On the double of the (restricted) super Jordan plane
aa r X i v : . [ m a t h . QA ] A ug ON THE DOUBLE OF THE (RESTRICTED) SUPERJORDAN PLANE
NICOLÁS ANDRUSKIEWITSCH AND HÉCTOR PEÑA POLLASTRI
Abstract.
We consider the super Jordan plane, a braided Hopf algebraintroduced–to the best of our knowledge–in [AAH1], and its restrictedversion in odd characteristic introduced in [AAH2]. We show that theirDrinfeld doubles give rise naturally to Hopf superalgebras justifying aposteriori the adjective super given in loc. cit . These Hopf superalgebrasare extensions of super commutative ones by the enveloping, respectivelyrestricted enveloping, algebra of osp (1 | . Contents
Introduction 11. Recollections 32. Preliminaries 63. The double of the super Jordan plane 134. The double of the restricted super Jordan plane 19References 26
Introduction
The context.
Let k be an algebraically closed field. The super Jordan planeis the graded algebra B presented by generators x , x with defining relations x , x x − x x − x x , (0.1)where x = x x + x x . It is known that B has Gelfand-Kirillov dimension2. The braided vector space ( V, c ) with basis { x , x } and braiding c ( x i ⊗ x ) = − x ⊗ x i , c ( x i ⊗ x ) = ( − x + x ) ⊗ x i , i = 1 , , (0.2)determines a structure of braided Hopf algebra on B in the sense of [T].The Jordan plane and the super Jordan plane play a central role in thestudy of pointed Hopf algebras over abelian groups with finite Gelfand-Kirillov dimension in [AAH1], assuming char k = 0 . This material is based upon work supported by the National Science Foundation underGrant No. DMS-1440140 while N. A. was in residence at the Mathematical SciencesResearch Institute in Berkeley, California, in the Spring 2020 semester. The work of N.A. and H. P. P. was partially supported by CONICET and Secyt (UNC).
Let p > be a prime. Assume that char k = p . Motivated by [CLW] thatdeals with the restricted Jordan plane, it is shown in [AAH2] that manyanalogues of the Nichols algebras from [AAH1] have finite dimension (see[ABDF] for examples in characteristic 2). Among them the restricted superJordan plane i.e. the algebra presented by x , x with relations (0.1) and x p , x p . (0.3)We began the study of the Drinfeld doubles of (suitable bosonizations of)the Jordan plane and its restricted analogue in [AP] showing among otherresults that they fit into exact sequences relating the enveloping algebra of sl ( k ) , the algebras of functions on some algebraic groups and their restrictedanalogues. In the present paper we carry out a similar analysis of the superJordan plane and its restricted version. We start in Section 1 with briefdiscussions on various topics needed later: Yetter-Drinfeld supermodules,restricted Lie superalgebras and Nichols algebras. In Section 2 we first recordpresentations of duals and doubles of finite-dimensional Hopf algebras arisingas bosonizations; this is essentially straightforward but useful for furthercomputations. Then we present the objects of our interest: the super Jordanplane, its restricted version and their duals. Finally we deal with differentdescriptions as bosonization of the same Hopf algebra. This allows to definealternatively the Hopf superalgebras e D and D discussed below. The double of the super Jordan plane.
Here we just need that char k = 2 .As in [AAH1] we realize the braided vector space ( V, c ) with c given by (0.2)as Yetter-Drinfeld module over the group algebra kZ , hence we have theHopf algebra H := B kZ . Then we consider the dual super Jordan plane B d described just before Lemma 2.6. The Sweedler dual of kZ is spanned bythe characters of Z and the Lie algebra of the one-dimensional torus. Henceits smallest Hopf subalgebra realizing B d is isomorphic to k [ ζ ] ⊗ k C . Here C N stands for the cyclic group of order N and ζ is an indeterminate. Thuswe may consider K := B d k [ ζ ] ⊗ k C ) and define D = H ⊲⊳ K op with respectto a suitable pairing between H and K op . It turns out that there existsa Hopf superalgebra e D such that D ≃ e D k C ; this justifies a posteriorithe adjective super given to B . Thus the study of D reduces to that of e D .We present basic properties of e D in Proposition 3.3 including the definingrelations and a PBW-basis. Next we show in Theorem 3.5 that e D fits into anexact sequence of Hopf superalgebras O ( G ) ֒ → e D ։ U ( osp (1 | , where G isan algebraic supergroup explicitly described. For the next result, Theorem3.7, we need char k = p > ; then e D is free module of finite rank over a centralHopf subalgebra Z = O ( B ) where B is a solvable connected algebraic group.We close Section 3 establishing some basic ring-theoretical properties of e D . The double of the restricted super Jordan plane.
In Section 4 we assume that char k = p > . We realize ( V, c ) with the braiding (0.2) in k C p k C p YD . Let D ( H ) be the Drinfeld double of the bosonization H = B ( V ) k C p . Again N THE DOUBLE OF THE SUPER JORDAN PLANE 3 there exists a Hopf superalgebra D such that D ( H ) ≃ D k C , thus we mayfocus on D . We present basic properties of D in Proposition 4.4 and show inTheorem 4.6 an exact sequence of Hopf superalgebras R ι ֒ −→ D π −→→ u ( osp (1 | where R is a local commutative Hopf algebra and u ( osp (1 | is the restrictedenveloping algebra. We conclude that the simple D -modules are the sameas those of u ( osp (1 | and we present them as quotients of Verma modules.See Theorem 4.7 and Proposition 4.12.The extensions of Hopf superalgebras mentioned above fit into a 9-termcommutative diagram where all columns and rows are exact sequences: O ( G ) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) O ( B ) (cid:127) _ (cid:15) (cid:15) / / / / O ( G a ) (cid:127) _ (cid:15) (cid:15) O ( G ) (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) e D / / / / (cid:15) (cid:15) (cid:15) (cid:15) U ( osp (1 | (cid:15) (cid:15) (cid:15) (cid:15) R (cid:31) (cid:127) / / D / / / / u ( osp (1 | (0.4)See Theorem 4.14. This is an analogue of [AP, (0.2)] for the Jordan plane.1. Recollections
Conventions.
We denote by F p the field of p elements. If ℓ < n ∈ N ,then we set I ℓ,n = { ℓ, ℓ + 1 , . . . , n } , I n = I ,n . Let A be an algebra and a , . . . , a n ∈ A , n ∈ N . We denote by k h a , . . . , a n i the subalgebra generatedby a , . . . , a n . We identify V ∗ ⊗V ∗ ≃ ( V ⊗V ) ∗ , where V is a finite-dimensionalvector space, by h f ⊗ f ′ , x ⊗ y i = h f, x ih f ′ , y i , f, f ′ ∈ V ∗ , x, y ∈ V . (1.1)The cyclic group of order n is denoted by C n and the infinite cyclic groupby Γ . They are always written multiplicatively. The center of a group G isdenoted Z ( G ) ; similarly for the center of an algebra.Let L be a Hopf algebra. The kernel of the counit ε is denoted L + , theantipode (always assumed bijective) by S or by S L , the space of primitiveelements of L is denoted by P ( L ) and the group of group-likes by G ( L ) . Thespace of g, h -primitives is P g,h ( L ) = { x ∈ L : ∆( x ) = x ⊗ h + g ⊗ x } where g, h ∈ G ( L ) . We assume that the reader has some familiarity with the theoryof Hopf algebras particularly with the notions of Yetter-Drinfeld module andbosonization (or biproduct), see e.g. [R]. The category of Yetter-Drinfeldmodules over L is denoted by LL YD .The category of super vector spaces is denoted by sVec . If X ∈ sVec , and x ∈ X i , then we write | x | = i . We set |X | := X ∪ X . The symmetrictensor category sVec is identified with a full tensor subcategory of k C k C YD .An object M ∈ k C k C YD is in sVec if the two following conditions are satisfied:(0) For every a ∈ M such that ǫ − ⇀ a = a , δ ( a ) = 1 ⊗ a (then a is even). NICOLÁS ANDRUSKIEWITSCH AND HÉCTOR PEÑA POLLASTRI (1) For every a ∈ M such that ǫ − ⇀ a = − a , δ ( a ) = ǫ ⊗ a (then a is odd).Let A be a superalgebra, i.e. an algebra in sVec ; A is super commutativeif ab = ( − | a || b | ba for all a, b ∈ |A| .The algebra of regular functions on an (affine) algebraic super group G isdenoted O ( G ) . This is by definition a finitely generated super commutativeHopf algebra. See [M] for the definitions. As usual, G a is the additivealgebraic group ( k , +) and G m is the multiplicative algebraic group ( k × , · ) .Let R be the polynomial algebra k [ ζ ] with the unique Hopf algebra struc-ture such that ζ ∈ P ( R ) . Let R p be the quotient Hopf algebra R/ ( ζ p − ζ ) .Let V = k { X , . . . , X m } be a vector space of dimension m ∈ N . Wedenote by Λ( V ) = Λ( X , . . . , X m ) the exterior algebra of V .We denote by [ k ] [ n ] := Q ni =1 ( k + i − k ∈ k , n ∈ N the raising factorial,we also define [ k ] [0] := 1 for every k ∈ k . The unsigned Stirling numbers (cid:2) nk (cid:3) are defined as the coefficients of the ‘raising factorial’ polynomial: [ X ] [ n ] = n Y i =1 ( X + i −
1) = n X k =0 (cid:20) nk (cid:21) X k ∈ Z [ X ] . Recall that a sequence of morphisms of Hopf (super)algebras A ι ֒ −→ C π −→→ B is exact, and C is an extension of A by B , if(i) ι is injective.(ii) π is surjective. (iii) ker π = Cι ( A ) + .(iv) ι ( A ) = C co π .We write B = C//A i.e. B is the Hopf algebra quotient of C by A . Remark . If A ι ֒ −→ C is faithfully flat and ι ( A ) is stable by the left adjointaction of C then (i), (ii) and (iii) imply (iv), see [AD, 1.2.5, 1.2.14], [Sch].1.2. A brief review of Yetter-Drinfeld supermodules.
In this Subsec-tion, char k = 2 . See [AAY] for more details. Let A be a Hopf superalgebra.The category of Yetter-Drinfeld supermodules over A , denoted by AA YDS ,consists of super vector spaces X such that:(i) X is a left supermodule over A with action ⇀ .(ii) X is a left supercomodule over A with coaction δ .(iii) For every a ∈ |A| and u ∈ |X | : δ ( a ⇀ x ) = ( − | v ( − | ( | a (2) | + | a (3) | )+ | a (2) || a (3) | a (1) x ( − S ( a (3) ) ⊗ a (2) ⇀ x (0) . Then AA YDS is a braided tensor category. Namely, if X , Z ∈ AA YDS , thenthe super vector space
X ⊗ Z is an object in AA YDS via a ⇀ ( x ⊗ z ) = ( − | a (2) || x | a (1) ⇀ x ⊗ a (2) ⇀ z,δ ( x ⊗ z ) = ( − | x (0) || z ( − | x ( − z ( − ⊗ ( x (0) ⊗ z (0) ) , ∀ x ∈ |X | , z ∈ |Z| . The braiding c : X ⊗ Z −→ Z ⊗ X is given by: c ( x ⊗ z ) = ( − | x (0) || z | x ( − ⇀ z ⊗ x (0) . (1.2) N THE DOUBLE OF THE SUPER JORDAN PLANE 5
There is an embedding functor of braided tensor categories i : AA YDS −→ A k C A k C YD , X 7−→ i ( X ) (1.3)where i ( X ) = X as vector space with action ⇀ and coaction δ as: a ǫ k ⇀ x = ( − | x | k a ⇀ x,δ ( x ) = x ( − ǫ | x (0) | ⊗ x (0) , ∀ a ∈ A , x ∈ |X | , k ∈ I , . The super bosonization [AAY] . Let R be a Hopf algebra in AA YDS . Thecomultiplication is written ∆ R ( h ) = h (1) ⊗ h (2) for h ∈ R . The Hopf super-algebra R ♮ A is R ⊗ A as a super vector space with structure given by: ( h ♮ a )( f ♮ b ) := ( − | a (2) || f | h ( a (1) ⇀ f ) ♮ a (2) b, ∆( h ♮ a ) := ( − | ( h (2) ) (0) || a (1) | h (1) ♮ ( h (2) ) ( − a (1) ⊗ ( h (2) ) (0) ♮ a (2) ,ε ( h ♮ a ) = ε R ( h ) ε A ( a ) , R ♮ A , S ( h ♮ a ) = ( − | h (0) || a | (1 ♮ S A ( h ( − a ))( S R ( h (0) ) ♮ . There is a canonical isomorphism ( R ♮ A ) k C ≃ i ( R ) A k C ) .1.3. Restricted Lie superalgebras.
A Lie superalgebra g = g ⊕ g isrestricted if g is a restricted Lie algebra with p -operation x x [ p ] such that [ x [ p ] , z ] = (ad x ) p ( z ) for all x ∈ g , z ∈ g . We refer to [BMPZ, Chapter 3] for a detailed exposition. The restrictedenveloping algebra of the restricted Lie superalgebra g is defined as u ( g ) := U ( g ) / h x [ p ] − x p : x ∈ g i , where U ( g ) is the enveloping algebra. Assume for simplicity that dim g < ∞ and let { x , . . . , x r } , { y , . . . , y k } be bases of g and g respectively. Thenthe Hopf superalgebra u ( g ) has dimension p r k ; indeed it has a PBW-basis { x n · · · x n r r y m · · · y m k k : n , . . . , n r ∈ I ,p − , m , . . . , m k ∈ I , } . The ortho-symplectic Lie superalgebra g = osp (1 | . For our purposeswe recall its structure: g ≃ sl ( k ) with Cartan generators { e, f, h } ; g = k { ψ + , ψ − } is the natural sl ( k ) -module, hence the bracket is [ e, ψ + ] = 0 , [ h, ψ + ] = ψ + , [ f, ψ + ] = ψ − , [ e, ψ − ] = ψ + , [ h, ψ − ] = − ψ − , [ f, ψ − ] = 0 , [ ψ + , ψ + ] = 2 e, [ ψ − , ψ − ] = − f, [ ψ + , ψ − ] = − h. Assume now that char k = p . The algebra g has a p -structure given by e [ p ] = 0 , h [ p ] = h, f [ p ] = 0 . Below we consider the enveloping algebra U ( osp (1 | and its restricted ver-sion u ( osp (1 | which are Hopf superalgebras with suitable PBW-bases. NICOLÁS ANDRUSKIEWITSCH AND HÉCTOR PEÑA POLLASTRI
Nichols algebras.
A braided vector space is a pair ( V , c ) where V is avector space and c ∈ GL ( V ⊗ ) satisfies the braid equation ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) . Then the braid group B n acts on V ⊗ n ; the Nichols algebra B ( V ) is definedtaking the quotient of the quantum symmetrizer in each degree. See [A] fordetails. A realization of ( V , c ) over a Hopf algebra L is a structure of Yetter-Drinfeld L -module on V such that c coincides with the categorical braiding.Then B ( V ) is a Hopf algebra in LL YD and the bosonization B ( V ) L is aHopf algebra. If V is finite-dimensional, then the dual vector space V ∗ isbraided with the transpose braiding c ∗ –recall the identification (1.1): h c ∗ ( f ⊗ f ′ ) , x ⊗ y i = h f ⊗ f ′ , c ( x ⊗ y ) i , f, f ′ ∈ V ∗ , x, y ∈ V . (1.4)Similarly a realization of ( V , c ) over a Hopf superalgebra A is a structureof Yetter-Drinfeld A -supermodule on V such that c coincides with the cate-gorical braiding. Then B ( V ) is a Hopf algebra in AA YDS and the bosoniza-tion B ( V ) A is a Hopf superalgebra. Notice that there is an isomorphism i ( B ( V )) ≃ B ( i ( V )) where i is as in (1.3), see [AAY, §1.7].If ( V , c ) admits a realization over a Hopf superalgebra, then necessarily V is C -graded and c preserves the grading of V ⊗ V ; such ( V , c ) might becalled a super braided vector space, a concept already present in [KS, (53),p. 1610]. Indeed there is bijective correspondence between super braidedvector spaces and solutions of the super Yang-Baxter equation, see loc. cit. Preliminaries
The dual and the double of a bosonization.
Let L be a Hopfalgebra and V ∈ LL YD ; we assume that dim L < ∞ and dim B ( V ) < ∞ .In this subsection we describe explicitly the dual and the double of thebosonization A := B ( V ) L . In particular we show that A ∗ ≃ B ( V ∗ ) L ∗ ,and that V ∗ has the transpose braiding (1.4). We need some notation. First,we have morphisms of Hopf algebras A π ⇄ ι L such that πι = id . Next we fix ◦ a basis { v , . . . , v n } of V ; its dual basis is denoted by { w , . . . , w n } ; ◦ a basis { h , . . . , h m } of L ; its dual basis is denoted by { f , . . . , f m } .The braided tensor categories LL YD ≃ L ∗ L ∗ YD are equivalent via the functor F : LL YD −→ L ∗ L ∗ YD defined as follows. If X ∈ LL YD , then F ( X ) = X with structure f ⇀ v = h f, S ( v ( − ) i v (0) , δ ( v ) = m X i =0 S − ( f i ) ⊗ h i ⇀ v, v ∈ X , f ∈ L ∗ , see [AG, Prop. 2.2.1]. Let W := F ( V ∗ ) . Lemma 2.1. [G, Lemma 2.6]
As a braided vector space,
W ∈ L ∗ L ∗ YD isisomorphic to ( V ∗ , c ∗ ) . Hence dim B ( W ) = dim B ( V ) < ∞ . (cid:3) N THE DOUBLE OF THE SUPER JORDAN PLANE 7
Proposition 2.2. A ∗ ≃ B ( W ) L ∗ .Proof. By transposing the maps π and ι above, we get maps A ∗ ι ∗ ⇄ π ∗ L ∗ with ι ∗ π ∗ = id . Then A ∗ ≃ R L ∗ , where R = ( A ∗ ) co ι ∗ is a braided Hopf algebrain L ∗ L ∗ YD . Since A = L n ∈ N A n is graded, so are A ∗ = L n ∈ N ( A ∗ ) n and R = L n ∈ N R n . We proceed in three steps: (i) Find a basis of R ∈ L ∗ L ∗ YD . Since A ≃ V ⊗ L , ( A ∗ ) ≃ V ∗ ⊗ L ∗ as avector space. Given w ∈ V ∗ and f ∈ L ∗ we set w f := w ⊗ f ∈ ( A ∗ ) by h w f, v h i = h w, v ih f, h i , ∀ v ∈ V , h ∈ L. Then { w i f j : i ∈ I n , j ∈ I m } is a basis of ( A ∗ ) . Let w i := w i ε , i ∈ I n .We claim that { w , . . . , w n } is a basis of R . For this, we compute h ∆( w i ) , h ℓ ⊗ v j h k i = h w i , h ℓ ⇀ v j i ε ( h k ) , h ∆( w i ) , v j h k ⊗ h ℓ i = ε ( h k h ℓ ) δ i,j , for all i, j ∈ I n , k, ℓ ∈ I m . Since ∆ preserves the grading, we have ∆( w i ) = m X ℓ,k =1 ε ( h k h ℓ ) w i f k ⊗ f ℓ + m X ℓ,k =1 n X j =1 h w i , h ℓ ⇀ v j i ε ( h k ) f ℓ ⊗ w j f k = w i ⊗ m X ℓ =1 n X j =1 h w i , h ℓ ⇀ v j i f ℓ ⊗ w j . Then w i ∈ R because ι ∗ ( w j ) = 0 for all j ∈ I n . Since the w i ’s are linearlyindependent and dim R = dim B ( V ) = dim V , the claim follows. (ii) R ≃ W in L ∗ L ∗ YD . To compute the action and coaction of R we needthe multiplication of elements of the form ( w f ) f ′ and f ′ ( w f ) for f, f ′ ∈ L ∗ , w ∈ V ∗ . Since A ≃ L , ( A ∗ ) ≃ L ∗ , we see that h ( w f ) f ′ , v h i = h w f f ′ , v h i , h f ′ ( w f ) , v h i = h f ′ (1) ⇀ w f ′ (2) f, v h i for all v ∈ V , h ∈ L . Then f ′ ( w f ) = f ′ (1) ⇀ w f ′ (2) f and ( w f ) f ′ = w f f ′ for w ∈ V ∗ , f, f ′ ∈ L ∗ . Hence the action of L ∗ in R is given by f ⇀ w i = f (1) ( w i ε ) S ( f (2) ) = f (1) ⇀ w i f (2) S ( f (3) )= f (1) ⇀ w i ε ( f (2) ) ε = f ⇀ w i ε = n X j =1 h f, ( v j ) ( − ih w i , ( v j ) (0) i w j , i ∈ I n , f ∈ L ∗ . NICOLÁS ANDRUSKIEWITSCH AND HÉCTOR PEÑA POLLASTRI
The formula for the coaction follows from the comultiplication above: δ ( w i ) = ( ι ∗ ⊗ id) ◦ ∆( w i ) = m X ℓ =1 n X j =1 h w i , h ℓ ⇀ v j i f ℓ ⊗ w j , i ∈ I n . Thus
W ≃ R in L ∗ L ∗ YD . (iii) R ≃ B ( W ) ≃ B ( V ∗ ) . Let R ′ be the braided Hopf subalgebra of R generated by R . Then R ′ is a pre-Nichols algebra of R ≃ W . Thus dim B ( W ) ≤ dim R ′ ≤ dim R = dim B ( V ) = dim B ( W ); hence R ′ = R = B ( W ) and the result follows. (cid:3) Next we describe the relations of the Drinfeld double D ( A ) of A . Recallthat this is the Hopf algebra whose underlying coalgebra is A ⊗ A ∗ op andwith multiplication defined as follows. Let a ⊲⊳ r := a ⊗ r in D ( A ) . Then ( a ⊲⊳ r )( a ′ ⊲⊳ r ′ ) = (cid:10) r (1) , a ′ (1) (cid:11)(cid:10) r (3) , S ( a ′ (3) ) (cid:11) ( aa ′ (2) ⊲⊳ r ′ r (2) ) ,a, A ′ ∈ A , r, r ′ ∈ A ∗ op , where rr ′ = m ( r ⊗ r ′ ) is in A ∗ , not in A ∗ op . Remark . [DT] If A is not finite-dimensional, we may define its doublewith respect to another Hopf algebra B provided with a skew-pairing, i. e.a linear map τ : A ⊗ B → k satisfying τ ( a e a ⊗ b ) = τ ( a ⊗ b (1) ) τ ( e a ⊗ b (2) ) , τ (1 ⊗ b ) = ε ( b ) , a, e a ∈ A,τ ( a ⊗ e bb ) = τ ( a (1) ⊗ b ) τ ( a (2) ⊗ e b ) , τ ( a ⊗
1) = ε ( a ) , b, e b ∈ B. (2.1)Let σ : ( A ⊗ B ) ⊗ ( A ⊗ B ) → k be the 2-cocycle associated to τ , where A ⊗ B has the structure of tensor product Hopf algebra, that is σ ( a ⊗ b, e a ⊗ e b ) = ε ( a ) ε ( e b ) τ ( e a ⊗ b ) , a, e a ∈ A, b, e b ∈ B. Then the double of A (with respect to B and τ ) is the Hopf algebra A ⊗ B twisted by σ , i. e. A ⊲⊳ B =: ( A ⊗ B ) σ .We fix the following presentations for B ( V ) , B ( W ) op and D ( L ) : B ( V ) = k h v , . . . , v n | r , . . . , r n i , B ( W ) op = k h w , . . . , w n | r ′ , . . . , r ′ n i ,D ( L ) = k h s , . . . , s m , s ′ , . . . , s ′ m | e r , . . . , e r n i , (2.2)with s , . . . , s m and s ′ , . . . , s ′ m generators of L and ( L ∗ ) op respectively.Then we have the following presentation for D ( A ) . Proposition 2.4.
The algebra D ( A ) is presented by generators v , . . . , v n , w , . . . , w n , s , . . . , s m , s ′ , . . . , s ′ m with relations (2.2) and N THE DOUBLE OF THE SUPER JORDAN PLANE 9 s ′ i v j = h ( s ′ i ) (1) , ( v j ) ( − i ( v j ) (0) ( s ′ i ) (2) , ( i, j ) ∈ I m × I n ,w i s j = n X ℓ =1 h w i , ( s j ) (1) ⇀ v ℓ i ( s j ) (2) w ℓ , ( i, j ) ∈ I n × I m ,w i v j = h w i , v j i n X ℓ =1 h w i , ( v j ) ( − ⇀ v ℓ i ( v j ) (0) w ℓ + m X k =1 n X ℓ,t =1 h w i , ( v j ) ( − ⇀ v ℓ ih w ℓ , h k ⇀ v t ih w t , S (( v j ) (0) ) i ( v j ) ( − f k , ( i, j ) ∈ I n × I n . (2.3) Hence D ( A ) admits a triangular decomposition D ( A ) ≃ B ( V ) ⊗ D ( L ) ⊗ B ( W ) op . (2.4) Proof.
Let A be the algebra presented by generators and relations as above.By definition of D ( A ) and the formulas above there exists an algebra map A ։ D ( A ) . Hence dim A ≥ D ( A ) . By definition of A there exist morphisms ψ : B ( V ) −→ A, ψ : B ( W ) op −→ A, ψ : D ( L ) −→ A. Let m : A ⊗ A −→ A be the multiplication. We have a linear map φ : B ( V ) ⊗ D ( L ) ⊗ B ( W ) op ψ ⊗ ψ ⊗ ψ −−−−−−−→ A ⊗ A ⊗ A m ◦ ( m ⊗ id) −−−−−−→ A (2.5)which is surjective by (2.3), since every product of the generators can berewritten to get a sum of elements of the form abc with a ∈ B ( V ) , b ∈ D ( L ) and c ∈ B ( W ) op . Thus dim D ( A ) = dim B ( V ) · dim D ( L ) · dim B ( W ) op ≥ dim A, hence D ( A ) ≃ A . As for (2.4), (2.5) provides the desired isomorphism. (cid:3) Objects of interest.
In this Subsection we introduce the braided vec-tor spaces of our interest and their Nichols algebras, cf. [AAH1, AAH2].
Braided vector spaces.
The braided vector space ( V ( − , , c ) =: ( V, c V ) ,called the − -block in [AAH1], has a basis { x , x } and braiding c ( x i ⊗ x ) = − x ⊗ x i , c ( x i ⊗ x ) = ( − x + x ) ⊗ x i , i = 1 , . (0.2)For simplicity we denote ( W, c W ) := ( V ( − , ∗ , c ∗ ) , cf. (1.4). Let { u , u } be the basis of W given by h u i , x j i = 1 − δ i,j , i, j ∈ I . In this basis c W is c W ( u ⊗ u i ) = − u i ⊗ u , c W ( u ⊗ u i ) = u i ⊗ ( u − u ) , i = 1 , . (2.6)Then ( W, c W ) ≃ ( V, c − V ) as braided vector spaces, via u x , u
7→ − x . The super Jordan plane.
This is the graded algebra B = k h x , x | x , x x − x x − x x i with the braided Hopf algebra structure extending the braiding (0.2), cf. [T]. Lemma 2.5. (a) [AAH1] ( char k = 2 ) The ordered monomials x n x n x n (2.7) with ( n , n , n ) ∈ I , × N × N form a basis of B and GKdim B = 2 . (b) ( char k = 0 ) [AAH1] B is isomorphic to B ( V ) as braided Hopf algebras. (c) ( char k = p ) [AAH2] The restricted super Jordan plane B / h x p , x p i isisomorphic to B ( V ) as braided Hopf algebras. The ordered monomials (2.7) with ( n , n , n ) ∈ I , × I ,p − × I , p − form a basis of B ( V ) . (cid:3) Set u := u u + u u . We define the dual super Jordan plane as thealgebra B d presented by generators u and u with defining relations u = 0 , u u = u u − u u (2.8)with the braided Hopf algebra structure extending the braiding (2.6), cf.[T]. Observe that B d is not isomorphic to B as braided Hopf algebras. The restricted dual Jordan plane is the quotient of B d by the relations u p = 0 , u p = 0 . (2.9) Lemma 2.6. (a) ( char k = 0 ) B d ≃ B ( W ) as braided Hopf algebras. (b) ( char k = p ) The restricted dual Jordan plane is isomorphic to B ( W ) as braided Hopf algebras. It has dimension p ; indeed the monomials u n u n u n (2.10) with ( n , n , n ) ∈ I , × I ,p − × I , p − form a basis of B ( W ) .Proof. Since B d ≃ B as algebras, the monomials (2.10) with ( n , n , n ) ∈ I , × N × N form a linear basis of B d . It is easy to see that u and u u − u u + u u are primitive in T ( W ) , hence B d is a pre-Nicholsalgebra of W . Now B ( V ) and B ( W ) have the same graded dimension by[G, Lemma 2.6]. Then (a) follows. For (b) we prove by induction that ∆( u n ) = X ℓ =0 n − ℓ X k =0 (cid:18) n − ℓk (cid:19) n ℓ u ℓ u n − k − ℓ ⊗ u ℓ u k , ∆( u n ) = X ℓ =0 n − ℓ X k =0 k X t =0 (cid:18) n − ℓk (cid:19)(cid:18) kt (cid:19) n ℓ [ k − n + ℓ ] [ t ] u n − k ) − ℓ ⊗ u ℓ u t u k − t )2 , (2.11)for all n ∈ N . So u p , u p ∈ P ( B d ) and B d / ( u p , u p ) is a pre-Nichols algebraof W ; hence B d / ( u p , u p ) ≃ B ( W ) by dimension counting. (cid:3) N THE DOUBLE OF THE SUPER JORDAN PLANE 11
Change of bosonization.
Let L be a Hopf algebra and let C be agroup provided with a group homomorphism C → Aut
Hopf ( L ) . Then L is aHopf algebra in k C k C YD (with trivial coaction) and we may consider the smashproduct L ⋊k C := L k C . Let L ⋊ k C π / / k C ι o o be the natural projection andinclusion. We show that under certain conditions, L ⋊k C can be alternativelydescribed as L F k C for a genuine braided Hopf algebra L F in k C k C YD .Namely, suppose that L = R U where U is a Hopf algebra and R isa Hopf algebra in UU YD . Let Aut
Hopf UU ( R ) be the group of automorphismsof Hopf algebras in UU YD , i.e. preserving multiplication, comultiplication,action and coaction. There is a morphism of groups Aut
Hopf UU ( R ) → Aut
Hopf ( L ) , Aut
Hopf UU ( R ) ∋ ς ς ⊗ id ∈ Aut
Hopf ( L ) . We fix a group C and a morphism of groups C → Aut
Hopf ( L ) factorizingthrough Aut
Hopf UU ( R ) . Furthermore we assume that U = U ′ ⊗ k G where U ′ a Hopf algebra and G is a group. Proposition 2.7. (a)
Given F ∈ Hom gps ( G , Z ( C )) , there exists a Hopf al-gebra L F in k C k C YD such that L ⋊ k C ≃ L F k C . (b) Let ̟ F : G × C → C be the morphism of groups given by ̟ F ( γ, c ) = F ( γ ) c , γ ∈ G , c ∈ C , and let G F := ker ̟ F = { ( γ, F ( γ ) − ) : γ ∈ G } . Then L F decomposes as L F ≃ R F ♯ ( U ′ ⊗ k G F ) (braided bosonization in k C k C YD )where R F = R as a subalgebra and U ′ -module but with G F ≃ G acting by γ ⇀ r = F ( γ ) − · ( γ · r ) , r ∈ R, γ ∈ G . (c) Assume that (i) C and G are abelian and F has a section ϑ ∈ Hom gps ( C , G ) , (ii) For every r ∈ R and c ∈ C , c · r = ϑ ( c ) · r .Let N F := ker F × { e } . Then the subalgebra R F ♯U ′ ⊗ k N F has a structure ofHopf algebra denoted L F such that k C ֒ → L F ։ L F is exact. Furthermore R F is a Hopf algebra in U ′ ⊗ k N F U ′ ⊗ k N F YD .Proof. (a): Let π F : L ⋊ k C → k C be the Hopf algebra projection given by π F ( r u ⊗ γ ⊗ c ) = ε ( r ) ε ( u ) F ( γ ) c, r ∈ R, u ∈ U ′ , γ ∈ G , c ∈ C . Then π F ι = id k C and the claim follows from [R] taking the subalgebra L F := ( L ⋊ k C ) co π F = R · U ′ ⊗ k G F . (b) The Hopf algebra maps L F p F / / U ′ ⊗ k G F ı F o o given by p F ( r u ⊗ γ F ( γ ) − ) = ε ( r ) ε ( u )(1 γ F ( γ ) − ) ,ı F ( u ⊗ γ F ( γ ) − ) = 1 u ⊗ γ F ( γ ) − , r ∈ R, u ∈ U ′ , γ ∈ G , satisfy p F ı F = id k G F . Hence R F := (cid:0) L F (cid:1) co p F and the claim follows.(c) By (i), G F ≃ N F × C ϑ where C ϑ = { ( ϑ ( c ) − , c ) : c ∈ C } and using also (i), C ϑ is central in L F . Now the multiplication provides a linear isomorphism L F ≃ R F ♯ ( U ′ ⊗ k G F ) ≃ R F ♯ (cid:0) U ′ ⊗ ( k N F × k C ϑ ) (cid:1) ≃ (cid:0) R F ♯U ′ ⊗ k N F (cid:1) ⊗ k C ϑ . Then the quotient L F ≃ L F / k ( C ϑ ) + L F is isomorphic as an algebra to thesubalgebra R F ♯U ′ ⊗ k N F . The rest is clear. (cid:3) The situation that we have in mind is when C = C N and R = ⊕ i ∈ C N R i has a C N -grading of Hopf algebras in UU YD . For instance, let V ∈ k G k G YD be C N -graded and take R = B ( V ) (or any pre-Nichols algebras whose definingrelations are C N -homogeneous). We fix ω ∈ G N (what amounts in this caseto fix a quasi-triangular structure on k C N ) and let act C N on R i by ω i .We discuss two specific examples of interest in this paper. In the restof this Section char k = 2 , in particular char k = 0 is allowed. Recall that R = k [ ζ ] with ζ primitive and Γ ≃ Z . We fix generators C = h ǫ i , Γ = h g i . (2.12) Example 2.8.
We realize ( V, c ) , with braiding (0.2), in k Γ k Γ YD by g ⇀ x = − x , g ⇀ x = − x + x , δ ( x i ) = g ⊗ x i , i = 1 , . (2.13)Since the ideal of T ( V ) generated by the relations (0.1) belongs to k Γ k Γ YD , B is a Hopf algebra in k Γ k Γ YD hence we have H := B k Γ . (2.14)Now thinking on V as purely odd and taking C = C , G = Γ , L = H , U = k Γ and F : Γ → C the standard projection, we have H ⋊ k C ≃ e H k C where e H ≃ B ♮ k Γ , (2.15)that corresponds to the realization of V in k Γ k Γ YDS by g ⇀ x = x , g ⇀ x = x − x , δ ( x i ) = g ⊗ x i , | x i | = 1 , i ∈ I . (2.16)Since F admits no section, there is no further decomposition. Example 2.9.
Analogously we realize ( W, c W ) , cf. (2.6), in R ⊗ k C R ⊗ k C YD by ζ ⇀ u i = u i , ǫ ⇀ u i , = − u i ,δ ( u ) = ǫ ⊗ u , δ ( u ) = ǫ ⊗ u − ζǫ ⊗ u , i = 1 , . (2.17)As before B d is a Hopf algebra in R ⊗ k C R ⊗ k C YD hence we have K := B d R ⊗ k C ) . (2.18)Now thinking on W as a purely odd super vector space and taking C = C ≃ G , L = K , U = R ⊗ k C and F = id C , we have K ⋊ k C ≃ f K k C , where f K ≃ B d ♮ ( R ⊗ k C ) ; furthermore k C is central and e K := f K // k C ≃ B d ♮ R. (2.19) N THE DOUBLE OF THE SUPER JORDAN PLANE 13
Here the last corresponds to the realization of W in RR YDS by ζ ⇀ u i = u i , | u i | = 1 , i = 1 , ,δ ( u ) = 1 ⊗ u , δ ( u ) = 1 ⊗ u − ζ ⊗ u . (2.20)The superalgebras e H and e K admit a suitable Hopf pairing that allows analternative characterization of the Hopf superalgebra e D , see Remark 3.4.3. The double of the super Jordan plane
The definition.
We define the Drinfeld double D of H = B k Γ , see(2.14), with respect to a suitable pairing with K = B d R ⊗ k C ) , see (2.18).Then we show that there exists a Hopf superalgebra e D such that D ≃ e D k C .In this way e D is fundamental to the study of the Drinfeld double D ; amongother characteristics, it bears a triangular decomposition (3.14).To start with the Hopf algebras H and K have PBW-basis, denoted by B H , B K , given by the ordered monomials (2.7) or (2.10) multiplied accordinglyby elements of the groups Γ and C , or powers of ζ . The comultiplicationsof H , respectively K satisfy x , x ∈ P g , ( H ) , u ∈ P ǫ, ( K ) and ∆( u ) = u ⊗ ǫ ⊗ u − ǫζ ⊗ u . (3.1) Lemma 3.1.
The algebra H is presented by generators x , x , g , g − andrelations (0.1) , g x = − x g , g x = − x g + x g , g ± g ∓ = 1 . (3.2) Also, K is presented by generators u , u , ζ, ǫ and relations (0.1) (in the u i ’s), ǫ = 1 , ǫζ = ζǫ, (3.3) ǫu = − u ǫ, ǫu = − u ǫ, (3.4) ζu = u ζ + u , ζu = u ζ + u . (cid:3) (3.5)We define the Drinfeld double of H as D := H ⊲⊳ K op , see Remark 2.3, withrespect to the unique skew-pairing τ : H ⊗ K op → k such that τ ( x ⊗ u ) = 0 , τ ( x ⊗ u ) = 1 , τ ( x ⊗ ζ ) = 0 , τ ( x ⊗ ǫ ) = 0 ,τ ( x ⊗ u ) = 1 , τ ( x ⊗ u ) = 0 , τ ( x ⊗ ζ ) = 0 , τ ( x ⊗ ǫ ) = 0 ,τ ( g ± ⊗ u ) = 0 , τ ( g ± ⊗ u ) = 0 , τ ( g ± ⊗ ζ ) = ± , τ ( g ± ⊗ ǫ ) = − . Proposition 3.2.
The algebra D is presented by generators x , x , ζ , g , ǫ , u , u with relations (0.1) (in the x i ’s and in the u i ’s), (3.2) , (3.3) , (3.4) , u ζ = ζu + u , u ζ = ζu + u , (3.6) ǫ g = g ǫ, ζ g = g ζ, (3.7) ǫx = − x ǫ, ǫx = − x ǫ, (3.8) ζx = x ζ + x , ζx = x ζ + x , (3.9) u g = − g u + g u , u g = − g u , (3.10) u x = − x u , u x = − x u + (1 − g ǫ ) , (3.11) u x = − x u + g ǫζ + x u , u x = − x u + (1 − g ǫ ) + x u , (3.12) The family B D consisting of the monomials x n x n x n g n ζ m u m u m u m ǫ k (3.13) with ( k, n , m , n, n , n , m, m , m ) ∈ I , × Z × N is a basis of D .Proof. Let A be the algebra presented as in the statement; then A ։ D . As B D is a basis of D by construction, the corresponding monomials are linearlyindependent in A . Let S = { x , x , x , g , ζ, u , u , u , ǫ } with x and u as before, ordered by x < x < x < g < ζ < u < u < u < ǫ . Fromthe defining relations we have u x = x u , ǫu = u ǫ, u x = x u , ǫx = x ǫ,u x = x u + ( g ǫ + 1) u , ζx = x ζ + 2 x ,u x = x u − x u + x ( g ǫ + 1) . If a > b ∈ S , then ab is a linear combination of monomials c · · · c s with c ≤ c ≤ · · · ≤ c s ∈ S . Thus the monomials (3.13) generate A and A ≃ D . (cid:3) Let e g := g ǫ and e D := k h x , x , u , u , e g , ζ i ֒ → D . Proposition 3.3. (i)
A basis of e D is given by the family B consisting ofmonomials x n x n x n e g n ζ m u m u m u m , ( n , m , n, n , n , m, m , m ) ∈ I , × Z × N . (ii) e D has a triangular decomposition i.e. a linear isomorphism induced bymultiplication e D ≃ B ⊗ ( k Γ ⊗ R ) ⊗ ( B d ) op . (3.14)(iii) The algebra e D is presented by generators x , x , u , u , e g ± , ζ withdefining relations (0.1) (in the x i ’s and the u i ’s), (3.6) , (3.9) , (3.11) (with e g instead of g ǫ ) and e g ± e g ∓ = 1 , ζ e g = e g ζ, e g x = x e g , e g x = x e g − x e g , u e g = e g u − e g u , u e g = e g u . (iv) e D k C ≃ D as Hopf algebras; here e D is a Hopf superalgebra with co-multiplication given by e g ∈ G ( e D ) , ζ, u ∈ P ( e D ) , x , x ∈ P e g , ( e D ) and ∆ e D ( u ) = u ⊗ ⊗ u − ζ ⊗ u , (3.15) Proof. (i) Argue as in the proof of Proposition 3.2; in turn (i) implies (ii).(iii) Let A be the algebra as in the statement. Using the commutationrelations we have a surjective map of algebras φ : A ։ e D . Clearly the family B ′ consisting of the monomials analogous to those in B generates A and islinearly independent because φ ( B ′ ) = B . Hence A ≃ e D . N THE DOUBLE OF THE SUPER JORDAN PLANE 15 (iv) Let us identify k C with the subalgebra of D generated by ǫ ∈ G ( D ) by ι : k C ֒ → D . There is a Hopf algebra map π : D ։ k C given by x , x , u , u , g ǫ, ζ , ǫ ǫ. Then πι = id and D ≃ D co π k C . Clearly e D ⊆ D co π because every generatorof e D is coinvariant, and e D is a braided Hopf algebra in k C k C YD with comul-tiplication ∆ e D as in (3.15). But e D k C contains B D , thus e D = D co π . Sinceevery generator of e D is either even or odd, e D is a Hopf superalgebra. (cid:3) Remark . Recall the Hopf superalgebras e H (2.15) and e K (2.19). Thereare Hopf superalgebra maps e H ֒ → e D and e K op ֒ → e D (by the presentationof e D ) and e D is isomorphic to the double of e H with respect to a suitableskew-pairing, cf. [GZB].3.2. The double as a super abelian extension.
We show that e D fitsinto an exact sequence of Hopf superalgebras R ֒ → e D ։ U with R supercommutative and U super cocommutative.Let G be the super algebraic group such that its algebra of functions isthe commutative Hopf superalgebra O ( G ) := k [ X , X , T ± ] ⊗ Λ( Y , Y ) with | X | = | X | = | T | = 0 , | Y | = | Y | = 1 , and comultiplication ∆( X ) = X ⊗ T ⊗ X + Y T ⊗ Y , ∆( T ) = T ⊗ T, ∆( X ) = X ⊗ ⊗ X + Y ⊗ Y , ∆( Y ) = Y ⊗ ⊗ Y , ∆( Y ) = Y ⊗ T ⊗ Y . (3.16) Theorem 3.5.
There is an exact sequence of Hopf superalgebras O ( G ) ֒ → e D ։ U ( osp (1 | . (3.17) Proof.
From the defining relations of e D we deduce that x x = x x , x u = u x , u u = u u , e g x = x e g , x u = u x , e g u = u e g . Hence the map ι : O ( G ) ֒ → e D given by Y x , Y u , X x , X u , T e g is a well defined injective morphism of Hopf superalgebras. Next the map π : e D −→ U ( osp (1 | given by x , x ψ − , u , u ψ + , e g , ζ
7→ − h. is a well defined surjective morphism of Hopf superalgebras, since π ( x ) = π ( u ) = 0 , π ( − x ) = f and π ( u ) = e . We check that ker π = e D ι ( O ( G )) + using the PBW-basis of e D . Thus (3.17) is exact. (cid:3) A central Hopf subalgebra.
In this Subsection char k = p > . Weshow that e D has a central Hopf subalgebra Z = O ( B ) , with B a solvablealgebraic group. We shall need the following commutation relations in e D . Lemma 3.6.
The following equalities are valid for all n, m ∈ N : x m x n = m X k =0 (cid:18) mk (cid:19) [ n ] [ k ] x n + k x m − k )2 ,x m +12 x n = X ℓ =0 m X k =0 (cid:18) mk (cid:19) [ n ] [ k + ℓ ] x ℓ x n + k x m − k ) − ℓ +12 ,x m x = m X k =0 ( − k [ − m ] [ k ] x x k x m − k )2 ,x m +12 x = X ℓ =0 m X k =0 ( − k + ℓ [ − m ] [ k ] x ℓ x k − ℓ +121 x m − k )+ ℓ e g n x m = m X k =0 (cid:18) mk (cid:19) [ − n ] [ k ] x k x m − k )2 e g n , e g n x m +12 = X ℓ =0 m X k =0 (cid:18) mk (cid:19) ( − n +1 [ − n ] [ k + ℓ ] x ℓ x k x m − k ) − ℓ +12 e g n ,x m x n = m X k =0 (cid:18) mk (cid:19) [ n ] [ k ] x n + k x m − k )2 ,x m +12 x n = X ℓ =0 m X k =0 (cid:18) mk (cid:19) [ n ] [ k + ℓ ] x ℓ x n + k x m − k ) − ℓ +12 ,ζ n x m = n X ℓ =0 (cid:18) nℓ (cid:19) m n − ℓ x m ζ ℓ ,ζ n x m = n X ℓ =0 (cid:18) nℓ (cid:19) (2 m ) n − ℓ x m ζ ℓ ,u m u = m X k =0 ( − k [ − m ] [ k ] u u k u m − k )2 ,u m +12 u = X ℓ =0 m X k =0 ( − k + ℓ [ − m ] [ k ] x ℓ u k − ℓ +121 u m − k )+ ℓ u m e g n = m X k =0 (cid:18) mk (cid:19) [ − n ] [ k ] e g n u k u m − k )2 ,u m +12 e g n = X ℓ =0 m X k =0 (cid:18) mk (cid:19) ( − n +1 [ − n ] [ k + ℓ ] x ℓ e g n u k u m − k ) − ℓ +12 , N THE DOUBLE OF THE SUPER JORDAN PLANE 17 u m ζ n = n X ℓ =0 (cid:18) nℓ (cid:19) m n − ℓ ζ ℓ u m ,u m ζ n = n X ℓ =0 (cid:18) nℓ (cid:19) (2 m ) n − ℓ ζ ℓ u m ,u x n = x u − nx n u + nx x n − (1 + e g ) ,u n x = x u n + n ( e g + 1) u u n − ,u n x = x u n − nx u u n − + nu u n − ,u n x = x u n − nx u u n − + n e g u n − − n e g u u n − − n ( n − e g u u n − − ,u x n = x n u + nx n − x e g − n ( n − x x n − x e g ,u x n = x n u − nx n u + nx n − − nx n − x e g ζ + n ( n − x x n − x e g ζ − n (2 n − x n − x e g + n ( n − n − x x n − x e g , e g n x = x e g n , e g n x m = x m e g n , x x n = x n x ,u n x = x u n , x n u = u x n . Proof.
Straightforward by induction. (cid:3)
For our next statement we need to set up the notation. Let B := (( G a × G a ) ⋊ G m ) × H (3.18)be the algebraic group that in the first factor has the semidirect productwhere G m acts on G a × G a by λ · ( r , r ) = ( λ r , λ r ) , λ ∈ k × , r , r ∈ k while in the second factor appears the Heisenberg group H i.e. the group ofupper triangular matrices with ones in the diagonal. Let ζ ( p ) := ζ p − ζ and Z := k h x p , x p , u p , u p , e g p , ζ ( p ) i ֒ → e D . Note that Z is an even subalgebra. Theorem 3.7. (i) Z is a central Hopf subalgebra of e D . (ii) e D is a finitely generated free Z -module. (iii) Z ≃ k [ T ± , X , . . . , X ] as an algebra. In particular Z is a domain. (iv) Z ≃ O ( B ) as Hopf algebras.Proof. (i) By Lemma 3.6 Z is a even central subalgebra of e D . We need toverify that is a subcoalgebra and invariant by the antipode. We have thefollowing comultiplication formulas for every n ∈ N ∆( x n ) = X ℓ =0 n − ℓ X k =0 (cid:18) n − ℓk (cid:19) n ℓ x ℓ x k e g n − k ) − ℓ ⊗ x ℓ x n − k − ℓ , ∆( x n ) = X ℓ =0 n − ℓ X k =0 k X t =0 (cid:18) n − ℓk (cid:19)(cid:18) kt (cid:19) n ℓ [ k − n + ℓ ] [ t ] x ℓ x t x k − t )2 e g n − k ) − ℓ ⊗ x n − k ) − ℓ , ∆( u n ) = X ℓ =0 n − ℓ X k =0 (cid:18) n − ℓk (cid:19) n ℓ u ℓ u k ⊗ u ℓ u n − k − ℓ . Hence u p , ζ ( p ) ∈ P ( e D ) , x p , x p are (1 , e g p ) -primitive and e g p ∈ G ( e D ) . Itonly remains to calculate ∆( u p ) . Recall the formula (2.11) and δ ( u n ) = n X k =0 k X j =0 (cid:18) nk (cid:19)(cid:20) kj (cid:21) ( − k ζ j ⊗ u k u n − k )2 , n ∈ N , where (cid:2) kj (cid:3) are the Stirling numbers. Since [ ζ ] [ p ] = Q pi =1 ( ζ + i −
1) = P pk =0 (cid:2) pk (cid:3) ζ k = ζ p − ζ , we have (cid:20) pk (cid:21) = 0 , k = 2 , . . . , p − , (cid:20) pp (cid:21) = 1 , and (cid:20) p (cid:21) = − . (3.19)Then we get ∆( u p ) = u p ⊗ ⊗ u p − ζ ( p ) ⊗ u p .(ii) To prove this we consider another basis of e D using a different basis of k [ ζ ] . The family of polynomials ( ζ ( p ) ) k ζ j , k ∈ N , j ∈ I ,p − , is a basis of k [ ζ ] , see the proof of [AP, Prop. 2.6]. Thus the elements x n x n x n e g n ( ζ ( p ) ) k ζ j u m u m u m , with ( n , m , n, n , n , k, m , m , j ) ∈ I , × N × I ,p − form a basis of e D .Hence a basis of e D as a Z -module is given by x n x n x n e g n ζ j u m u m u m , with ( n , m , n , m , n , m , j, n ) ∈ I , × I , p − × I ,p − .(iii) The map φ : k [ T ± , X , . . . , X ] −→ Z given by T e g p , X x p , X x p , X
7→ − ζ ( p ) , X u p , X u p , is the desired isomorphism of algebras.(iv) It is easy to see that O ( B ) ≃ k [ T ± , X , . . . , X ] with comultiplicationdetermined by T ∈ G ( O ( B )) , X , X ∈ P T , ( O ( B )) , X , X ∈ P ( O ( B )) and ∆( X ) = X ⊗ ⊗ X + X ⊗ X . The claim follows. (cid:3) Ring theoretical properties of the double.Proposition 3.8. (i)
The algebra e D admits an exhaustive ascending fil-tration ( e D n ) n ∈ N such that gr e D ≃ k [ X , . . . , X , T ± ] ⊗ Λ( Y , . . . , Y ) . (ii) The algebras D and e D are noetherian. (iii) If char k = p > then e D is a PI-algebra. N THE DOUBLE OF THE SUPER JORDAN PLANE 19
Proof. (i) Let A be the algebra presented by generators x , x , u , u , ζ , e g ± and Z i , i ∈ I , with relations e g ± e g ∓ = 1 . The algebra A is graded with deg x = deg u = 3 , deg x = deg u = 4 , deg e g ± = ± , deg Z = deg Z = deg ζ = 1 , deg Z = deg Z = 2 . The filtration associated to this grading induces a filtration on e D via theepimorphism A ։ e D given by Z x , Z u , Z x , Z u , the remaining generators being mapped to their homonyms. The relationsof e D imply that gr e D is super conmutative with the same parity as e D . Then φ : k [ X , . . . , X , T ± ] ⊗ Λ( Y , . . . , Y ) → gr e D given by X i Z i , i ∈ I , and T e g , Y x , Y x , Y u , Y u , X ζ is an isomorphism of algebras by comparison of the Hilbert series.(ii) It is well-known that k [ X , . . . , X , T ± ] ⊗ Λ( Y , . . . , Y ) is noetherian,hence so is e D by (i) and a fortiori D which is a finitely generated e D -module.(iii) follows from Theorem 3.7, Proposition 4.4 and [MR, Corollary 1.13]. (cid:3) The double of the restricted super Jordan plane
In this Section, char k = p > .4.1. The bosonizations.
Recall that R p = k [ ζ ] / ( ζ p − ζ ) is a Hopf algebrawith ζ primitive. Besides (2.12) we also fix the generators C p = h g i , C p = h γ i . (4.1)It is well-known that k C p ≃ R p , see e.g. [AP, 1.3]. Hence k C p ≃ R p ⊗ k C and the algebra k C p is presented by generators ǫ and ζ with relations ǫ = 1 , (4.2) ζ p = ζ, (4.3) ǫζ = ζǫ. (4.4)We consider the realizations of V in k C p k C p YD and W in k C p k C p YD given by γ ⇀ x = − x , γ ⇀ x = − x + x , δ ( x i ) = γ ⊗ x i , i = 1 , (4.5) ζ ⇀ u i = u i , ǫ ⇀ u i = − u i , i = 1 , ,δ ( u ) = ǫ ⊗ u , δ ( u ) = ǫ ⊗ u − ζǫ ⊗ u , (4.6)Thus we have the Hopf algebras H := B ( V ) k C p , K := B ( W ) k C p . They have PBW-basis, denoted by B H and B K , given by the ordered mono-mials (2.7) or (2.10) with ( n , n , n ) ∈ I , × N × N , multiplied accordinglyby elements of the groups C p , C p , or powers of ζ . Then dim H = dim K = 8 p . The comultiplications are given by x , x ∈ P γ, ( H ) , u ∈ P ǫ, ( K ) and ∆ K ( u ) by (3.1) Therefore there are surjective Hopf algebra maps H ։ H, K ։ K. The basic properties of H and K ≃ H ∗ follow without difficulties.4.2. The double.
In this subsection we show that the Drinfeld double of H fits into an exact sequence of Hopf algebras k C ֒ → D ( H ) ։ D .We first give a presentation of D ( H ) ; for this we need that of D ( k C p ) which follows easily since D ( k C p ) ≃ k C p ⊗ k C p ≃ k C p ⊗ R p ⊗ k C . Proposition 4.1. D ( H ) is generated by x , x , ζ, ǫ, γ, u , u with relations (0.1) , (0.3) , (2.9) , (3.8) , (3.9) , (4.2) , (4.3) , (4.4) and γ p = 1 , ǫγ = γǫ, ζγ = γζ, (4.7) γx = − x γ, γx = ( − x + x ) γ, (4.8) u = 0 , u u = u u + u u , (4.9) u i ζ = ζu i + u i , i = 1 , . (4.10) u x = − x u , u x = − x u + (1 − γǫ ) , (4.11) u x = − x u + (1 − γǫ ) + x u , u x = − x u + γζǫ + x u , (4.12) u γ = − γu , u γ = − γu + γu . (4.13) The comultiplication is determined by γ, ǫ ∈ G ( D ( H )) , ζ ∈ P ( D ( H )) , x , x ∈P γ, ( D ( H )) , u ∈ P ǫ, ( D ( H )) and (3.1) . The monomials x n x n x n γ i ǫ j ζ k u m u m u m with ( n , m , j, n , k, m , i, n , m ) ∈ I , × I ,p − × I , p − is a PBW-basisof D ( H ) .Proof. This is a direct application of Proposition 2.4. (cid:3)
Let t := γ p ǫ and g := γ p +1 . The Hopf subalgebra generated by t isisomorphic to k C and the one generated by g to k C p . By the definingrelations t is a central element. Let Z = k h t i and D := D ( H ) /D ( H ) Z +0 .We use the same symbol for an element in D ( H ) and its class in D . Proposition 4.2. (a)
The algebra D is generated by x , x , g, ζ, u , u , ǫ with relations (0.1) , (0.3) , (2.9) , (3.8) , (3.9) , (4.9) , (4.10) , (4.2) , (4.3) , (4.4) ζg = gζ, g p = 1 , (4.14) gx = x g, gx = ( x − x ) g, (4.15) u g = gu − gu , u g = gu , (4.16) u x = − x u + (1 − g ) , u x = − x u + gζ + x u , (4.17) u x = − x u + (1 − g ) + x u , u x = − x u , (4.18) ǫg = gǫ. (4.19) N THE DOUBLE OF THE SUPER JORDAN PLANE 21 (b)
The sequence of Hopf algebras Z := k C ֒ → D ( H ) ։ D is exact. (c) The algebra D has dimension p and basis consisting in the monomials x n x n x n g n ζ m u m u m u m ǫ k with ( n , m , k, n , m, m , n, n , m ) ∈ I , × I ,p − × I , p − Proof. (a): Very similar to the proof of Proposition 3.2. (b) Conditions (i)and (ii) above are evident while (iii) is proved considering the basis of D ( H ) x n x n x n g n ζ m u m u m u m ǫ k (1 − t ) ℓ with ( n , m , ℓ, k, n, m, n , m , n , m ) ∈ I , × I ,p − × I , p − . Now Remark1.1 gives (iv) since Z is central. The proof of (c) is direct. (cid:3) Remark . There is an exact sequence of Hopf algebras O ( B ) ֒ −→ D −→→ D .4.3. The Hopf superalgebra D . Here we show that D is the bosonizationof the Hopf superalgebra D := k h x , x , u , u , g, ζ i ֒ → D and that D is arestricted analogue of the Hopf superalgebra e D . Proposition 4.4. (i)
A PBW-basis of D is given by the monomials x n x n x n g n ζ m u m u m u m , ( n , m , n, m, n , m , n , m ) ∈ I , × I ,p − × I , p − . Thus dim D = 16 p . (ii) D has a triangular decomposition i.e. a linear isomorphism induced bymultiplication D ≃ B ( V ) ⊗ ( k C p ⊗ R p ) ⊗ B ( W ) op . (4.20)(iii) D is generated by x , x , u , u , g , ζ with relations (0.1) , (0.3) , (2.9) , (3.9) , (4.3) , (4.15) , (4.9) , (4.10) , (4.14) , (4.16) , (4.17) , (4.18) . (iv) D is a Hopf superalgebra with comultiplication given by g ∈ G ( D ) , ζ, u ∈ P ( D ) , x , x ∈ P g, ( D ) and (3.15) . Indeed D k C ≃ D asHopf algebras. (v) There is an exact sequence of Hopf superalgebras O ( B ) ι ֒ −→ e D π −→→ D . Proof. (i), (ii), (iii) are analogous to the ones of Proposition 3.3. (iv) As inProposition 3.3, show that
D ≃ D co ̟ where ̟ : D ։ k C is given by x , x , u , u , g , ζ , ǫ ǫ. (v) By the presentations of e D and D there exist π : e D ։ D such that g g, ζ ζ, x i x i , u i u i , i = 1 , . The map ι : O ( B ) ֒ → e D is defined as the φ in the proof of Theorem 3.7.Clearly ker π = e D ι ( O ( B )) + by the PBW bases. Since ι ( O ( B )) is central and e D is a free module over ι ( O ( B )) , the claim follows from Remark 1.1. (cid:3) Remark . We may realize V in k C p k C p YDS and W in R p R p YDS by g ⇀ x = x , g ⇀ x = x − x , δ ( x i ) = g ⊗ x i , | x i | = 1 , i = 1 , ζ ⇀ u i = u i , | u i | = 1 , i = 1 , ,δ ( u ) = 1 ⊗ u , δ ( u ) = 1 ⊗ u − ζ ⊗ u . Thus we have the Hopf superalgebras H := B ( V ) ♮ k C p , K := B ( W ) ♮ R p ;clearly dim H = dim K = 4 p . Their comultiplications are determined by x , x ∈ P g, ( H ) , u ∈ P ( K ) , while ∆ K ( u ) is given by (3.15). Thereforethere are surjective Hopf algebra maps e H ։ H , e K ։ K . The basic prop-erties of H and K follow at once. Also, there are isomorphisms of Hopfalgebras H ≃ H k C , K ≃ K k C . Finally, there are morphisms of Hopfsuperalgebras H ֒ → D , K op ֒ → D , hence D ≃ D ( H ) , see [GZB].4.4. D as an extension. We next show that D fits into an exact sequenceof Hopf superalgebras R ֒ → D ։ u with R super commutative and u supercocommutative. First let R be the super commutative Hopf superalgebra R := k [ X , X , T ] / ( X p , X p , T p − ⊗ Λ( Y , Y ) with | X | = | X | = | T | = 0 , | Y | = | Y | = 1 and comultiplication (3.16).Arguing as in Proposition 3.17, we have: Theorem 4.6.
There exist Hopf superalgebra maps ι and π such that R ι ֒ −→ D π −→→ u ( osp (1 | (4.21) is an exact sequence of Hopf superalgebras. (cid:3) The simple D -supermodules can be determined from the previous result. Theorem 4.7.
There are exactly p isomorphism classes of simple D -moduleswhich have dimensions , , , . . . , p − .Proof. Being nilpotent, the ideal D R + = h x , x , u , u , g − i is containedin the Jacobson radical of D ; thus Irrep
D ≃
Irrep u ( osp (1 | and [WZ,Prop. 6.3] applies. (cid:3) Simple modules.
We describe the simple D -modules as quotients ofVerma modules reproving Theorem 4.7. Let D = ⊕ n ∈ Z D n be Z -graded by deg x = deg x = − , deg u = deg u = 1 , deg g = deg ζ = 0 . Recall that D ( k C p ) ≃ R p ⊗ k C p . Consider the triangular decomposition(4.20) and the graded subalgebras D > := B ( W ) op , D < := B ( V ) . Then(1) D > ⊆ ⊕ n ∈ N D n , D < ⊆ ⊕ n ∈− N D n and D ( k C p ) ⊆ D .(2) ( D > ) = k = ( D < ) .(3) D ≥ := D ( k C p ) D > and D ≤ := D < D ( k C p ) are subalgebras of D . N THE DOUBLE OF THE SUPER JORDAN PLANE 23
In this context the simple modules of D arise inducing from D ≥ . Theelements of Λ := Irrep D ( k C p ) are called weights . Since D > is local, the(homogeneous) projection D ≥ ։ D ( k C p ) allows to identify Λ ≃ Irrep D ≥ .The Verma module associated to λ ∈ Λ is M ( λ ) = Ind DD ≥ λ = D ⊗ D ≥ λ. By a standard argument, M ( λ ) is indecomposable. Let L ( λ ) be the head of M ( λ ) . The following result is well-known, see for instance [V, Theorem 2.1]. Lemma 4.8.
The map λ L ( λ ) gives a bijection Λ ≃ Irrep D . (cid:3) The set Λ is easy to compute since D ( k C p ) ≃ k C p ⊗ R p and k C p is local.Given k ∈ F p , let λ k = k w k be the one-dimensional vector space with action g · w k = w k , ζ · w k = kw k . Lemma 4.9.
The map k λ k provides a bijection F p ≃ Λ . (cid:3) . We fix k ∈ F p and compute L ( λ k ) . Since M ( λ k ) is free as a D < -modulewith basis w k , the elements w ( n ,n ,n ) k := x n x n x n · w k , n ∈ I , , n ∈ I ,p − , n ∈ I , p − , form a linear basis of M ( λ k ) . This makes M ( λ k ) a graded module by deg w ( n ,n ,n ) k = deg( x n x n x n ) , so M ( λ k ) = ⊕ n ≤ M ( λ k ) n and M ( λ k ) = k w k . Any proper submodule of M ( λ k ) is then contained in ⊕ n ≤− M ( λ k ) n .Since L ( λ k ) is the unique simple quotient of M ( λ k ) , we divide the later byproper submodules until we get a simple one. We start by the submodule N k of M ( λ k ) generated by S k := { w (1 , , k , w (0 , , k } . Lemma 4.10.
The submodule N k is proper.Proof. The action of u and u gives u x · w k = − x u · w k = 0 , u x · w k = x u · w k = 0 ,u x · w k = − x u · w k + (1 − g ) · w k + x u · w k = 0 ,u x · w k = x u · w k − x u · w k + x (1 + g ) · w k = 2 x · w k . So D > · S k ⊆ S k . Then N k = D ≤ · S k ⊆ ⊕ n ≤− M ( λ k ) n is proper. (cid:3) Let V k = M ( λ k ) /N k and let y j be the class of w (0 , ,j ) k in V k . Lemma 4.11.
The family ( y j ) j ∈ I , p − generates linearly V k and g , u , u , x and x act trivially on V k .Proof. We claim that the class of w ( n ,n ,n ) k = 0 in V k if ( n , n ) = (0 , .It suffices to show that w (0 , ,n ) k = 0 and w (1 , ,n ) k = 0 in V k for every n .This follows by induction on n using the analogues for D of the relations inLemma 3.6. Since x and x commute, they both act trivially on V k : x · w ( n ,n ,n ) k = x n x n · w (0 , ,n ) k = 0 , x · w ( n ,n ,n ) k = x n x n · w (1 , ,n ) k = 0 . Then ( y j ) j ∈ I , p − generates linearly V k . Also g , u and u act trivially onthese generators by Lemma 3.6. (cid:3) Set y − = y p = 0 . The action of D on V k can be computed inductively: ζ · y j = ( k − j ) y j , g · y j = y j , x · y j = y j +1 , x · y j = 0 ,u · y j = 0 , u · y j = ( j y j − if j is even, ( j − − k ) if j is odd, j ∈ I , p − . (4.22)Now e V k := D y k +1 is a proper submodule of V k because D > · y k +1 = 0 . Proposition 4.12.
The module L k = V k / e V k is simple of dimension k + 1 . It follows that L k = L ( λ k ) , the head of the Verma module M ( λ k ) . Proof.
Let z j be the class of y j in L k ; the action of D on the z j ’s is still givenby (4.22). Then ( z j ) j ∈ I , k is a basis of L k . To see that L k is simple, we showthat every = z ∈ L k generates L k . Let z = P mj =0 c j z j with m ≤ k and c m = 0 . Then u m · z ∈ k × z , and D · z = L k . (cid:3) O ( G ) as an extension. Let G = ( G a × G a ) ⋊ G m be the semidirectproduct where G m acts on G a × G a by λ · ( t , t ) = ( λ t , t ) , λ ∈ k × , t , t ∈ k . Then O ( G ) is isomorphic to A := k [X , X , T ± ] with comultiplicationgiven by T ∈ G ( A ) , X ∈ P T , ( A ) , X ∈ P ( A ) . Proposition 4.13.
There is a short exact sequence of Hopf superalgebras O ( G ) ι ֒ −→ O ( G ) π −→→ R , (4.23) with morphisms ι : O ( G ) ֒ → O ( G ) , π : O ( G ) −→ R given by ι (X ) = X p , ι (X ) = X p , ι (T) = T p ,π ( X ) = X , π ( X ) = X , π ( T ) = T, π ( Y ) = Y , π ( Y ) = Y . Proof.
Clearly π is of Hopf superalgebras by the definition of comultiplicationin O ( G ) and R . In O ( G ) we have the following comultiplication formulas ∆( X n ) = X ℓ =0 n − ℓ X k =0 (cid:18) n − ℓk (cid:19) n ℓ Y ℓ X k T n − k ) − ℓ ⊗ Y ℓ X n − k − ℓ , ∆( X n ) = X ℓ =0 n − ℓ X k =0 (cid:18) n − ℓk (cid:19) n ℓ Y ℓ X k ⊗ Y ℓ X n − k − ℓ ,n ∈ N . Thus X p ∈ P T p , ( O ( G )) , X p ∈ P ( O ( G )) and ι is of Hopf superalge-bras. Then ker π = O ( G ) ι ( O ( G )) + , ι is injective and π is surjective, usingthe PBW bases. Since O ( G ) is a free O ( G ) -module, Remark 1.1 applies. (cid:3) N THE DOUBLE OF THE SUPER JORDAN PLANE 25
A commutative square.
We summarize the relationship between theHopf superalgebras studied so far in the commutative diagram O ( G ) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) O ( B ) (cid:127) _ (cid:15) (cid:15) / / / / O ( G a ) (cid:127) _ (cid:15) (cid:15) O ( G ) (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) e D / / / / (cid:15) (cid:15) (cid:15) (cid:15) U ( osp (1 | (cid:15) (cid:15) (cid:15) (cid:15) R (cid:31) (cid:127) / / D / / / / u ( osp (1 | (0.4) Proposition 4.14.
All columns and rows in (0.4) are exact sequences.Proof.
Theorems 3.5 and 4.6, and and Propositions 4.4 and 4.13 covers ev-erything except the topmost row and the rightmost column.For the rightmost column we need to prove that the even Hopf subalgebra Z ′ = h e p , f p , h p − h i of U ( osp (1 | is O ( G a ) ≃ k [ X , X , X ] . Taking thebasis of U ( osp (1 | consisting of monomials f n ( h p − h ) k h ℓ e m ψ i + ψ j − with ( n, m, k, ℓ, i, j ) ∈ N × I ,p − × I , , we see that the assignment X f p , X h p − h, X e p , gives an algebra isomorphism Z ′ ≃ k [ X , X , X ] ≃ O ( G a ) . Comparingcomultiplications, the previous isomorphism is of Hopf algebras. O ( G a ) isstable by the adjoint action of U ( osp (1 | and a free module over O ( G a ) using the previous basis, then Remark 1.1 applies and the column is exact.We next describe explicitly the top row. φ : O ( G ) → O ( B ) is given by X X , X X , T T. Recall that O ( B ) ≃ k [ T ± , X , . . . , X ] and O ( G ) ≃ k [X , X , T ± ] , cf. The-orem 3.7 and Proposition 4.13. Take ψ : O ( B ) −→ O ( G a ) given by T , X , X
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