ qs , λ P P ` form a basis for ¯ R p U q p g qq . Define S p U q p g qq to be the subalgebra of ¯ R p U q p g qq which has a basis consisting of isomor-phism classes r L p λ qs , λ P M ` .Note that for any λ, µ P M ` there is a unique decomposition L p λ q b L p µ q – À ν c νλ,µ L p ν q , where ν P M ` and c νλ,µ is the multiplicity of L p ν q . Therefore, givenany isomorphism class r V s P S p U q p g qq , we have Π p V q Ď M and hence can definea corresponding central element C V . This gives rise to the following commutativediagram:(4.2) S p U q p g qq Z p U q p g qq p U q W . ξγ ´ ρ ˝ π Lemma 4.4.
There is an algebra isomorphism ξ : S p U q p g qq Ñ p U q W defined by ξ pr V sq “ ÿ µ P Π p V q m V p µ q K µ , where m V p µ q “ dim L p λ q µ .Proof. First, since ξ pr V sr W sq “ ξ pr V b W sq “ ξ pr V sq ξ pr W sq (see, e.g., [Hum72, § ξ is an algebra homomorphism. If ξ pr V sq “ ξ pr W sq , then V and W have the same character and hence V – W . Therefore ξ is injective. Itremains to show the surjectivity.Recall from Lemma 4.2 that p U q W has a basis av p λ q “ ř w P W K wλ , λ P M ` .It suffices to show that for any λ P M ` there exists a U q p g q -module V such that ξ pr V sq “ av p λ q . We use induction on λ .For the base case, if λ P M ` and there are no dominant weights lower than λ ,we have ÿ w P W K wλ “ | W || W λ | ÿ µ P W λ K µ “ | W || W λ | ξ pr L p λ qsq , where W λ : “ t wλ | w P W u denotes the W -orbit of λ . For general λ P M ` , wehave ÿ w P W K wλ “ | W || W λ | p ξ pr L p λ qsq ´ ÿ µ P Π p λ qz W λ dim L p λ q µ K µ q“ | W || W λ | ξ pr L p λ qsq ´ ÿ µ ă λ, µ P P ` | W µ || W λ | dim L p λ q µ ÿ w P W K wµ , where µ ă λ and µ P P ` imply that µ P M ` . By induction hypothesis, each sum ř w P W K wµ has a preimage in S p U q p g qq and so does av p λ q “ ř w P W K wλ . Thiscompletes the proof. (cid:3) The following is a consequence of the commutative diagram (4.2) and Lemma 4.4.
Corollary 4.5.
The algebra S p U q p g qq is isomorphic to Z p U q p g qq , with each iso-morphism class r V s P S p U q p g qq assigned to the central element C V . Proof of the main theorem.
Recall from Section 3.3 the monoid algebra C r M ` s generated by X λ , λ P Hilb p M ` q . The explicit relations among these gen-erators are given in Theorem 3.15 and Theorem 3.16 for the Lie algebra g oftype I and II, respectively. In the sequel, we shall consider the monoid algebra C p q qr M ` s “ C p q q b C C r M ` s , of which the generators and relations remain thesame. Lemma 4.6.
The monoid algebra C p q qr M ` s is isomorphic to S p U q p g qq , with eachgenerator X λ mapped to the isomorphism class r T p λ qs for λ P Hilb p M ` q .Proof. Recall that as a vector space C p q qr M ` s has a basis X λ with λ P M ` .If S p U q p g qq has a basis r T p λ qs , λ P M ` then there exists a bijective linear mapsending X λ to r T p λ qs for all λ P M ` . Moreover, this linear map preserves thealgebra structure since r T p λ qsr T p µ qs “ r T p λ q b T p µ qs “ r T p λ ` µ qs .Now it suffices to show that r T p λ qs , λ P M ` make up a basis for S p U q p g qq . Notethat there is the following decomposition(4.3) T p λ q – L p λ q ‘ à µ ă λ, µ P M ` L p µ q ‘ m λ,µ . Recall that by definition r L p λ qs , λ P M ` form a basis for S p U q p g qq . For any γ P M ` ,the set I p γ q “ t µ P M ` | µ ď γ u is finite. Then the elements r T p λ qs , λ P I p γ q arelinearly independent, since we may arrange I p γ q non-decreasingly under the partialorder and the transformation matrix between r T p λ qs , λ P I p γ q and r L p λ qs , λ P I p γ q is non-singular by (4.3). Moreover, r T p λ qs , λ P M ` is a spanning set of S p U q p g qq ,since each element of S p U q p g qq is a finite linear combination of r L p λ qs , λ P M ` and hence a finite linear combination of r T p λ qs , λ P M ` by using (4.3). Therefore, r T p λ qs , λ P M ` form a basis for S p U q p g qq . (cid:3) Corollary 4.7.
We have the following:(1) The isomorphism classes r T p λ qs , λ P Hilb p M ` q generate the algebra S p U q p g qq .(2) The elements C T p λ q , λ P Hilb p M ` q generate the centre Z p U q p g qq .Proof. Part (1) is a consequence of Lemma 4.6, and part (2) follows from thecombination of part (1) and Corollary 4.5. (cid:3)
We are in a position to prove our main theorem.
XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 19
Proof of Theorem 2.5.
By Corollary 4.7, Z p U q p g qq is generated by C T p λ q for all λ P Hilb p M ` q . Combining Corollary 4.5 and Lemma 4.6, we have the algebraisomorphism Z p U q p g qq – C p q qr M ` s , with each generator C T p λ q assigned to X λ for λ P Hilb p M ` q . Now the theorem follows from Theorem 3.15 and Theorem 3.16 byreplacing the ground field C with C p q q . (cid:3) Remark . In part (2) of Theorem 2.5, one can show that C L p λ q , λ P Hilb p M ` q also form a generating set of Z p U q p g qq , but they do not obey the same relations.For examples of these relations, refer to Example 3.17, where x λ should be replacedwith C T p λ q for λ P Hilb p M ` q . Appendix A. Proofs of commutative relations
In this appendix, we shall prove commutative relations in Lemma 2.1 and Propo-sition 2.3. First we prove Lemma 2.1.
Proof of Lemma 2.1.
For succinctness, we just prove that C p k q V commutes with E i and the other cases can be treated similarly. Since r Γ V , ∆ p x qs “
0, then we have rp Γ V q k , ∆ p x qs “ x P U. Assuming that p Γ V q k “ ř j A j b B j , we have0 “ Tr pp K ´ i K ρ b qrp Γ V q k , ∆ p E i qsq“ Tr p ÿ j p K ´ i K ρ b qr A j b B j , K i b E i ` E i b sq“ Tr p ÿ j p K ´ i K ρ b qp A j K i b B j E i ` A j E i b B j ´ K i A j b E i B j ´ E i A j b B j qq . This can be written as a sum of two terms: the first term isTr p ÿ j p K ´ i K ρ b qp A j K i b B j E i ´ K i A j b E i B j qq“ ÿ j Tr p K ρ A j qp B j E i ´ E i B j q“r C V , E i s , and the second term isTr p ÿ j p K ´ i K ρ b qp A j E i b B i ´ E i A j b B j qq“ ÿ j p Tr p K ´ i K ρ A j E i q ´ Tr p K ´ i K ρ E i A j qq B j , which is equal to 0 sinceTr p K ´ i K ρ A j E i q “ Tr p E i K ´ i K ρ A j q“ q ´p ρ ´ α i ,α i q Tr p K ´ i K ρ E i A j q “ Tr p K ´ i K ρ E i A j q . Therefore, we have r C p k q V , E i s “ (cid:3) Now we turn to prove Proposition 2.3. Let us start with the following lemma.
Lemma A.1.
Let ζ “ ζ V : U q p g q Ñ GL p V q be the linear representation associatedto V , and let K V be as defined in (2.6) . (1) For ď i, j ď n , we have K V p ζ p K ˘ i q b K ˘ j q “ p ζ p K ˘ i q b K ˘ j q K V K V p ζ p E i q b q “ p ζ p E i q b K i q K V , K V p b E i q “ p ζ p K i q b E i q K V , K V p ζ p F i q b q “ p ζ p F i q b K ´ i q K V , K V p b F i q “ p ζ p K ´ i q b F i q K V . (2) For any x P U q p g q , we have K V φ p ∆ p x qq “ ∆ p x q K V , where the first tensor factors in both ∆ p x q and φ p ∆ p x qq are regarded aselements in End p V q via the linear representation ζ .Proof. For part (1), we only prove the second equation; the others can be treatedsimilarly. Recall that P η : V Ñ V η is the linear projection onto the weight space V η of V . It can be verified easily that P η ζ p K ˘ i q “ ζ p K ˘ i q P η ,P η ζ p E i q “ ζ p E i q P η ´ α i ,P η ζ p F i q “ ζ p F i q P η ` α i , where P η ´ α i : “ P η ` α i : “
0) if η ´ α i R Π p V q (resp. η ` α i R Π p V q ). Usingthe second equality, we have K V p ζ p E i q b q “ ÿ η P Π p V q P Vη ζ p E i q b K η “ ÿ η P Π p V q ζ p E i q P η ´ α i b K η “ p ζ V p E i q b K i q K V . Part (2) is a direct consequence of part (1), and we take x “ E i as an example.Using φ p ∆ p E i qq “ K ´ i b E i ` E i b K ´ i , we have K V φ p ∆ p E i qq “ K V p ζ p K ´ i q b E i ` ζ p E i q b K ´ i q“ p ζ p K i q b E i ` ζ p E i q b q K V “ ∆ p x q K V . This completes the proof. (cid:3)
Now we are in a position to prove Proposition 2.3.
Proof of Proposition 2.3.
By (2.3) there are equations R V ∆ p x q “ φ p ∆ p x qq R V and R T ∆ p x q “ φ p ∆ p x qq R T . Applying the algebra homomorphism φ to the latter, wehave φ p R T ∆ p x qq “ φ p ∆ p x qq φ p R TV q . Then it follows thatΓ V ∆ p x q “ K V φ p R TV q R V ∆ p x q“ K V φ p R TV q φ p ∆ p x qq R V “ K V φ p ∆ p x qq φ p R TV q R V “ ∆ p x q Γ V , where the last equation follows from part (2) of Lemma A.1. (cid:3) XPLICIT GENERATORS AND RELATIONS OF THE CENTRE 21
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J. Phys. A. (1991), no. 5, 937–943.(Y Dai) School of Mathematical Sciences, University of Science and Technology ofChina, Heifei, 230026, China
Email address : [email protected] (Y Zhang) School of Mathematics and Statistics, University of Sydney, NSW 2006,Australia
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