Classification of Hopf superalgebras associated with quantum special linear superalgebra at roots of unity using Weyl groupoid
aa r X i v : . [ m a t h . QA ] J un Weyl groupoid of quantum superalgebra sl (2 |
1) at roots of unity
Alexander Mazurenko e-mail: [email protected]
Vladimir A. Stukopin
1) Moscow Institute of Physics and Technology (MIPT), e-mail: [email protected]) SMI of VSC RAS (South Mathematical Institute of Vladikavkaz Scientific Center of Russian Academy of Sciences)3) HSE (Higher School of Economics)
Abstract
We summarize the definition of the Weyl groupoid in order to investigate quantum superalgebras. The Weylgroupoid of sl (2 |
1) is constructed to this end. We prove that in this case quantum superalgebras associated withDynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopfsuperalgebras are connected via twists and isomorphisms. We build a PBW basis for each quantum superalgebra,and investigate how quantum superalgebras are connected with their classical limits, i. e. Lie superbialgebras.We find explicit multiplicative formulas for universal R -matrices and describe relations between them for eachrealization. Keywords:
Dynkin diagram, Lie superalgebras, Lie superbialgebras, Lusztig isomorphisms, PBW basis, quantumsuperalgebras, universal R-matrix, quantum Weyl groupoid.MSC Primary 16W35, Secondary 16W55, 17B37, 81R50, 16W30
1. Introduction
In this paper we investigate quantum deformation of the Lie superalgebra sl (2 |
1) at roots of unity. Our con-siderations are based on a Weyl groupoid, see Definition 3.2. We show how to associate quantum superalgebrasat roots of unity to Dynkin diagrams in Section 4.2. One of our main results is Theorem 4.2 where we show thatthe two realizations are isomorphic as superalgebras. We investigate how to build a PBW basis for each realizationin Theorem 4.3. In Theorem 4.5 we show how the two realizations are connected as Hopf superalgebras. We alsocompute universal R -matrices and describe relations between them for each realization.Our work is motivated by results obtained in [13] and reformulated in [7]. In these papers is defined a Weylgroupoid. In [24] is investigated a Weyl groupoid related to the Lie superalgebras. The case of quantum superal-gebras is considered in [14]. We were inspired also by results obtained in [20] and [21].We investigate only the Weyl groupoid of the Lie superalgebra sl (2 | sl ( m | n ), where m = n and m, n >
0, and can be extended to the more general caseof an arbitrary basic Lie superalgebra. Thus, our definition is based on the definition of the Weyl groupoid givenin [13], [7], [24] and contains the classical and quantum versions of the Weyl groupoid. We also give an explicitconstruction of the Weyl quantum groupoid using Lustig automorphisms in the spirit of the [21] and [22]. Usingthis explicit description of the Weyl quantum groupoid we investigate Hopf superalgebras structures and triangularstructures associated with Dynkin diagrams and show how they are connected via twists and isomorphisms.We will now give an outline of this paper. In Section 2 we recall basic facts about Lie superalgebras, and remindsome categorical definitions about supercategories. Next we describe Lie superalgebra sl (2 |
1) and show how toendow it with the Lie superbialgebra structure.Section 3 is divided in three parts. In Subsection 3.1 we give the definition of Cartan scheme, use it to constructa category called Weyl groupoid and show how to build W ( C ) the Weyl groupoid of the Lie superalgebra sl (2 | W ( C ) to the categoryof Lie superalgebras associated with Dynkin diagrams. Moreover, we show how to endow these Lie superalgebraswith the structure of Lie superbialgebras and investigate how they are related to each other.Section 4 is divided in six parts. In Subsection 4.1 we recall the definition of the quantized universal envelopingsuperalgebras. Next in Subsection 4.2 it is shown how to associate with Dynkin diagram quantum superalgebra at June 12, 2020 oots of unity. Subsection 4.3 contains auxiliary categorical definitions and results about Hopf superalgebras. InSubsection 4.4 we construct a faithful covariant functor from the W ( C ) to the category of superalgebras associatedwith Dynkin diagrams and prove that these superalgebras are isomorphic. In Subsection 4.5 we show how to builda PBW basis for these superalgebras. In Subsection 4.6 we investigate braided Hopf superalgebras associated withDynkin diagrams and show how they are connected via twists and isomorphisms.In this paper we use the following notation. Let N , Z and Q denote the sets of natural numbers, integers andrational numbers, respectively. Let k be an algebraically closed field of characteristic zero. We also use Iversonbracket defined by [ P ] = ( P is true;0 otherwise , where P is a statement that can be true or false.
2. Special Lie superalgebra sl (2 | As for the terminology concerning Lie superalgebras, we refer to [16], [9].A super vector space (superspace) V over field k is a k -vector space endowed with a Z -grading, in other words,it writes as a direct sum of two vector spaces V = V ¯0 ⊕ V ¯1 such as V ¯0 is the even part and V ¯1 is the odd part. Definea parity function |·| : V → Z for a homogeneous element x in a superspace by | x | = ¯ a , where v ∈ V ¯ a and ¯ a ∈ Z .A superalgebra A over the field k is a Z -graded algebra A = A ¯0 ⊕ A ¯1 over k . A Lie superalgebra is a superalgerba g = g ¯0 ⊕ g ¯1 with the bilinear bracket (the super Lie bracket) [ · , · ] : g × g → g which satisfies the following axioms,with homogeneous x, y, z ∈ g : [ x, y ] = − ( − | x || y | [ y, x ] , [ x, [ y, z ]] = [[ x, y ] , z ] + ( − | x || y | [ y, [ x, z ]] . A Lie superbialgebra ( g , [ · , · ] , δ ) (see [12], [18]) is a Lie superalgebra ( g , [ · , · ]) with a skew-symmetric linear map δ : g → g ⊗ g that preserves the Z -grading and satisfies the following conditions:( δ ⊗ id g ) ◦ δ − ( id g ⊗ δ ) ◦ δ = ( id g ⊗ τ g , g ) ◦ ( δ ⊗ id ) ◦ δ, (2.1) δ ([ x, y ]) = [ δ ( x ) , y ⊗ ⊗ y ] + [ x ⊗ ⊗ x, δ ( y )] , (2.2)where x, y ∈ g , id g is the identity map on g , 1 denotes the identity element in the universal enveloping algebra of g and τ V,W : V ⊗ W → W ⊗ V is the linear function given by τ V,W ( v ⊗ w ) = ( − | v || w | w ⊗ v (2.3)for homogeneous v ∈ V and w ∈ W .We use the well-known result (for more detail see [28], [29]). Proposition 2.1.
Let g be a Lie superalgebra of type A with associated Cartan matrix ( A = ( a ij ) i,j ∈ I , τ ) , where τ is a subset of I = { , , ..., n } . Then g is generated by h i , e i and f i for i ∈ I (whose parities are all even except for e t and f t , t ∈ τ , which are odd), where the generators satisfy the relations [ h i , h j ] = 0 , [ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j , [ e i , f j ] = δ ij h i and the ”super classical Serre-type” relations [ e i , f j ] = 0 , if a ij = 0 , [ e i , e i ] = [ f i , f i ] = 0 , if i ∈ τ, ( ad e i ) | a ij | e j = ( ad f i ) | a ij | f j = 0 , if i = j, and i = τ, [[[ e m − , e m ] , e m +1 ] , e m ] = [[[ f m − , f m ] , f m +1 ] , f m ] = 0 , if m − , m, m + 1 ∈ I and a mm = 0 , where for x ∈ g the linear mapping ad x : g → g is defined by ad x ( y ) = [ x, y ] for all y ∈ g . Denote by n + (resp. n − ) and h the subalgebra of g ( A, τ ) generated by e , ... , e n (resp. f , ... , f n ) and h , ... , h n . Then define by b + = h ⊕ n + (resp. b − = h ⊕ n − ) the positive Borel subalgebra (resp. the negative Borelsubalgebra) of g ( A, τ ).We remind some categorical definitions. Our notations here follow [5] (see also [6]). Let SV ec denote thecategory of superspaces and all (not necessarily homogeneous) linear maps. Set SV ec to be the subcategory of SV ec consisting of all superspaces but only the even linear maps (superspace morphisms). The tensor productequips SV ec with a monoidal structure, and the map u ⊗ v → ( − | u || v | v ⊗ u makes SV ec into a strict symmetricmonoidal category. 2 efinition 2.1.
1. A supercategory means a category enriched in SV ec , i. e. each morphism space is a super-space and composition induces an even linear map. A superfunctor between categories is a SV ec -enrichedfunctor, i. e. a functor F : A → B such that the function Hom A ( λ, µ ) → Hom B ( F λ, F µ ), f → F f is an evenlinear map for all λ, µ ∈ Obj( B ).2. For any supercategory A , the underlying category A is the category with the same objects as A but only itseven morphisms.3. Let sLieAlg be the supercategory which objects are Lie superalgebras over field k . A morphism f ∈ Hom sLieAlg ( V, W ) between Lie superalgebras ( V, [ · , · ] V ) and ( W, [ · , · ] W ) is a linear map of the underlyingvector spaces such that f ([ x, y ] V ) = [ f ( x ) , f ( y )] W for all x, y ∈ V .4. Let sBiLieAlg be the supercategory which objects are Lie superbialgebras over field k . A morphism f ∈ Hom sLieAlg ( V, W ) between Lie superbialgebras ( V, [ · , · ] V , δ V ) and ( W, [ · , · ] W , δ W ) is a linear map of the under-lying vector spaces such that f ([ x, y ] V ) = [ f ( x ) , f ( y )] W and ( f ⊗ f ) ◦ δ V ( x ) = δ W ◦ f ( x ) for all x, y ∈ V .We also need the following general result. Proposition 2.2.
Let f ∈ Hom sLieAlg ( g , g ) be an isomorphism. Suppose that g is a Lie superbialgebra with askew-symmetric even linear map δ g : g → g ⊗ g which satisfies (2.1) - (2.2) . Then f induces a Lie superbialgebrastructure on g , where a skew-symmetric even linear map δ g : g → g ⊗ g which satisfies (2.1) - (2.2) is definedby δ g := ( f ⊗ f ) ◦ δ g ◦ f − . The special Lie superalgebra sl (2 |
1) over k is the algebra M , ( k ) of 3 × k , Z -graded as sl (2 | ¯0 ⊕ sl (2 | ¯1 , where sl (2 | ¯0 = { X = diag ( A, D ) | Str ( X ) := tr ( A ) − tr ( D ) = 0 , A ∈ M , ( k ) , D ∈ M , ( k ) } , and sl (2 | ¯1 = { ( BC ) | B ∈ M , ( k ) , C ∈ M , ( k ) } , with the bilinear super bracket [ x, y ] = xy − ( − ab yx for x ∈ sl (2 | ¯ a , y ∈ sl (2 | ¯ b , ¯ a, ¯ b ∈ Z , on sl (2 | sl (2 |
1) over k the following elements: h = e , − e , , h = e , + e , ,e = e , , f = e , , e = e , , f = e , , e = [ e , e ] = e , , f = [ f , f ] = − e , , where e i,j ∈ M , ( k )denotes matrix with 1 at ( i, j )-position and zeros elsewhere. The elements h , h , e , f are even and e , f , e , f are odd. We have [ h i , h j ] = 0, [ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j , [ e i , f j ] = δ ij h i , [ e , e ] = [ f , f ] = 0, [ e , [ e , e ]] =[ f , [ f , f ]] = 0 with ( a ij ) the matrix A = (cid:18) − − (cid:19) . The Cartan subalgebra of sl (2 |
1) is the k -span h = h h , h i . Denote by h ∗ the dual space of h . sl (2 |
1) decomposesas a direct sum of root spaces h ⊕ L α ∈ h ∗ sl (2 | α , where sl (2 | α = { X | [ h, X ] = α ( h ) X, ∀ h ∈ h } . An α ∈ h ∗ − { } is called a root if the root space sl (2 | α is not zero. The root system for sl (2 |
1) is definedto be ∆ = { α ∈ h ∗ | sl (2 | α = 0 , α = 0 } . Define sets of even and odd roots, respectively, to be ∆ ¯0 = { α ∈ ∆ | sl (2 | α ∩ sl (2 | ¯0 = 0 } , ∆ ¯1 = { α ∈ ∆ | sl (2 | α ∩ sl (2 | ¯1 = 0 } . Thus we can define a parity function |·| ∆ : ∆ → Z by | x | ∆ = ¯ a if x ∈ ∆ ¯ a , where ¯ a ∈ Z .Consider the k -span d = h e , e , e i and it’s dual space d ∗ = h ǫ , ǫ , δ i . We define a non-degenerate symmetricbilinear form ( · , · ) : d ∗ × d ∗ → k by ( ǫ i , ǫ j ) = δ ij , ( ǫ i , δ ) = 0 , ( δ , δ ) = − i, j ∈ I , where I := { , } . We will make a convention to parameterize this basis by the set I (2 |
1) = { , , ¯1 } .Thus ǫ ¯1 := δ . We also need a set I S (2 |
1) = { , , } and a convention ǫ := δ .Notice that h ∗ ⊂ d ∗ . Then the root system ∆ ⊆ h ∗ has the form ∆ = ∆ ¯0 ⊕ ∆ ¯1 , where ∆ ¯0 = {± ( ǫ − ǫ ) } , ∆ ¯1 = {± ( ǫ − δ ) , ± ( ǫ − δ ) } . Accordingly, we also have the decomposition ∆ = ∆ + ∪ ∆ − , where ∆ + = { ǫ − ǫ , ǫ − δ , ǫ − δ } and ∆ − = { ǫ − ǫ , δ − ǫ , δ − ǫ } . We choose the basis τ = { α := ǫ − ǫ , α := ǫ − δ } . Theform ( · , · ) on d ∗ induces a non-degenerate symmetric bilinear form on h ∗ , which will be denoted by ( · , · ) as well. We3efine a natural pairing h· , ·i : h × h ∗ → k by linearity with h h i , α i = α ( h i ) for all i ∈ I, α ∈ τ . Introduce the totalorder on the root system ∆: δ − ǫ < δ − ǫ < ǫ − ǫ < < ǫ − ǫ < ǫ − δ < ǫ − δ . (2.4)The Cartan matrix is A = ( a ij = α j ( h i ); α j ∈ τ, i, j ∈ I ). One could describe A by the corresponding Dynkindiagram. Join vertex i with vertex j if a ij = 0. We need two types of vertices: ◦ if a ii = 2 and | α i | ∆ = 0; ⊗ if a ii = 0 and | α i | ∆ = 1, where i ∈ I .Define the linear function δ sl (2 | : sl (2 | → sl (2 | ⊗ sl (2 |
1) on the generators by δ sl (2 | ( h i ) = 0 , δ sl (2 | ( e i ) = 12 ( h i ⊗ e i − e i ⊗ h i ) , δ sl (2 | ( f i ) = 12 ( h i ⊗ f i − f i ⊗ h i )for i ∈ I , and extend it to all the elements of sl (2 |
1) using equation (2.2) and by linearity. Then sl (2 |
1) becomes aLie superbialgebra.
3. Weyl groupoid
We give a categorical definition of a Weyl groupoid. This enable us to describe the Weyl groupoid of sl (2 |
1) bygenerators and relations. We mention how it is connected with the category of Lie superalgebras.
We adopt to our purposes the definition of a Weyl groupoid which was introduced in [13] and reformulated in[7] (see also [24], [15], [21]). Thus we define Weyl groupoid as a supercategory. In Section 3.2 we give the examplehow Lie superalgebra sl (2 |
1) fits in our definition.In order to define Weyl groupoid we need auxiliary data. In this way we associate with an object of the Weylgroupoid a non-empty set which labels its Dynkin diagram, root basis, reflections which act on this basis, mapswhich indicate the direction of the action and integer coefficients used to define reflections. Conditions imposed onthe coefficients are analogous to that in the definition of a generalized Cartan matrix [17].
Definition 3.1.
Let A and D be non-empty sets, where A = ( a d ) d ∈ D , V = V ¯0 + V ¯1 a super vector space, τ d and γ d non-empty subspaces of V , where τ d ⊆ γ d for all d ∈ D , ρ dα : A → A a (partial) map for all α ∈ γ d and d ∈ D ,and C d = { c dα,β ∈ Z } α ∈ γ d ,β ∈ τ d for all d ∈ D . The tuple C = C ( A, D, V, ( τ d ) d ∈ D , ( γ d ) d ∈ D , ( ρ dα ) α ∈ γ d ,d ∈ D , ( C d ) d ∈ D )is called a Cartan scheme if for all d ∈ D ∃ ! ρ d ′ β for ρ dα : ρ d ′ β ρ dα = id , ρ dα ρ d ′ β = id , if ρ d ′ β ρ dα and ρ dα ρ d ′ β are defined, for all α ∈ γ d , where β ∈ γ d ′ , d ′ ∈ D ,2. c dα,α = 2 and c dα,β ≤
0, where α, β ∈ γ d with α = β ,3. if c dα,β = 0, then c dβ,α = 0, where α, β ∈ γ d ,4. c dα,β = c d ′ α,β , where ρ dα ( a d ) = a d ′ ∈ A , for all α, β ∈ τ d .Now we are able to formulate the definition of a Weyl groupoid where morphisms are generalizations of reflections. Definition 3.2.
Let C = C ( A, D, V, ( τ d ) d ∈ D , ( γ d ) d ∈ D , ( ρ dα ) α ∈ γ d ,d ∈ D , ( C d ) d ∈ D ) be a Cartan scheme. For all d ∈ D , α ∈ γ d and β ∈ τ d define σ dα ∈ Aut ( V ) by σ dα ( β ) = β − c dα,β α. (3.1)The Weyl groupoid of C is the supercategory W ( C ) such that Obj( W ( C )) = A and the morphisms are compositionsof maps σ dα with d ∈ D and α ∈ γ d , where σ dα is considered as an element in Hom W ( C ) ( a d , ρ dα ( a d )). The cardinalityof D is the rank of W ( C ). Definition 3.3.
A Cartan scheme is called connected if its Weyl groupoid is connected, that is, if for all a, b ∈ A there exists w ∈ Hom W ( C ) ( a, b ). The Cartan scheme is called simply connected, if it is connected and Hom W ( C ) ( a, a ) = { id a } for all a ∈ A .We characterize root systems in axiomatic way and also add explicit conditions that are imposed on reflections.4 efinition 3.4. Let C = C ( A, D, V, ( τ d ) d ∈ D , ( γ d ) d ∈ D , ( ρ dα ) α ∈ γ d ,d ∈ D , ( C d ) d ∈ D ) be a Cartan scheme. For all a d ∈ A let R a d ⊆ V , and define m a d α,β = | R a d ∩ ( N α + N β ) | for all α, β ∈ γ d and d ∈ D . We say that R = R ( C , ( R a ) a ∈ A )is a root system of type C , if it satisfies the following axioms:1. exists decomposition R a = R a + ∪ − R a + , for all a ∈ A ;2. R a d ∩ Z α = { α, − α } for all α ∈ γ d and d ∈ D ;3. σ dα ( R a d ) = R ρ dα ( a d ) for all α ∈ γ d and d ∈ D ;4. for all α ∈ γ d , β ∈ γ d ′ , where d, d ′ ∈ D , α = β , if α, β ∈ V , m a d α,β = 2 or α, β ∈ V ¯0 , m a d α,β is finite and ρ dα ρ d ′ β ρ dα is defined, then ( ρ d ′ β ρ dα ) m adα,β = id .The elements of the set R a , where a ∈ A , are called roots. The root system R is called finite if for all a ∈ A theset R a is finite. If R is a root system of type C , then we say that W ( R ) := W ( C ) is the Weyl groupoid of R . sl (2 | sl (2 | I = { , } and D = { , , ..., } . The elements of the set D will be used to label different Dynkin diagramsfor sl (2 | τ d = { α d := ǫ i − ǫ i , α d := ǫ i − ǫ i | { i , i , i } = I (2 | } ) d ∈ D . We require that τ d is thebasis of ∆ for all d ∈ D . Set τ = { α := ǫ − ǫ , α := ǫ − δ } . Consider the family of symmetric matrices A d = (( α di , α dj )) i,j ∈ I . Define a family of tuples A = ( a d = ( G d A d , τ d )) d ∈ D , where G d is a diagonal matrix for all d ∈ D which diagonal elements belong to {− , } .Let c α,β := − max { k ∈ Z | β + kα ∈ ∆ } for α, β ∈ ∆. Set ( γ d = τ d ∪ { α d := α | α ∈ ∆ ¯0 , ± α / ∈ τ d , c α,β ≤ β ∈ τ d } ) d ∈ D . Introduce a family of sets ( C d = { c dα,β := c α,β | α ∈ γ d , β ∈ τ d } ) d ∈ D .Denote the (usual) left action of the symmetric group S on I S (2 |
1) by ⊲ : S × I S (2 | → I S (2 |
1) and on ∆ by (cid:9) : S × ∆ → ∆, where s (cid:9) ( ǫ j − ǫ j ) = ǫ s⊲j − ǫ s⊲j , for s ∈ S , j , j ∈ I S (2 | ρ dα : A → A for all α ∈ γ d , d ∈ D , such that ρ dα ( a d ) = a b , where b ∈ D , α = ǫ j − ǫ j and τ b = { α ′ k := ( j , j ) (cid:9) α k = σ dα ( α k ) | α k ∈ τ d , k ∈ I } .Consider the simply connected Cartan scheme C = C ( A, D, h ∗ , ( τ d ) d ∈ D , ( γ d ) d ∈ D , ( ρ dα ) α ∈ γ d ,d ∈ D , ( C d ) d ∈ D ). Wecall W ( C ) the Weyl groupoid of sl (2 |
1) (see Fig. 1). Notice that R = R ( C , (∆ = ∆ a ) a ∈ A ) is the root system oftype C . (cid:13) ǫ − ǫ N ǫ − δ N ǫ − δ N δ − ǫ N δ − ǫ (cid:13) ǫ − ǫ (cid:13) ǫ − ǫ N ǫ − δ N ǫ − δ N δ − ǫ N δ − ǫ (cid:13) ǫ − ǫ d =1 σ ǫ − δ σ δ − ǫ d =3 σ ǫ − δ σ δ − ǫ d =5 d =2 σ ǫ − δ σ δ − ǫ d =4 σ ǫ − δ σ δ − ǫ d =6 σ ǫ − ǫ σ ǫ − ǫ σ ǫ − ǫ σ ǫ − ǫ σ ǫ − ǫ σ ǫ − ǫ Figure 1:
Dynkin Diagrams of sl (2 | W ( C ) is the category generated by morphisms (recall (3.1)) B = { σ d ± ( ǫ − ǫ ) , σ d ± ( ǫ − δ ) ∈ Hom( W ( C )) | d ∈ D } (3.2)and by conditions and relations: for all σ dα ∈ B there exists unique σ d ′ β ∈ B , where α ∈ γ d , β ∈ γ d ′ and d, d ′ ∈ D ,such that m a d α,β = 2, a d ′ = ρ dα ( a d ), a d = ρ d ′ β ( a d ′ ) and σ d ′ β σ dα = id a d , σ dα σ d ′ β = id a d ′ ; (3.3) σ ǫ − δ = σ ǫ − ǫ σ ǫ − δ σ ǫ − ǫ ; σ ǫ − δ = σ ǫ − ǫ σ ǫ − δ σ ǫ − ǫ . (3.4)It is easy to see that an element σ d ′ β σ dα is undefined if a d ′ = ρ dα ( a d ), where α ∈ γ d and β ∈ γ d ′ .5 .3. Connection with the category of Lie superalgebras Recall the definition of the category sLieAlg (see Definition 2.1). We are able to construct the covariant faithfulfunctor F : W ( C ) → sLieAlg.Fix G d A d = ( g α di ,d ( α di , α dj )) i,j ∈ I and τ d = { α d := ǫ i − ǫ i , α d := ǫ i − ǫ i } ∈ Obj( W ( C )) for d ∈ D , i , i , i ∈ I (2 |
1) and g α di ,d ∈ {− , } for i ∈ I . Recall Proposition 2.1 and define a Lie superalgebra g ( A d , τ d ) to be a Liesuperalgebra generated by { h β,d , e β,d , f β,d | β ∈ τ d } and by relations[ h α di ,d , e α dj ,d ] = g α di ,d ( α di , α dj ) e α dj ,d , [ h α di ,d , f α dj ,d ] = − g α di ,d ( α di , α dj ) f α dj ,d , [ e α,d , f β,d ] = δ α,β h α,d , (3.5)[ e α,d , e α,d ] = [ f α,d , f α,d ] = 0 , if | α | = 1 , (3.6)( ad e α,d ) | ( α,β ) | e β,d = ( ad f α,d ) | ( α,β ) | f β,d = 0 , if α = β, and | α | 6 = 1 , (3.7)where α di , α dj ∈ τ d , i, j ∈ I , α, β ∈ τ d and δ α,β denotes the Kronecker delta. Thus the action on objects is given bythe formula F (( G d A d , τ d )) = g ( G d A d , τ d ) , (3.8)where A d = (( α di , α dj )) i,j ∈ I , G d is a diagonal matrix which diagonal elements belong to {− , } and d ∈ D . Noticethat sl (2 |
1) = g ( A , τ ).Consider a generator σ d α ∈ Hom W ( C ) ( a d , a d ) (3.2) and fix a free isomorphism L d ,d ∈ Hom sLieAlg ( g ( G d A d , τ d ) , g ( G d A d , τ d )) , where α ∈ τ d and d , d ∈ D . Define F ( σ d α ) = L d ,d and F ( σ d − α ) = L d ,d , where L d ,d := L − d ,d . It iseasy to see that F is indeed the covariant faithful functor. We give an example of the family of isomorphisms { F ( σ ) ∈ Hom(sLieAlg) } σ ∈B . For any α ∈ ∆ and l , l ∈ Z let r α ;( l ,l ) := [ α > l + [ α < l . We use thenotations introduced in this section to formulate Proposition 3.1.
There exist the unique covariant faithful functor F : W ( C ) → sLieAlg which satisfies equation (3.8) and for all σ d α ∈ B F ( σ d α ) = L d ,d , (3.9) where σ d α ∈ Hom W ( C ) ( a d , a d ) , α ∈ γ d , d , d ∈ D , and L d ,d : g ( G d A d , τ d ) → g ( G d A d , τ d ) are unique isomorphisms in sLieAlg satisfying equations (3.10) - (3.17) below. L d ,d ( h α,d ) = − g α,d g − α,d h − α,d , L d ,d ( h β,d ) = g β,d ( g − α,d h − α,d + g σ d α ( β ) ,d h σ d α ( β ) ,d ) , (3.10) L d ,d ( e α,d ) = ( − r α ;( | α | , g α,d f − α,d , (3.11) L d ,d ( f α,d ) = ( − r α ;(0 , | α | ) g − α,d e − α,d , (3.12) L d ,d ( e β,d ) = ( − | σ d α ( α ) || σ d α ( β ) | g β,d g x,d g y,d [ e x,d , e y,d ] , (3.13) L d ,d ( f β,d ) = [ f y,d , f x,d ] , (3.14) where x = σ d α ( β ) , y = σ d α ( α ) , if | α | = 1 and α > , otherwise x = σ d α ( α ) , y = σ d α ( β ) ; α = β and α, β ∈ τ d , L d ,d ( h β,d ) = − g β,d g σ d α ( β ) ,d h σ d α ( β ) ,d , (3.15) L d ,d ( e β,d ) = ( − r α ;(0 , g β,d f σ d α ( β ) ,d , (3.16) L d ,d ( f β,d ) = ( − r α ;(1 , g σ d α ( β ) ,d e σ d α ( β ) ,d , (3.17) where α / ∈ τ d and β ∈ τ d .One has L d ,d = ( L d ,d ) − . roof. The proof follows from the considerations preceding the statement and from the direct computations.We can endow Lie superalgebras b d = g ( G d A d , τ d ) with the structure of a Lie superbialgebra. Recall that G d A d = ( g α di ,d ( α di , α dj )) i,j ∈ I , where g α di ,d ∈ {− , } for i ∈ I . Define the linear function δ b d : b d → b d ⊗ b d on thegenerators by δ b d ( h α,d ) = 0 , δ b d ( e α,d ) = g α,d h α,d ⊗ e α,d − e α,d ⊗ h α,d ) , δ b d ( f α,d ) = g α,d h α,d ⊗ f α,d − f α,d ⊗ h α,d ) , (3.18) δ b d ( e α + β,d ) = h α + β,d ⊗ e α + β,d − e α + β,d ⊗ h α + β,d + t α,β (( − | e α,d || e β,d | e β,d ⊗ e α,d − e α,d ⊗ e β,d ) , (3.19) δ b d ( f α + β,d ) = h α + β,d ⊗ f α + β,d − f α + β,d ⊗ h α + β,d + t α,β (( − | f α,d || f β,d | f α,d ⊗ f β,d − f β,d ⊗ f α,d ) , (3.20)where h α + β,d = ( g α,d h α,d + g β,d h β,d ), e α + β,d = [ e α,d , e β,d ], f α + β,d = [ f β,d , f α,d ], t α,β = (( α, β ) + ( β, α )) = ( α, β ), α = β and α, β ∈ τ d . Extend δ b d to all the elements of b d by linearity. Then b d becomes a Lie superbialgebra. Proposition 3.2.
There exist unique isomorphisms W d ,d ∈ Hom sLieAlg ( g ( G d A d , τ d ) , g ( G d A d , τ d )) such that W d ,d ( h α d i ,d ) = g α d i ,d g α d i ,d h α d i ,d , W d ,d ( e α d i ,d ) = g α d i ,d g α d i ,d e α d i ,d , W d ,d ( f α d i ,d ) = f α d i ,d , where α d i ∈ τ d , α d i ∈ τ d , i ∈ I and ( d , d ) ∈ { (1 , , (2 , , (3 , , (4 , , (5 , , (6 , } . Also W d ,d ∈ Hom sBiLieAlg ( g ( G d A d , τ d ) , g ( G d A d , τ d )) and, moreover, are the isomorphisms in sBiLieAlg.Proof. The proof follows from the direct computations.
Remark 3.1.
1. It is easy to see by direct computations that positive (negative) Borel subalgebras of g ( G d A d , τ d )and g ( G d A d , τ d ) are not isomorphic for d ∈ { , , , } and d ∈ { , } .2. Notice that g ( G d A d , τ d ) and g ( G d A d , τ d ) are isomorphic as Lie superbialgebras for d ∈ { , } and d ∈{ , } . Indeed, there exists the unique isomorphism W d ,d ∈ Hom sBiLieAlg ( g ( G d A d , τ d ) , g ( G d A d , τ d ))such that W d ,d ( h α d i ,d ) = g α d i ,d g α d j ,d h α d j ,d , W d ,d ( e α d i ,d ) = g α d i ,d g α d j ,d e α d j ,d , W d ,d ( f α d i ,d ) = f α d j ,d , where i = j and i, j ∈ I . Also W − d ,d is defined by W − d ,d ( h α d i ,d ) = g α d j ,d g α d i ,d h α d j ,d , W − d ,d ( e α d i ,d ) = g α d j ,d g α d i ,d e α d j ,d , W − d ,d ( f α d i ,d ) = f α d j ,d , where i = j and i, j ∈ I .3. It is easy to see by direct computations that g ( G d A d , τ d ) and g ( G d A d , τ d ) are not isomorphic as Lie su-perbialgebras for d ∈ { , , , } and d ∈ { , } . Indeed, let f ∈ Hom sBiLieAlg ( g ( G d A d , τ d ) , g ( G d A d , τ d ))be an isomorphism. Then δ a d ◦ f ( e λ,d ) = g λ,d g α,d g β,d δ a d ( γ [ e α,d , e β,d ] + γ [ f β,d , f α,d ]) == g λ,d g α,d g β,d ( h α + β,d ⊗ ( γ [ e α,d , e β,d ] + γ [ f β,d , f α,d ]) −− ( γ [ e α,d , e β,d ] + γ [ f β,d , f α,d ]) ⊗ h α + β,d −− t α,β ( γ e α,d ⊗ e β,d + γ f β,d ⊗ f α,d )++ t α,β (( − | e α,d || e β,d | γ e β,d ⊗ e α,d + ( − | f α,d || f β,d | γ f α,d ⊗ f β,d ));( f ⊗ f ) ◦ δ a d ( e λ,d ) = g λ,d g α,d g β,d ( h α + β,d ⊗ ( γ [ e α,d , e β,d ] + γ [ f β,d , f α,d ]) −− ( γ [ e α,d , e β,d ] + γ [ f β,d , f α,d ]) ⊗ h α + β,d ) , where α = α d , β = α d , h α + β,d = ( g α,d h α,d + g β,d h β,d ), t α,β = (( α, β ) + ( β, α )) = ( α, β ), | λ | = 0, λ ∈ τ d and γ , γ ∈ k . Notice that t α,β = 0. Thus we get the contradiction.Notice that the image of the functor F : W ( C ) → sLieAlg defined above is the subcategory SL in the categorysLieAlg. Recall that objects of SL (3.8) are also Lie superbialgebras defined by (3.18) - (3.20). Thus it followsfrom Proposition 2.2 that morphisms in SL (3.9) are also morphisms in category sBiLieAlg. Consequently, SL isthe subcategory in the category sBiLieAlg. 7 . Weyl groupoid of quantum superalgebra sl (2 |
1) at roots of unity
Here we recall the notion of quantized universal enveloping superalgebras (for more detail see [28], [29], [10],[11]).Let K = k [[ h ]], where h is an indeterminate and view K as a superspace concentrated in degree ¯0. Let M be amodule over K . Consider the inverse system of K -modules p n : M n /h n M → M n − = M/h n − M. Let ˆ M = lim ←− M n be the inverse limit. Then ˆ M has the natural inverse limit topology (called the h -adic topology).Let V be a k -superspace. Let V [[ h ]] to be the set of formal power series. The superspace V [[ h ]] is naturally a K -module and has a norm given by || v n h n + v n +1 h n +1 + ... || = 2 − n , where v n = 0 and v i ∈ V for i ≥ n . The topology defined by this norm is complete and coincides with the h -adictopology. We say that a K -module M is topologically free if it is isomorphic to V [[ h ]] for some k -module V .Let M and N be topologically free K -modules. We define the topological tensor product of M and N to be \ M ⊗ K N which we denote by M ⊗ N . It follows that M ⊗ N is topologically free and that V [[ h ]] ⊗ W [[ h ]] = ( V ⊗ W )[[ h ]]for k -module V and W .We say a (Hopf) superalgebra defined over K is topologically free if it is topologically free as a K -module andthe tensor product is the above topological tensor product.A quantized universal enveloping (QUE) superalgebra A is a topologically free Hopf superalgebra over k [[ h ]] suchthat A/hA is isomorphic as a Hopf superalgebra to universal enveloping superalgebra U ( g ) for some Lie superalgebra g . We use the following result proved in the non-super case in [8] and in the super case in [2]. Proposition 4.1.
Let A be a QUE superalgebra: A/hA ∼ = U ( g ) . Then the Lie superalgebra g has a natural structureof a Lie superbialgebra defined by δ ( x ) = h − (∆(˜ x ) − ∆ op (˜ x )) mod h, (4.1) where x ∈ g , ˜ x ∈ A is a preimage of x , ∆ is a comultiplication in A and ∆ op := τ U ( g ) ,U ( g ) ◦ ∆ (for the definition of τ U ( g ) ,U ( g ) see (2.3) ). Definition 4.1.
Let A be a QUE superalgebra and let ( g , [ · , · ] , δ ) be the Lie superbialgebra defined in Proposition4.1. We say that A is a quantization of the Lie superbialgebra g .Let t be an indeterminate. Set (cid:20) m + nn (cid:21) t = n − Y i =0 t m + n − i − t − m − n + i t i +1 − t − i − ∈ k [ t ] , where m, n ∈ N . Denote by e ht = X n ≥ t n h n n ! ∈ k [[ h ]] . (4.2)Put q = e h/ and recall notations introduced in Section 2. We need the following result, see [19] and [29]. Theorem 4.1.
Let ( g , A, τ ) be a Lie superalgebra of type A , where Cartan matrix A is symmetrizable, i. e. thereare nonzero rational numbers g i for i ∈ I such that d i a ij = d j a ji . There exists an explicit QUE Hopf superalgebra U DJh ( g , A, τ ) . The Hopf superalgebra U DJh ( g , A, τ ) is defined as the k [[ h ]] -superalgebra generated by the elements h i , e i and f i , where i ∈ I (all generators are even except e i and f i for i ∈ τ which are odd), and the relations: [ h i , h j ] = 0 , [ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j , [ e i , f j ] = δ i,j q g i h i − q − g i h i q g i − q − g i , nd the quantum Serre-type relations e i = f i = 0 for i ∈ I such that a ii = 0 , [ e i , e j ] = [ f i , f j ] = 0 for i, j ∈ I such that a ij = 0 and i = j, | a ij | X v =0 ( − v (cid:20) | a ij | v (cid:21) q gi e | a ij |− vi e j e vi = | a ij | X v =0 ( − v (cid:20) | a ij | v (cid:21) q gi f | a ij |− vi f j f vi = 0 for i = j , i / ∈ τ and i, j ∈ I , [[[ e m − , e m ] q , e m +1 ] q − , e m ] = [[[ f m − , f m ] q , f m +1 ] q − , f m ] = 0 , if m − , m, m + 1 ∈ I and a mm = 0 . [ · , · ] v is the bilinear form defined by [ x, y ] v = xy − ( − | x || y | vyx on homogeneous x, y and v ∈ k [[ h ]] . The comulti-plication, counit and antipode are given by ∆( h i ) = h i ⊗ ⊗ h i , ∆( e i ) = e i ⊗ q g i h i ⊗ e i , ∆( f i ) = f i ⊗ q − g i h i + 1 ⊗ f i ; ǫ ( h i ) = ǫ ( e i ) = ǫ ( f i ) = 0; S ( h i ) = − h i , S ( e i ) = − q − g i h i e i , S ( f i ) = − f i q g i h i , where i ∈ I .4.2. Definition of quantum superalgebra at roots of unity We introduce the quantum superalgebra of sl (2 |
1) for any Dynkin diagram using notations from section3.2 and3.3 (see [4], [30]). Let q be an algebraically independent and invertible element over Q . Consider Lie superalgebra( g , G d A d , τ d ), where G d A d = ( g α di ,d ( α di , α dj )) i,j ∈ I and τ d = { α d , α d } ∈ Obj( W ( C )), d ∈ D and g α di ,d ∈ {− , } for i ∈ I . Let U dq := U q ( g , G d A d , τ d ) for any d ∈ D be the associative superalgebra over Q ( q ) with 1, generated by { e i,d , f i,d , k i,d , k − i,d | i ∈ I } , satisfying XY = Y X for
X, Y ∈ { k i,d , k − i,d | i ∈ I } , (4.3) k i,d k − i,d = k − i,d k i,d = 1 , e i,d k j,d = q − g αdj ,d ( α dj ,α di ) k j,d e i,d , k j,d f i,d = q − g αdj ,d ( α dj ,α di ) f i,d k j,d , (4.4)[ e i,d , f j,d ] = e i,d f j,d − ( − | α di || α dj | f j,d e i,d = δ i,j k g αdi ,d i,d − k − g αdi ,d i,d q g αdi ,d − q − g αdi ,d , (4.5) e i,d = f i,d = 0 , if | α di | = 1 , (4.6)[ e i,d , [ e i,d , e j,d ] q − ] q = [ f i,d , [ f i,d , f j,d ] q − ] q = 0 , if | α di | = 0 , (4.7)where δ i,j is the Kronecker delta, α di , α dj ∈ τ d and i, j ∈ I ; [ · , · ] v is the bilinear form defined by [ x, y ] v = xy − ( − | x || y | vyx on homogeneous x, y and v ∈ Q ( q ). The parity function is defined by | k i,d | = 0 and | e i,d | = | f i,d | = | α di | ,where α di ∈ τ d and i ∈ I .Also U dq ( g ) is a Hopf superalgebra which comultiplication ∆, counit ǫ and antipode S are∆ d ( k i,d ) = k i,d ⊗ k i,d , ∆ d ( e i,d ) = e i,d ⊗ k g αdi ,d i,d ⊗ e i,d , ∆ d ( f i,d ) = f i,d ⊗ k − g αdi ,d i,d + 1 ⊗ f i,d ; (4.8) ǫ d ( k i,d ) = 1 , ǫ d ( e i,d ) = ǫ d ( f i,d ) = 0; S d ( k ± i,d ) = k ∓ i,d , S d ( e i,d ) = − k − g αdi ,d i,d e i,d , S d ( f i,d ) = − f i,d k g αdi ,d i,d , (4.9)where i ∈ I . Proposition 4.2.
There exists the unique injective morphism of Hopf superalgebras for d ∈ Df : U q ( g , G d A d , τ d ) → U DJh ( g , G d A d , τ ) , where τ = { α | | α | = 1 , α ∈ τ d } , such that for i ∈ If ( q ) = e h , f ( k i ) = e hhi , f ( k − i ) = e − hhi , f ( e i ) = e i , f ( f i ) = f i . roof. The result follows from the direct computations.Fix Hopf superalgebra U q ( g , G d A d , τ d ) for d ∈ D . It follows from Proposition 4.2 that we can consider U q asa supersubalgebra in U DJh ( g , G d A d , τ ). Thus we are able to apply equation (4.1) to U q . Then it easy to see thatthe Lie superalgebra ( g , G d A d , τ d ) has a natural structure of a Lie superbialgebra defined by equation (3.18) andextended to all the elements of g by requiring (2.2).From now on let q be a root of unity of odd order p . Then it is easy to see that U dq can be defined in the sameway. Now we introduce some auxiliary notations. Notation 4.1.
Define for all d ∈ D and i ∈ Ik α di := k i,d , e α di := e i,d and f α di := f i,d , where α di ∈ τ d . Set a total order ≤ on τ d in the following way α d < α d . We put e γ,d := [ e α,d , e β,d ] q − gα and f γ,d := [ f β,d , f α,d ] q gα , (4.10)where α ∈ τ d : | α | = 0, β ∈ τ d : | β | = 1; α = α d and β = α d , if | α d | = | α d | = 1; γ = α + β ∈ ∆. Denote τ de := τ d ∪ { α + β ∈ ∆ | α, β ∈ τ d } . Introduce a total order ≤ on τ de in the following way α d < α d + α d < α d .Thus define a total order ≤ on the generators of U dq and elements defined by (4.10): set k α ≤ k β , if α ≤ β , where α, β ∈ τ d ; e α ≤ e β and f α ≤ f β , if α ≤ β , where α, β ∈ τ de ; f α < k λ < e β , where α, β ∈ τ de and λ ∈ τ d .Let H denote the set of all functions h : τ de → { , , ..., p − } such that h ( α ) ≤ | α | = 1. Define for any d ∈ De h,d := Y β ∈ τ de e h ( β ) β,d and f h,d := Y β ∈ τ de f h ( β ) β,d with h ∈ H, (4.11)where the product is taken with respect to the selected order (in ascending order).Let H denote the set of all functions g : τ d → { , , ..., p − } . In the same way we use the standard order onnatural numbers to define the product (taken in ascending order) for any d ∈ Dk g,d := Y β ∈ τ d k g ( β ) β,d with g ∈ H . (4.12)For any α ∈ ∆ and l , l ∈ Z let r α ;( l ,l ) := [ α > l + [ α < l . For all n ∈ Z set [ n ] := q n − q − n q − q − .Denote by exp q ( x ) := ∞ X n =0 x n ( n ) q ! , where x is an indeterminate and for all k ∈ N we set ( k ) q := q k − q − and (0) q ! := 1, ( n ) q ! := (1) q (2) q ... ( n ) q , if n ∈ Z + .Now we define the quantum superalgebras U dq := U q ( g , G d A d , τ d ) of sl (2 |
1) at roots of unity for the Dynkindiagrams labeled by d ∈ D , see also [23], Proposition 3.1. Definition 4.2.
For any d ∈ D let U dq be the quotient of the Hopf superalgebra U dq by the two-sided Z -gradedHopf ideal I generated by the following elements: e pα,d , f pα,d , where | α | = 0 and α ∈ τ de , (4.13) k pi,d − , where i ∈ I. (4.14)For convenience we preserve the same notations for U dq as for U dq , where d ∈ D . Notice that Proposition 4.1 isnot true for U dq , as when we specialize to a root of unity q then the equation (4.1) doesn’t hold.10 .3. Category of Hopf superalgebras and twists We consider some categorical definitions and general results about Hopf superalgebras. Our notations herefollow [1].
Definition 4.3.
1. Let sAlg be the strict monoidal supercategory ([5], Definition 1.4) of unital associative super-algebras over field Q ( q ). A morphism f ∈ Hom sAlg ( V, W ) between superalgebras (
V, µ V , η V ) and ( W, µ W , η W )is a linear map of the underlying vector spaces such that f ◦ µ V = µ W ◦ ( f ⊗ f ) and f ◦ η V = η W .2. Let HAlg be the strict monoidal category of Hopf algebras over field Q ( q ).3. Let sHAlg be the strict monoidal supercategory of Hopf superalgebras over field Q ( q ). A morphism f ∈ Hom sHAlg ( V, W ) between Hopf superalgebras (
V, µ V , η V , ∆ V , ǫ V , S V ) and ( W, µ W , η W , ∆ W , ǫ W , S W ) is a linearmap of the underlying vector spaces such that f ◦ µ V = µ W ◦ ( f ⊗ f ), f ◦ η V = η W , ( f ⊗ f ) ◦ ∆ V = ∆ W ◦ f , ǫ W ◦ f = ǫ V and f ◦ S V = S W ◦ f .Let ( H, µ, η, ∆ , ǫ, S ) be a Hopf superalgebra in sHAlg. Recall some results about twists, see [20], [3], [27]. Definition 4.4.
A twist for H is an invertible even element J ∈ H ⊗ H which satisfies(∆ ⊗ id H )( J )( J ⊗
1) = ( id H ⊗ ∆)( J )(1 ⊗ J ) , (4.15)( ǫ ⊗ id H )( J ) = ( id H ⊗ ǫ )( J ) = 1 , (4.16)where id H is the identity map of H . Proposition 4.3.
Let ( H, µ, η, ∆ , ǫ, S ) be a Hopf (super)algebra in HAlg (sHAlg) and let J be a twist for H . Thenthere is a new Hopf (super)algebra H J := ( H, µ, η, ∆ J , ǫ, S J ) defined by the same (super)algebra and counit, and ∆ J ( h ) := J (∆( h )) J − , S J ( h ) := U ( S ( h )) U − for all h ∈ H . Here U = ( id H ⊗ S )( J ) and is invertible. Moreover, U − = ( S ⊗ id H )( J − ) . If H is a quasi-cocommutative (braided) Hopf (super)algebra with an universal R -matrix R then H J is also quasi-cocommutative(braided) with the universal R -matrix R J : R J := τ H,H ( J ) R H J − , where τ H,H is defined by (2.3) .Proof.
The result follows from the definition and properties of a comultiplication, antipode and universal R -matrix.The Hopf (super)algebra H J := ( H, µ, η, ∆ J , ǫ, S J ) (4.17)is called the twisted Hopf (super)algebra by the twist J . The same notation we use for the quasi-cocommutative(braided) Hopf (super)algebra H J := ( H, µ, η, ∆ J , ǫ, S J , R J ). We call J the twist of type 1.Fix χ ∈ Hom
HAlg ( V, W ) (Hom sHAlg ( V, W )). If χ is an isomorphism we call it the twist of type 2. Proposition 4.4.
Let χ ∈ Hom
HAlg ( V, W ) (Hom sHAlg ( V, W ) ) be the twist of type for objects ( V, µ V , η V , ∆ V , ǫ V , S V ) and ( W, µ W , η W , ∆ W , ǫ W , S W ) . Let for any w ∈ W ∆ χW ( w ) := ( χ ⊗ χ ) ◦ ∆ V ( χ − ( w )) , ǫ χW ( w ) := ǫ V ◦ χ − ( w ) , S χW ( w ) := χ ◦ S V ( χ − ( w )) . Then V χ := ( W, µ W , η W , ∆ χW , ǫ χW , S χW ) is a Hopf (super)algebra isomorphic to V . If V is a quasi-cocommutative(braided) Hopf (super)algebra with an universal R -matrix R V then W is also quasi-cocommutative (braided) withthe universal R -matrix R χW : R χW = ( χ ⊗ χ )( R V ) . Proof.
The result follows from the definition of a Hopf (super)algebra morphism and direct computations.The Hopf (super)algebra V χ := ( W, µ W , η W , ∆ χW , ǫ χW , S χW ) (4.18)is called the twisted Hopf (super)algebra by the isomorphism χ . The same notation we use for the quasi-cocommutative(braided) Hopf (super)algebra V χ := ( W, µ W , η W , ∆ χW , ǫ χW , S χW , R χW ).11 .4. Lusztig type isomorphisms In this section we show that morphisms of category W ( C ) can be represented by isomorphisms between thequantum superalgebras U dq , where d ∈ D , in category sAlg. Compare with the Section 3.3, see also [14], [22].We introduce the covariant faithful functor F q : W ( C ) → sAlg. Fix ( G d A d , τ d ) ∈ Obj( W ( C )) for d ∈ D . Theaction on objects is given for all d ∈ D by the formula F q (( G d A d , τ d )) = U dq . (4.19)Consider a generator σ d α ∈ Hom W ( C ) ( a d , a d ) (3.2) and fix a free isomorphism T d ,d ∈ Hom sAlg ( U d q , U d q ), where α ∈ γ d and d , d ∈ D . Define F q ( σ d α ) = T d ,d and F q ( σ d − α ) = T d ,d , where T d ,d := T − d ,d . It is easy to see that F q is indeed the covariant faithful functor.We give an example of the family of isomorphisms { F q ( σ ) ∈ Hom(sAlg) } σ ∈B . Call them Lusztig type iso-morphisms. We use notations introduced in 4.1. Remind that G d A d = ( g α di ,d ( α di , α dj )) i,j ∈ I , where α di ∈ τ d and g α di ,d ∈ {− , } for i ∈ I . Theorem 4.2.
There exist the unique covariant faithful functor F q : W ( C ) → sAlg which satisfies equation (4.19) and for all σ d α ∈ B F q ( σ d α ) = T d ,d , (4.20) where σ d α ∈ Hom W ( C ) ( a d , a d ) , α ∈ γ d , d , d ∈ D , and T d ,d : U d q → U d q are unique isomorphisms in sAlg satisfying equations (4.21) - (4.28) below. T d ,d ( k α,d ) = k − g α,d g − α,d − α,d , T d ,d ( k β,d ) = k g β,d g − α,d − α,d k g β,d g σd α ( β ) ,d σ d α ( β ) ,d , (4.21) T d ,d ( e α,d ) = ( − r α ;( | α | , q ( α,α )2 r α ;( − , g α,d f − α,d k r α ;(1 , − g − α,d − α,d , (4.22) T d ,d ( f α,d ) = ( − r α ;(0 , | α | ) q ( α,α )2 r α ;(1 , − g − α,d k r α ;( − , g − α,d − α,d e − α,d , (4.23) T d ,d ( e β,d ) = ( − | σ d α ( α ) || σ d α ( β ) | g β,d g x,d g y,d [ e x,d , e y,d ] q z , (4.24) T d ,d ( f β,d ) = [ f y,d , f x,d ] q − z , (4.25) where x = σ d α ( β ) , y = σ d α ( α ) , if | α | = 1 and α > , otherwise x = σ d α ( α ) , y = σ d α ( β ) ; z = r α ;(1 , − (2[ α > | α || β | − , α = β and α, β ∈ τ d , T d ,d ( k β,d ) = k − g β,d g σd α ( β ) ,d σ d α ( β ) ,d , (4.26) T d ,d ( e β,d ) = ( − r α ;(0 , g β,d f σ d α ( β ) ,d k r α ;(1 , − g σd α ( β ) ,d σ d α ( β ) ,d , (4.27) T d ,d ( f β,d ) = ( − r α ;(1 , g σ d α ( β ) ,d k r α ;( − , g σd α ( β ) ,d σ d α ( β ) ,d e σ d α ( β ) ,d , (4.28) where α / ∈ τ d and β ∈ τ d .One has T d ,d = ( T d ,d ) − , where d , d ∈ D .Proof. The proof follows from the considerations preceding the statement and from the direct computations.Notice that the image of the functor F q : W ( C ) → sAlg defined above is the subcategory QS in the categorysAlg. Recall that objects of QS (4.19) are also Hopf superalgebras defined by (4.8) - (4.9). Thus it follows fromProposition 4.4 that morphisms in QS (4.20) are also morphisms in category sHAlg. Consequently, QS is also thesubcategory in the category sHAlg. Recall that we defined in the analogous way the subcategory SL in the categorysBiLieAlg, see Section 3.3. Proposition 4.5.
Categories QS and SL are equivalent, where the equivalence H : QS → SL is defined on objectsby H ( U dq ) = g ( G d A d , τ d ) and on morphisms by H ( id U dq ) = id g ( G d A d ,τ d ) and H ( T d ,d ) = L d ,d , where d, d , d ∈ D .Proof. It is easy to see that the functor H is full, faithful and dense. The result follows.12 .5. PBW basis of U dq We build for any d ∈ D the PBW basis of U dq . Remind the notations and conventions introduced in 4.1. Seealso [25], [26]. Theorem 4.3.
The elements Y = { f h − ,d · k h ,d · e h + ,d | h − , h + ∈ H, h ∈ H } form a Q ( q ) -basis of the quantum superalgebra U dq , where d ∈ D .Proof. The statement immediately follows from the proof ([23], Theorem 3 . Q ( q ) super vector space L generated by X = { e α,d , f α,d , k β,d , k − β,d | α ∈ τ de , β ∈ τ d } . Introduce a pair( T ( L ) , i ) where T ( L ) is the tensor superalgebra of the vector superspace L and i is the canonical inclusion of L in T ( L ). We identify for convenience X and i ( X ). Rewrite equations (4.3) - (4.7) and (4.13) - (4.14) in T ( L ) in thefollowing way a ⊗ b − ( − | a || b | q δ ( a,b ) b ⊗ a − [ a, b ] q δ ( a,b ) = 0 , (4.29)where a, b ∈ X, [ a, b ] q δ ( a,b ) ∈ T ( L ) , δ : X × X → {− , − , , , } , a ⊗ p − c a = 0 , (4.30)where a ∈ X , | a | = 0 and c a ∈ Q ( q ) ⊂ T ( L ). Denote by J a Z -graded two-sided ideal in T ( L ) generated byrelations (4.29) and (4.30). Notice that U dq ∼ = T ( L ) /J .The index of x i ⊗ x i ⊗ ... ⊗ x i n ∈ T ( L ) is defined to be the number of pairs ( l, m ) with l < m but x i l > x i m ,where x i j ∈ X , i j ∈ τ de and j ∈ N . We adopt in a natural way the definition of the index on elements of U dq . Denoteby G the monomials having index 0. Notice that G = Y in U dq . Thus, we want to prove that G forms the basis of U dq considered as the Q ( q )-superspace.Notice that each element in U dq is a Q ( q )-linear combination of unit and standard monomials. Indeed, it is easyto prove by induction on degree and index of elements in U dq that this is the case.Further show that elements of G are linear independent in U dq . Let R be the polynomial ring R = Q ( q )[ z , ..., z | X | ].Endow R with the structure of the superalgebra by defining the parity function | z i | = | f α,d | , | z j + | τ de | | = | k α dj ,d | and | z i + | I | + | τ de | | = | e α,d | , where α ∈ τ de and α dj ∈ τ d follow in ascending order, i ∈ { , ..., | τ de |} and j ∈ { , ..., I } . Nowwe want to construct a morphism of superspaces U dq → R which restriction on G is a monomorphism that takes allthe elements of G to linear independent polynomials in R . Then the result follows. Thus, we proof that there is asuperspace morphism θ : T ( L ) → R which satisfies the following relations θ (1) = 1 , θ ( f α,d ) = z i , θ ( k α dj ,d ) = z j + | τ de | , θ ( e α,d ) = z i + | I | + | τ de | , where α ∈ τ de and α dj ∈ τ d follow in ascending order, i ∈ { , ..., | τ de |} and j ∈ { , ..., I } , θ ( x i ⊗ x i ⊗ ... ⊗ x i n ) = z i z i ...z i n , if x i ≤ x i ≤ ... ≤ x i n ,θ ( x i ⊗ x i ⊗ ... ⊗ x i k ⊗ x i k +1 ⊗ ... ⊗ x i n ) − ( − | x ik || x ik +1 | q δ ( x ik ,x ik +1 ) θ ( x i ⊗ x i ⊗ ... ⊗ x i k +1 ⊗ x i k ⊗ ... ⊗ x i n ) == θ ( x i ⊗ x i ⊗ ... ⊗ [ x i k , x i k +1 ] ⊗ ... ⊗ x i n )for all x i , x i , ..., x i n ∈ X and 1 ≤ k < n , where x i j ∈ X , i j ∈ τ de and j ∈ N , θ ( x ⊗ p ) = c x , where x ∈ X, | x | = 0 and c x ∈ Q ( q ).Recall that T ( L ) = Q ( q )1 and T n ( L ) = N ni =1 L , where n ∈ N . Denote by T n,j ( L ) a linear subspace T n ( L )spanned by all monomials x i ⊗ x i ⊗ ... ⊗ x i n , which have index less or equal to j . Thus, T n, ( L ) ⊂ T n, ( L ) ⊂ ... ⊂ T n ( L ) . We define θ : T ( L ) → R by θ (1) = 1. Suppose inductively that θ : T ( L ) ⊕ T ( L ) ... ⊕ T n − ( L ) → R has already beendefined satisfying the required conditions. We will show that θ can be extended to θ : T ( L ) ⊕ T ( L ) ... ⊕ T n ( L ) → R .We define θ : T n, ( L ) → R by θ ( x i ⊗ x i ⊗ ... ⊗ x i n ) = z i z i ...z i n n . We suppose θ : T n,i − → R has already been defined, thus giving a superspacemorphism from θ : T ( L ) ⊕ T ( L ) ... ⊕ T n − ( L ) ⊕ T n,i − ( L ) → R satisfying the required conditions. We wish todefine θ : T n,i ( L ) → R .Assume that the monomial x i ⊗ x i ⊗ ... ⊗ x i n has the index i ≥ x i k ≥ x i k +1 . Then define θ ( x i ⊗ ... ⊗ x i k ⊗ x i k +1 ⊗ ... ⊗ x i n ) = θ ( x i ⊗ ... ⊗ [ x i k , x i k +1 ] ⊗ ... ⊗ x i n )+ (4.31)+( − | x ik || x ik +1 | q δ ( x ik ,x ik +1 ) θ ( x i ⊗ ... ⊗ x i k +1 ⊗ x i k ⊗ ... ⊗ x i n ) . This definition is correct as both terms on the right side of the equation belong to a super vector space T ( L ) + T ( L ) + ... + T n − ( L ) + T n,i − ( L ). We state that the definition 4.31 doesn’t depend on the choise of the pair( x i k , x i k +1 ), where x i k > x i k +1 . Let ( x i j , x i j +1 ) be another pair, where x i j > x i j +1 . There are two different possiblesituations: 1. x i j > x i k +1 , 2. x i j = x i k +1 . It is easy to see that the statement is true in both cases.Further define θ ( x i ⊗ ... ⊗ x i k ⊗ x ⊗ p ⊗ x i k + p +1 ⊗ ... ⊗ x i n ) = c x θ ( x i ⊗ ... ⊗ x i k ⊗ x i k + p +1 ⊗ ... ⊗ x i n ) , (4.32)where p ≤ n , x ∈ X , | x | = 0 and c x ∈ Q ( q ). Let the monomial x i ⊗ ... ⊗ x i k ⊗ x ⊗ p ⊗ x i k + p +1 ⊗ ... ⊗ x i n have theindex i ≥
1. Then it is easy to see that the order of application of equations (4.31) and 4.32 doesn’t affect on result.Notice, in this connection, that θ ( x p ⊗ y ) = θ ( y ⊗ x p ) = c x θ ( y ) , if x > y , where x, y ∈ X , | x | = 0 and c x ∈ Q ( q ), θ ( y ⊗ x p ) = θ ( x p ⊗ y ) = c x θ ( y ) , if y > x , where x, y ∈ X , | x | = 0 and c x ∈ Q ( q ).Thus we have defined a map θ : T n,i ( L ) → R . A linear extension of this map gives us θ : P n − j =0 T j ( L ) ⊕ T n,i ( L ) → R , which satisfies the required conditions. Since T n = T n,r for sufficiently large r , we can consider amap θ : P nj =0 T j ( L ) → R . Since T ( L ) = T ⊕ P i ∈ N T i ( L ), we get a map θ : T ( L ) → R , which satisfies the requiredconditions. It is easy to see that θ : T ( L ) → R annihilates J . Thus, θ induces the required superspace morphism¯ θ : T ( L ) /J → R , that is ¯ θ : U dq → R . R -matrix We describe how the standard Hopf superalgebra structures associated with each Dynkin diagram are related.We begin with
Proposition 4.6.
Let σ d α ∈ Hom (( G d A d , τ d ) , ( G d A d , τ d )) , where σ d α ∈ B , d , d ∈ D and α ∈ τ d such that | α | = 0 . There exist unique isomorphism W d ,d ∈ Hom sAlg ( U d q , U d q ) defined by W d ,d ( k α d i ,d ) = k g αd i ,d g αd i ,d α d i ,d , W d ,d ( e α d i ,d ) = g α d i ,d g α d i ,d e α d i ,d , W d ,d ( f α d i ,d ) = f α d i ,d , where α d i ∈ τ d , α d i ∈ τ d and i ∈ I .Also W d ,d ∈ Hom sHAlg ( U d q , U d q ) and, moreover, is the isomorphism in sHAlg.Proof. The proof follows from the direct computations.Isomorphisms described in Theorem 4.2 induce Hopf superalgebra structures being twists of type 2, see Section4.3. We want to understand how the new Hopf superalgebra structure is related to the standard one defined byequations (4.8) - (4.9).Let F q ( σ d α ) = T d ,d , where σ d α ∈ B , α ∈ τ d and d , d ∈ D . Order the roots in ∆ using the equation (2.4).Introduce auxiliary maps Q T d ,d : U d q → U d q defined by Q T d ,d = W d ,d ◦ T − d ,d (4.33)for | α | = 0, and elements W T d ,d ∈ U d q ⊗ U d q defined by W T d ,d = exp q (( − g − α,d ( q − q − ) k − g − α,d − α,d e − α,d ⊗ f − α,d k g − α,d − α,d ) , (4.34)for | α | = 1 and α > W T d ,d = exp q (( − g − α,d ( q − q − ) f − α,d ⊗ e − α,d ) (4.35)for | α | = 1 and α <
0. 14 heorem 4.4. Q T d ,d and W T d ,d are twists of type or for U d q . Moreover, the Hopf superalgebra U d q coincideswith the Hopf superalgebra (( U d q ) T d ,d ) P Td ,d , where P T d ,d is equal to Q T d ,d or W T d ,d .Proof. It is easy to see that Q T d ,d (4.33) is the twist of type 1. One has to check that equations (4.15) -(4.16) are true for W T d ,d defined by (4.34) - (4.35). To prove the second statement build Hopf superalgebra(( U d q ) T d ,d ) P Td ,d , where P T d ,d is equal to Q T d ,d ( W T d ,d ), using Proposition 4.4 (Proposition 4.3) and theresult will follow.Let a d = ( G d A d , τ d ) and a d n = ( G d n A d n , τ d n ) be arbitrary objects in W ( C ) for d , d n ∈ D and n ∈ N . Itfollows from the definition of W ( C ) and equations (3.2) - (3.4) that there is a morphism θ ∈ Hom W ( C ) ( a d , a d n ).Let θ = σ d n − α in − ...σ d α i σ d α i , where σ d k α ik ∈ Hom W ( C ) ( a d k , a d k +1 ), i k ∈ I , d k ∈ D , α i k ∈ τ d k and k, n ∈ N . It followsfrom Theorem 4.2 that the functor F q : W ( C ) → sAlg induces a Lusztig type isomorphism T d ,d n : U d q → U d n q insAlg such that F q ( θ ) = T d ,d n and T d ,d n = T d n − ,d n ...T d ,d T d ,d . Thus we can consider Hopf superalgebra( U d q ) ω := ( U d q ) (( ... (((( T d ,d ) PTd ,d ) Td ,d ) PTd ,d ) ... ) Tdn − ,dn ) PTdn − ,dn , where P T di,dj is equal to Q T di,dj (4.33) or W T di,dj (4.34) - (4.35) for i, j ∈ N , see formulas (4.17) and (4.18) fornotations. Theorem 4.5.
Hopf superalgebra ( U d q ) ω coincides with the Hopf superalgebra U d n q .Proof. The result follows from Theorem 4.2 and Theorem 4.4.
Remark 4.1.
1. Notice that for all d ∈ D and α ∈ τ d we have S d ( k ± α,d ) = k ± α,d , S d ( e α,d ) = q − ( α,α ) e α,d , S d ( f α,d ) = q ( α,α ) f α,d . Then there is no isomorphism f ∈ Hom sHAlg ( U d q , U d q ) for ( d , d ) ∈ { (1 , , (6 , } . Indeed, suppose f is asuch isomorphism. Then f = S d ◦ f = S d ◦ f ◦ S d = f ◦ S d = ⇒ S d = id U d q . We get a contradiction.2. Notice that U d q and U d q are isomorphic as Hopf superalgebras for d ∈ { , } and d ∈ { , } . Indeed, thereexists the unique isomorphism W qd ,d ∈ Hom sHAlg ( U d q , U d q ) such that W qd ,d ( k α d i ,d ) = k g αd i ,d g αd j ,d α d j ,d , W qd ,d ( e α d i ,d ) = g α d i ,d g α d j ,d e α d j ,d , W qd ,d ( f α d i ,d ) = f α d j ,d , where i = j and i, j ∈ I . Also ( W qd ,d ) − is defined by( W qd ,d ) − ( h α d i ,d ) = k g αd j ,d g αd i ,d α d j ,d , ( W qd ,d ) − ( e α d i ,d ) = g α d j ,d g α d i ,d e α d j ,d , ( W qd ,d ) − ( f α d i ,d ) = f α d j ,d , where i = j and i, j ∈ I .Notice that we can construct new Hopf superalgebras using Proposition 4.4 and Theorem 4.2. Example 4.1.
For simplicity of notation, we assume that all Cartan matrices are symmetric, i. e. G d is theidentity matrix for all d ∈ D (see Section 3.2 for the definition of G d ).Consider T , : U q → U q defined in Theorem 4.2. Recall that τ = { α = ǫ − ǫ , α = ǫ − δ } and let α = α + α . Then we get a new Hopf superalgebra structure on U q :∆ T , ( k i, ) = k i, ⊗ k i, , ∆ T , ( e α , ) = e α , ⊗ k − , ⊗ e α , , ∆ T , ( e α , ) = ∆ ( e α , ) + ( q − q − ) f α , k , ⊗ [ e α , , e α , ] q , ∆ T , ( f α , ) = f α , ⊗ k , + 1 ⊗ f α , , ∆ T , ( f α , ) = ∆ ( f α , ) − ( q − q − )[ f α , , f α , ] q − ⊗ k − , e α , ; ǫ T , ( k i, ) = 1 , ǫ T , ( e α i , ) = ǫ T , ( f α i , ) = 0;15 T , ( k ± i, ) = k ∓ i, , S T , ( e α , ) = − k , e α , , S T , ( e α , ) = [[ e α , , e α , ] q , f α , k − , ] q − k − , ,S T , ( f α , ) = − f α , k − , , S T , ( f α , ) = [ k , e α , , [ f α , , f α , ] q − ] q k , , where ∆ T , = ∆ T , U q , ǫ T , = ǫ T , U q , S T , = S T , U q and i ∈ I .Consider T , : U q → U q . Remind that τ = { α = ǫ − δ , α = δ − ǫ } and let α = α + α . Then∆ T , ( k i, ) = k i, ⊗ k i, , ∆ T , ( e α , ) = ∆ ( e α , ) + (1 − q ) k − , e α , ⊗ f α , k , , ∆ T , ( e α , ) = e α , ⊗ k , + k , ⊗ e α , , ∆ T , ( f α , ) = ∆ ( f α , ) + (1 − q − ) k − , e α , ⊗ f α , k , , ∆ T , ( f α , ) = f α , ⊗ k − , + k − , ⊗ f α , ; ǫ T , ( k i, ) = 1 , ǫ T , ( e α i , ) = ǫ T , ( f α i , ) = 0; S T , ( k ± i, ) = k ∓ i, , S T , ( e α , ) = − ( q − q − ) f α , k − , k , e α , − q − k − , e α , , S T , ( e α , ) = − k − , e α , ,S T , ( f α , ) = ( q − q − ) f α , k , k − , e α , − q f α , k , , S T , ( f α , ) = − f α , k , , where ∆ T , = ∆ T , U q , ǫ T , = ǫ T , U q , S T , = S T , U q and i ∈ I .We know that the universal R -matrix ¯ R of U q ( sl (2 | R = ˜ RK, where ˜ R = exp q (( q − q − ) e α ⊗ f α ) exp q (( q − q − ) e α ⊗ f α ) exp q (( − q − q − ) e α ⊗ f α ) ×× exp q (( − q − q − q − ) e α e α ⊗ f α f α ) ,K = p − X ≤ i ,j ,i ,j ≤ p − q i (2 i − j ) − j i k i k j ⊗ k i k j . It follows from Theorem 4.5 and Corollary 4.6 that R -matrix ¯ R for U q with the standard Hopf superalgebrastructure defined by equations (4.8) - (4.9) is¯ R = ( τ U q ,U q ◦ W T , ) ¯ R T W − T , , where ¯ R T = ( T , ⊗ T , )( ¯ R ) = ˜ R T K T , , ˜ R T = exp q (( q − q − ) e α , ⊗ f α , ) exp q (( − q − q − ) f α , k α , ⊗ k − α , e α , ) exp q (( q − q − ) e α , ⊗ f α , ) ×× exp q (( − q − q − ) f α , k α , e α , ⊗ f α , k − α , e α , ) ,K T , = p − X ≤ i ,r ,i ,r ≤ p − q i r + i r k i α k r α ⊗ k i α k r α ,W T , = 1 ⊗ − ( q − q − ) k − α , e α , ⊗ f α , k α , ,W − T , = 1 ⊗ q − q − ) k − α , e α , ⊗ f α , k α , . It follows from the direct computations that¯ R = exp q (( q − q − ) e α , ⊗ f α , ) exp q (( q − q − ) e α , ⊗ f α , ) exp q (( q − q − ) e α , ⊗ f α , ) K T , . eferences [1] Aissaoui, S., Makhlouf, A.: On classification of finite-dimensional superbialgebras and Hopf superalgebras.arXiv e-prints (2014). arXiv:1301.0838[2] Andruskiewitsch, N.: Lie superbialgebras and Poisson-Lie supergroups. Abh. Math. Sem. Univ. Hamburg. 63,147163 (1993)[3] Andruskiewitsch, N., Etingof, P., Gelaki, S.: Triangular Hopf algebras with the Chevalley property. MichiganMath. J. 49(2), 277-298 (2001)[4] Benkart, G., Kang, S., Kashiwara, M.: Crystal bases for the quantum superalgebra U q ( gl ( m, n )). J. Amer.Math. Soc. 13, 295-331 (2000)[5] Brundan, J., Ellis, A.P.: Monoidal supercategories. Comm. Math. Phys. 351(3), 10451089 (2017)[6] Brundan, J., Ellis, A.P.: Super Kac-Moody 2-categories. Proc. Lond. Math. Soc. 115(5), 925973 (2017)[7] Cuntz, M., Heckenberger, I.: Weyl groupoids with at most three objects. J. Pure Appl. Algebra. 213(6),11121128 (2009)[8] Drinfeld, V.G.: Quantum groups. J Math Sci. 41, 898915 (1988)[9] Frappat, L., Sciarrino, A., Sorba, P.: Structure of basic Lie superalgebras and of their affine extensions.Commun. Math. Phys. 121, 457500 (1989)[10] Geer, N.: EtingofKazhdan quantization of Lie superbialgebras. Adv. Math. 207(1), 138 (2006)[11] Geer, N.: Some remarks on quantized Lie superalgebras of classical type. J. Algebra. 314(2), 565580 (2007)[12] Gould, M.D., Zhang, R.B., Bracken, A.J.: Lie bi-superalgebras and the graded classical YangBaxter equation.Rev. Math. Phys. 3(02), 223240 (1991)[13] Heckenberger, I., Yamane, H.: A generalization of Coxeter groups, root systems, and Matsumotos theorem.Math. Z. 259(2), 255276 (2008)[14] Heckenberger, I., Spill, F., Torrielli, A., Yamane, H.: Drinfeld second realization of the quantum affine super-algebras of D (1) (2 , x ) via the Weyl groupoid. Publ. Res. Inst. Math. Sci. Kyoto B. 8, 171216 (2008)[15] Hoyt, C.: Classification of finite-growth contragredient Lie superalgebras. arXiv e-prints (2009).arXiv:1606.05303[16] Kac, V.G.: Lie superalgebras. Advances in Math. 26, 8-96 (1977)[17] Kac, V.G.: Infinite dimensional Lie algebras. Cambridge Univ. Press (1990)[18] Karaali, G.: A New Lie Bialgebra Structure on sl (2 | R -matrix for quantized (super)algebras. Comm. Math. Phys. 141(3), 599617 (1991)[20] Khoroshkin, S.M., Tolstoy, V.N.: Twisting of quantum (super)algebras. Connection of Drinfelds and Cartan-Weyl realizations for quantum affine algebras. arXiv e-prints (1994). arXiv:hep-th/9404036[21] Levendorskii, S.Z., Soibel’man, Ya.S.: Quantum Weyl group and multiplicative formula for the R-matrix of asimple Lie algebra. Functional Analysis and Its Applications. 25(2), 143-145 (1991)[22] Lusztig, G.: Introduction to Quantum Groups. Modern Birkhuser Classics. Birkhuser/Springer, New York(2010)[23] Mazurenko, A., Stukopin, V.A.: R -matrix for quantum superalgebra sl (2 |
1) at roots of unity and its applicationto centralizer algebras. arXiv e-prints (2019). arXiv:1909.11613[24] Serganova, V.: Kac-Moody superalgebras and integrability. Progress in Mathematics. 288, 169218 (2011)1725] Tsymbaliuk, A.: PBWD bases and shuffle algebra realizations for U v ( L sl n ) , U v ,v ( L sl n ) , U v ( L sl ( m | nn