Quantized enveloping superalgebra of type P
aa r X i v : . [ m a t h . QA ] J a n Quantized enveloping superalgebra of type P Saber Ahmed, Dimitar Grantcharov, Nicolas Guay
Abstract
We introduce a new quantized enveloping superalgebra U q p n attached to the Lie superalgebra p n oftype P . The superalgebra U q p n is a quantization of a Lie bisuperalgebra structure on p n and we studysome of its basic properties. We also introduce the periplectic q -Brauer algebra and prove that it isthe centralizer of the U q p n -module structure on C ( n | n ) ⊗ l . We end by proposing a definition for a newperiplectic q -Schur superalgebra. Introduction
The simple finite-dimensional Lie superalgebras over C were classified by V. Kac in [K]. The list in loc. cit. contains three classes of Lie superalgebras: basic, strange and Cartan-type. There are two types of strangeLie superalgebras - P and Q - both of which are interesting due to the algebraic, geometric, and combi-natorial properties of their representations. The study of the representations of type P Lie superalgebras,which are also called periplectic in the literature, has attracted considerable attention in the last five years.Interesting results on the category O , the associated periplectic Brauer algebras, and related theories havebeen established in [BDEA + + P via the FRT formalism[FRT]. A similar approach was used by G. Olshanski in [Ol] to define quantum superalgebras of type Q .We prove that our quantized enveloping superalgebra U q p n quantizes a Lie bisuperalgebra structure on p n ,a periplectic Lie superalgebra.The fake Casimir element used in [BDEA + +
2] is a solution of the classical Yang-Baxter equa-tion and a quantum version of that fake Casimir element, denoted S , is a solution of the quantum Yang-Baxterequation which serves as an essential ingredient in the definition of U q p n . It follows that the tensor superspace C ( n | n ) ⊗ ℓ is a representation of U q p n and the centralizer of the action of U q p n is a quantum version of theperiplectic Brauer algebra. The classical setting corresponding to q = 1 was studied in [Mo]. A similar resultfor type Q Lie superalgebras was established in [Ol], where the centralizer of the action of the quantizedenveloping superalgebra was proven to be the Hecke-Clifford superalgebra of the symmetric group S ℓ . Hav-ing at our disposal the periplectic q -Brauer algebra, we can introduce the periplectic q -Schur superalgebrain a natural way. We conjecture that these are mutual cenralizers (that is, they satisfy a double-centralizerproperty).One immediate problem is to define U q p n in terms of Drinfeld-Jimbo generators and relations and studyits category O . For type Q Lie superalgebras, this problem was addressed in [GJKK]. Furthermore, in[GJKKK], a theory of crystal bases for the tensor representations of U q g was established. Unfortunately, itis unlikely that natural crystal bases exist in the type P case due to the nonsemisimplicity of the category oftensor modules, contrary to what happens in type Q . Another natural direction is to construct, using alsothe FRT formalism, quantum affine superalgebras of type P . (See [ChGu] for the type Q case.) Yangians oftype P and Q appeared already many years ago in the work of M. Nazarov [Na1, Na2]. We hope to returnto these questions in a future publication.After setting up the notation and basic definitions in the first section, we introduce the “butterfly” Liebisuperalgebra in Section 2 and define the quantized enveloping superalgebra of type P in the followingsection. The main result of Section 3 is Theorem 3.3, which states that S , the q -deformation of the fakeCasimir element, is a solution of the quantum Yang-Baxter equation. In Section 4, we prove that U q p n is1 quantization of the Lie bisuperalgebra structure from Section 2: see Theorem 4.3. The new periplectic q -Brauer algebra B q,ℓ and the new periplectic q -Schur algebra are introduced in the last section, where weprove that B q,ℓ can be defined equivalently either using generators and relations or as the centralizer of theaction of U q ( p n ) on the tensor space: see Theorem 5.5.The proofs of our results require extensive computations: further details for all the computations can befound in [AGG]. Acknowledgements:
The second named author is partly supported by the Simons Collaboration Grant358245. He also would like to thank the Max Planck Institute in Bonn (where part of this work was com-pleted) for the excellent working conditions. The third named author gratefully acknowledges the financialsupport of the Natural Sciences and Engineering Research Council of Canada provided via the DiscoveryGrant Program. We thank Patrick Conner, Robert Muth and Vidas Regelskis for help with certain com-putations in the preliminary stages of the present paper. We also thank Nicholas Davidson and JonathanKujawa for some useful discussions. P Let C ( n | n ) be the vector superspace C n ⊕ C n spanned by the odd standard basis vectors e − n , . . . , e − andthe even standard basis vectors e , . . . , e n . Let M n | n ( C ) be the vector superspace consisting of matrices A = ( a ij ) with a ij ∈ C and with rows and columns labelled using the integers − n, . . . , , , . . . , n , so i, j ∈ {± , ± , . . . , ± n } . Set p ( i ) = 1 ∈ Z if − n ≤ i ≤ − p ( i ) = 0 ∈ Z if 1 ≤ i ≤ n . The parityof the elementary matrix E ij is p ( i ) + p ( j ) mod 2. We denote by gl n | n the Lie superalgebra over C whoseunderlying vector space is M n | n ( C ) and which is equipped with the Lie superbracket[ E ij , E kl ] = δ jk E il − ( − ( p ( i )+ p ( j ))( p ( k )+ p ( l )) δ il E kj . Recall that the supertranspose ( · ) st on gl n | n is given by the formula ( E ij ) st = ( − p ( i )( p ( j )+1) E ji . Theinvolution ι on gl n | n which will be relevant for this paper is given by ι ( X ) = − π ( X st ) where π : gl n | n −→ gl n | n is the linear map given by π ( E ij ) = E − i, − j . Definition 1.1.
The Lie superalgebra p n of type P , which is also called the periplectic Lie superalgebra, isthe subspace of fixed points of gl n | n under the involution ι , that is, p n = { X ∈ gl n | n | ι ( X ) = X } . If X ∈ p n with (cid:18) A BC D (cid:19) and
A, B, C, D ∈ M n ( C ), then D = − A t , B = B t and C = − C t where t denotesthe transpose with respect to the diagonal i = − j . For convenience, we set E ij = E ij + ι ( E ij ) = E ij − ( − p ( i )( p ( j )+1) E − j, − i . The superbracket on p n is given by[ E ji , E lk ] = δ il E jk − ( − ( p ( i )+ p ( j ))( p ( k )+ p ( l )) δ jk E li − δ i, − k ( − p ( l )( p ( k )+1) E j, − l − δ − j,l ( − p ( j )( p ( i )+1) E − i,k (1)A basis of p n is provided by all the matrices E ij with indices i and j respecting one of the followinginequalities: 1 ≤ | j | < | i | ≤ n or 1 ≤ i = j ≤ n or − n ≤ i = − j ≤ − . Note that E ij = − ( − p ( i )( p ( j )+1) E − j, − i for all i, j ∈ {± , . . . , ± n } , hence E i, − i = 0 when 1 ≤ i ≤ n .2 Lie bisuperalgebra structure
To construct a Lie bisuperalgebra structure on p n , we define a Manin supertriple. We follow the idea in[Ol] for the case of the Lie superalgebra of type Q . Recall that a Manin supertriple ( a , a , a ) consists ofa Lie superalgebra a equipped with an ad-invariant supersymmetric non-degenerate bilinear form B alongwith two Lie subsuperalgebras a , a of a which are B -isotropic transversal subspaces of a . Note that such abilinear form B defines a non-degenerate pairing between a and a and a supercobracket δ : a → a ⊗ via B ⊗ ( δ ( X ) , Y ⊗ Y ) = B ( X, [ Y , Y ]) , where X ∈ a , Y , Y ∈ a . Definition 2.1.
The “butterfly” Lie superalgebra b n is the subspace of gl n | n spanned by E ij with ≤ | i | < | j | ≤ n and by E ii + E − i, − i , E i, − i for ≤ i ≤ n . Note that gl n | n = p n ⊕ b n . It is well-known that the bilinear form B ( · , · ) on gl n | n given by the super-trace, B ( A, B ) = Str( AB ), is ad-invariant, supersymmetric and non-degenrate.One easily checks that B ( X , X ) = 0 if X , X ∈ p n or if X , X ∈ b n . Hence we have the followingresult. Proposition 2.2. ( gl n | n , p n , b n ) is a Manin supertriple.Remark . A similar Manin supertriple is given in [LeSh], § gl n | n , p n , b n ).We extend the form B ( · , · ) to a non-degenerate pairing B ⊗ on gl n | n ⊗ C gl n | n by setting B ⊗ ( X ⊗ X , Y ⊗ Y ) = ( − | X || Y | B ( X , Y ) B ( X , Y )for all homogeneous elements X , X , Y , Y ∈ p n . The sign ( − | X || Y | is necessary to make this formad-invariant.Let s = X ≤| j | < | i |≤ n ( − p ( j ) E ij ⊗ E ji + 12 X ≤ i ≤ n E ii ⊗ ( E ii + E − i, − i ) + 12 X ≤ i ≤ n E − i,i ⊗ E i, − i (2)A scalar multiple of this element is called the fake Casimir element in [BDEA + Proposition 2.4. s is a solution of the classical Yang-Baxter equation: [ s , s ] + [ s , s ] + [ s , s ] = 0 . The proof of the above proposition follows from the lemma below, which should be well-known amongexperts.
Lemma 2.5.
Let p be a finite dimensional Lie superalgebra and suppose that ( p , p , p ) is a Manin triplewith respect to a certain supersymmetric, invariant, bilinear form B ( · , · ) . Let { X i } i ∈ I , { X ′ i } i ∈ I be bases of p and p , respectively, dual in the sense that B ( X ′ i , X j ) = δ ij . Set s = P i ∈ I X i ⊗ X ′ i . Then s is a solutionof the classical Yang-Baxter equation. We next compute the supercobracket δ using the identity B ( X, [ Y , Y ]) = B ( δ ( X ) , Y ⊗ Y ) for all X ∈ p n Y , Y ∈ b n . The formula for δ is (assuming, without loss of generality, that | j | ≤ | i | ): δ ( E ij ) = n X k = − n | j | < | k | < | i | ( − p ( k )+1 (cid:0) E ik ⊗ E kj − ( − ( p ( i )+ p ( k ))( p ( j )+ p ( k )) E kj ⊗ E ik (cid:1) −
12 (( − p ( i ) E ii − ( − p ( j ) E jj ) ⊗ E ij + 12 E ij ⊗ (( − p ( i ) E ii − ( − p ( j ) E jj ) − δ ( i < (cid:16) E i, − i ⊗ E − i,j − ( − p ( j ) E − i,j ⊗ E i, − i (cid:17) (3)+ δ ( j > (cid:16) ( − p ( i ) E − j,j ⊗ E i, − j + E i, − j ⊗ E − j,j (cid:17) Finally, the super cobracket is related to the element s . The following lemma is standard. Lemma 2.6.
The super cobracket can also be expressed as δ ( X ) = [ X ⊗ ⊗ X, s ] . (4) In this section, we define the quantized enveloping superalgebra U q p n following the approach used in [FRT]and [Ol]. We use a solution S of the quantum Yang-Baxter equation such that s is the classical limit of S .For simplicity, denote by C q the field C ( q ) of rational functions in the variable q and set C q ( n | n ) = C q ⊗ C C ( n | n ). Definition 3.1.
Let S ∈ End C q ( C q ( n | n ) ⊗ ) be given by the formula: S = 1 + X ≤ i ≤ n (cid:0) ( q − E ii + ( q − − E − i, − i (cid:1) ⊗ ( E ii + E − i, − i ) + q − q − X − n ≤ i ≤− E i, − i ⊗ E − i,i +( q − q − ) X ≤| j | < | i |≤ n ( − p ( j ) E ij ⊗ E ji (5) Remark . If we define S instead as an element of End C [[ ~ ]] ( C ~ ( n | n ) ⊗ ) by the same formula as in definition3.1 but with q, q − replaced by e ~ / , e − ~ / and C q ( n | n ) ⊗ replaced by C ~ ( n | n ) ⊗ , which equals C ( n | n ) ⊗ [[ ~ ]],then S = 1 + ~ s + O ( ~ ). Theorem 3.3. S is a solution of the quantum Yang-Baxter equation: S S S = S S S .Proof. The proof consists of verifying long computations. To simplify them, we have used the followingmethod. Set f ( q ) = S S S − S S S . The main idea is to consider f ( q ) as a Laurent polynomial P i = − f i q i with coefficients f i in End C (cid:16) C ⊗ n | n (cid:17) . Then one shows the eight relations f ( a ) = 0, f ′ ( b ) = 0, f ′′ ( c ) = 0 for a, b, c = ± b = ±√−
1. (Actually, just seven of those are enough.) We can then deducethat f ( q ) is a scalar multiple of ( q − q − ) and we show that the coefficient of q in f ( q ) is zero.Here are some more details.Let us set C = X ≤ i ≤ n ( E ii + E − i, − i ) ⊗ ( E ii + E − i, − i ) . S = 1 + ( q − q − ) s + (cid:18) q + q − − (cid:19) C. For convenience, we introduce the following notation:[ s C ] = s C + s C + s C + C s + C s + C s − s C − s C − s C − C s − C s − C s [ s CC ] = s C C + C s C + C C s − s C C − C S C − C C s [ ss C ] = s s C + C s s + s C s − s s C − C s s − s C s The relations f ( a ) = 0, f ′ ( b ) = 0, f ′′ ( c ) = 0 for a, b, c = ± b = ±√− Lemma 3.4. [ s C ] = 2[ s CC ] Lemma 3.5. [ ss C ] = 0For instance, f ′ ( −
1) = 0 follows from f ′ ( −
1) = − s C ] + 8[ s CC ] and the two lemmas. Furthermore, f ′′ ( −
1) = − s C ] + 8[ s CC ] − ss C ] + 8([ s , s ] + [ s , s ] + [ s , s ]) . Therefore, f ′′ ( −
1) = 0 thanks to Lemmas 2.5, 3.4, and 3.5. Similarly, the two lemmas above imply that f ′ ( √−
1) = 2 √− s C ] − √− s CC ] − ss C ]vanishes.The last step in the proof of Theorem 3.3 is to show the vanishing of the coefficient f of q . We have f = s s s − s s s + 14 [ sCC ] + 12 [ ssC ] + 18 C C C − C C C , which simplifies to s s s − s s s + 14 [ sCC ] (6)thanks to Lemma 3.5 and C C C − C C C = 0. Verifying that (6) vanishes follows by direct andextensive computations.With the aid of S , we can now define the main object of interest in this paper. Definition 3.6.
The quantized enveloping superalgebra of p n is the Z -graded C q − algebra U q p n generatedby elements t ij , t − ii with ≤ | i | ≤ | j | ≤ n and i, j ∈ {± , . . . , ± n } which satisfy the following relations: t ii = t − i, − i , t − i,i = 0 if i > , t ij = 0 if | i | > | j | ; (7) T T S = S T T (8) where T = P | i |≤| j | t ij ⊗ C E ij and the last equality holds in U q p n ⊗ C ( q ) End C ( q ) ( C q ( n | n )) ⊗ . The Z -degreeof t ij is p ( i ) + p ( j ) . U q p n is a Hopf algebra with antipode given by T T − and with coproduct given by∆( t ij ) = n X k = − n ( − ( p ( i )+ p ( k ))( p ( k )+ p ( j )) t ik ⊗ t kj . Limit when q and quantization We want to explain how Up n can be viewed as the limit when q U q p n and how the co-Poisson Hopfalgebra structure on Up n , which is inherited from the cobracket δ on p n , can be recovered from the coproducton U q p n .Set τ ij = t ij q − q − if i = j and set τ ii = t ii − q − . Let A be the localization of C [ q, q − ] at the ideal generatedby q −
1. Let U A p n be the A -subalgebra of U q p n generated by τ ij when 1 ≤ | i | ≤ | j | ≤ n . Theorem 4.1.
The map ψ : Up n −→ U A p n / ( q − U A p n given by ψ ( E ji ) = ( − p ( j ) τ ij for | i | < | j | , ≤ i = j ≤ n , and ψ ( E − i,i ) = − τ i, − i for ≤ i ≤ n , is an associative C -superalgebra isomorphism.Proof. First, we need to write down explicitly the defining relation (8). Comparing coefficients of E ij ⊗ E kl on both sides of relation (8), we obtain:( − ( p ( i )+ p ( j ))( p ( k )+ p ( l )) t ij t kl − t kl t ij + θ ( i, j, k ) (cid:0) δ | j | < | l | − δ | k | < | i | (cid:1) ǫt il t kj + ( − ( p ( i )+ p ( j ))( p ( k )+ p ( l )) (cid:0) δ j> ( q −
1) + δ j< ( q − − (cid:1)(cid:0) δ jl + δ j, − l (cid:1) t ij t kl − (cid:0) δ i> ( q −
1) + δ i< ( q − − (cid:1)(cid:0) δ ik + δ i, − k (cid:1) t kl t ij + θ ( i, j, k ) δ j> δ j, − l ǫt i, − j t k, − l − ( − p ( j ) δ i< δ i, − k ǫt − k,l t − i,j + ( − p ( j )( p ( i )+1) ǫ X − n ≤ a ≤ n (cid:0) ( − p ( i ) p ( a ) θ ( i, j, k ) δ j, − l δ | a | < | l | t i, − a t ka + ( − p ( − j ) p ( a ) δ i, − k δ | k | < | a | t al t − a,j (cid:1) = 0 (9)In the identity above, we set θ ( i, j, k ) = sgn(sgn( i ) + sgn( j ) + sgn( k )) and ǫ = q − q − . In order to check that ψ ([ E ji , E kl ]) = [ ψ ( E ji ) , ψ ( E kl )], we proceed as follows. We apply ψ on both sidesof (1). To show that the resulting right hand side coincides with [ ψ ( E ji ) , ψ ( E kl )], we use (9) and pass to thequotient U A p n / ( q − U A p n . This is done via a long case-by-case verification for i, j, k, l .From the way U A p n is defined, it follows that ψ is surjective. It remains to prove that it is injective.Since S is a solution of the quantum Yang-Baxter equation, the space C q ( n | n ) is a representation of U q p n via the assignment T S , hence also of U A p n by restriction. More explicitly, τ ij ( − p ( i ) E ji if | i | < | j | , and τ i, − i E − i,i , τ ii ( E ii − q − E − i, − i ) if 1 ≤ i ≤ n .Set C A ( n | n ) = A ⊗ C C ( n | n ). The space C A ( n | n ) is a U A p n -submodule and so are all the tensor powers C A ( n | n ) ⊗ ℓ . We thus have a superalgebra homomorphism φ ℓ : U A p n −→ End A ( C A ( n | n ) ⊗ ℓ ) for each ℓ ≥ π ℓ be the quotient homomorphismEnd A ( C A ( n | n ) ⊗ ℓ ) −→ End A ( C A ( n | n ) ⊗ ℓ ) / ( q − A ( C A ( n | n ) ⊗ ℓ ) ∼ = End C ( C ( n | n ) ⊗ ℓ ) . The composite π ℓ ◦ φ ℓ descends to a homomorphism π ℓ ◦ φ ℓ from U A p n / ( q − U A p n to End C ( C ( n | n ) ⊗ ℓ ). Thecomposite π ℓ ◦ φ ℓ ◦ ψ is the superalgebra homomorphism Up n −→ End C ( C ( n | n ) ⊗ ℓ ) induced by the natural p n -module structure on C ( n | n ) ⊗ ℓ twisted by the automorphism of p n given by E ij ( − p ( i )+ p ( j ) E ij .We can combine the homomorphisms π ℓ ◦ φ ℓ ◦ ψ for all ℓ ≥ U p n −→ Q ∞ ℓ =1 End C ( C ( n | n ) ⊗ ℓ ). This map is injective since C ( n | n ) is a faithful representation of p n . It follows that ψ is injective as well. 6e next show that a PBW-type theorem holds for U q p n . For this, we first introduce a total order ≺ onthe set of generators t ij , 1 ≤ | i | ≤ | j | ≤ n , of U q p n as follows. We declare that t ij ≺ t kl if(i) | i | > | k | , or(ii) | i | = | k | and | j | > | l | , or(iii) i = k and j = − l >
0, or(iv) i = − k > | j | = | l | .This order leads to a total lexicographic order on the set of words formed by the generators t ij . Namely, if A = A · · · A r and B = B · · · B s are two such words in the sense that each A k for 1 ≤ k ≤ r and each B l for1 ≤ l ≤ s is equal to some generator t ij , then A ≺ B if r < s or if r = s and there is a p such that A k = B k for 1 ≤ k ≤ p − A p ≺ B p . Note that, in this order, the generators t ij with i = j or i = − j are notgrouped together. We call a generator of the from t ii diagonal . Also, a word A k ...A k r r in the generators t ij is called a reduced monomial if A ≺ · · · ≺ A r , and k i ∈ Z > if A i is not diagonal, k i ∈ Z \ { } if A i isdiagonal, and k i = 1 if A i is odd. Theorem 4.2.
The reduced monomials form a basis of U q p n over C q .Proof. We first show that the set of reduced monomials spans U q p n . Note that it is enough to show that allquadratic monomials are in the span of this set. Let t ij t kl be a quadratic monomial which is not reduced.We have that either t kl = t ij , or i = k, j = l and t ij is odd. In the latter case, as shown in Case (21a),equation (9) implies t ij =0. In the former case, we proceed with a case-by-case reasoning considering sevenmutually exclusive subcases:(a) | i | < | k | and | j | 6 = | l | .(b) | i | < | k | and j = l .(c) | i | < | k | and j = − l .(d) | i | = | k | and | j | < | l | .(e) i = k and j = − l < i = − k < j = l .(g) i = − k < j = − l .Let’s consider in some details subcase (c). The remaining subcases are handled in a similar manner. Insubcase (c), (9) simplifies to:( − ( p ( i )+ p ( j ))( p ( k )+ p ( − j )) (cid:0) δ j> q + δ j< q − (cid:1) t ij t k, − j − t k, − j t ij + θ ( i, j, k ) δ j> ǫt i, − j t kj + ( − p ( j )( p ( i )+1) ǫ X − n ≤ a ≤ n ( − p ( i ) p ( a ) θ ( i, j, k ) δ | a | < | j | t i, − a t ka = 0 (10)Let us assume that | l | = | j | = 1. Then the previous equation reduces to( − ( p ( i )+ p ( j ))( p ( k )+ p ( − j )) (cid:0) δ j> q + δ j< q − (cid:1) t ij t k, − j + θ ( i, j, k ) δ j> ǫt i, − j t kj = t k, − j t ij Replacing j by − j leads to the equation( − ( p ( i )+ p ( − j ))( p ( k )+ p ( j )) (cid:0) δ j< q + δ j> q − (cid:1) t i, − j t kj + θ ( i, − j, k ) δ j< ǫt ij t k, − j = t kj t i, − j t k, − j t ij and t kj t i, − j are properly ordered and the previous two equations can be solved toexpress t ij t k, − j and t i, − j t kj in terms of the former.We then proceed by descending induction on | j | and show that t ij t k, − j can be expressed as a linearcombination of properly ordered monomials. The base case | j | = 1 was completed above. We use again (10)and the corresponding equation obtained after switching j and − j . In these two equations, by induction, themonomials t i, − a t ka with | a | < | j | can be expressed as linear combinations of properly ordered monomials.Moreover, t k, − j t ij and t kj t i, − j are already correctly ordered. As in the case | l | = | j | = 1, we can then solvethose two equations to express t ij t k, − j and t i, − j t kj in terms of properly ordered monomials.It remains to show that the reduced monomials form a linearly independent set. We follow the approachin [Ol]. Let M , . . . , M r be pairwise distinct reduced monomials in the generators τ ij such that a M + . . . + a r M r = 0 for some a , . . . , a r ∈ C q . Without loss of generality, we can assume that a i ∈ A . It is sufficientto prove that a , ..., a r ∈ A implies a , ..., a r ∈ ( q − A .Recall that there is a surjective homomorphism θ : U A p n → Up n More precisely, θ is the composite of ψ − from Theorem 4.1 and the projection U A p n → U A p n / ( q − U A p n from Theorem 4.1. Let M i = θ ( M i )and denote by ¯ a i the image of a i in A / ( q − A . Since M , . . . , M r are pairwise distinct reduced monomials, M , . . . , M r are pairwise distinct monomials in Up n . Then using that¯ a M + · · · + ¯ a r M r = θ ( a M + . . . + a r M r ) = 0and the (classical) PBW Theorem for Up n , we obtain ¯ a = . . . = ¯ a r = 0. Hence a , . . . , a r ∈ ( q − A asneeded.As mentioned in Remark 3.2, we may replace C ( q ) by C (( ~ )), q by e ~ / , and A by C [[ ~ ]], and an analogof Theorem 4.1 would hold true, implying that U C [[ ~ ]] p n is a flat deformation of Up n . Moreover, the nexttheorem states that U C [[ ~ ]] p n is a quantization of the co-Poisson Hopf superalgebra structure on Up n inducedby the Lie bisuperalgebra structure defined in Section 2. To be precise, the cobracket δ on p n extends to aPoisson co-bracket on Up n , which we also denote by δ . Let ( · ) ◦ be the involution on ( U C [[ ~ ]] p n ) ⊗ given by A ⊗ A ( − p ( A ) p ( A ) A ⊗ A where p ( A i ) is the Z / Z -degree of A i , i = 1 , A ∈ U C [[ ~ ]] p n , we denote by A both the image of A in U C [[ ~ ]] p n /h U C [[ ~ ]] p n and thecorresponding element in Up n via the isomorphism of the ~ -analogue of Theorem 4.1. Similarly, we identifythe corresponding elements in (cid:0) U C [[ ~ ]] p n /h U C [[ ~ ]] p n (cid:1) ⊗ (cid:0) U C [[ ~ ]] p n /h U C [[ ~ ]] p n (cid:1) and Up n ⊗ Up n . Theorem 4.3. If A ∈ U C [[ ~ ]] p n , we have ~ − (∆( A ) − ∆( A ) ◦ ) = δ ( A ) . Hence, U C [[ ~ ]] p n is a quantization ofthe co-Poisson Hopf superalgebra structure on Up n .Proof. We show that the identity above holds for the generators τ ij of U C [[ ~ ]] g , so let A = τ ij . We first notethat the identity is trivially satisfied for i = j , as both sides are zero. Assume henceforth that i = j . Then: ~ − (∆( τ ij ) − ∆( τ ij ) ◦ ) = (cid:18) e ~ / − e − ~ / ~ (cid:19) n X k = − n | i | < | k | < | j | (cid:16) ( − ( p ( i )+ p ( k ))( p ( j )+ p ( k )) τ ik ⊗ τ kj − τ kj ⊗ τ ik (cid:17) + (cid:18) e ~ / − ~ (cid:19) ( τ ii ⊗ τ ij − τ ij ⊗ τ ii + τ ij ⊗ τ jj − τ jj ⊗ τ ij ) − (cid:18) e ~ / − e − ~ / ~ (cid:19) δ i> (cid:16) ( − p ( j ) τ i, − i ⊗ τ − i,j + τ − i,j ⊗ τ i, − i (cid:17) + (cid:18) e ~ / − e − ~ / ~ (cid:19) δ j< (cid:16) ( − p ( i ) τ i, − j ⊗ τ − j,j − τ − j,j ⊗ τ i, − j (cid:17) U C [[ ~ ]] g / ~ U C [[ ~ ]] g , we have: ~ − (∆( τ ij ) − ∆( τ ij ) ◦ ) = n X k = − n | i | < | k | < | j | (cid:16) ( − ( p ( i )+ p ( k ))( p ( j )+ p ( k )) τ ik ⊗ τ kj − τ kj ⊗ τ ik (cid:17) + 12 ( τ ii ⊗ τ ij − τ ij ⊗ τ ii + τ ij ⊗ τ jj − τ jj ⊗ τ ij ) − δ i> (cid:16) τ − i,j ⊗ τ i, − i + ( − p ( j ) τ i, − i ⊗ τ − i,j (cid:17) + δ j< (cid:16) ( − p ( i ) τ i, − j ⊗ τ − j,j − τ − j,j ⊗ τ i, − j (cid:17) We next compute δ ( τ ij ) using the isomorphism of Theorem 4.1 and (3). δ ( τ ij ) = ( − p ( j ) δ ( E ji )= n X k = − n | i | < | k | < | j | ( − p ( j )+ p ( k ) (cid:16) ( − ( p ( i )+ p ( k ))( p ( j )+ p ( k )) E ki ⊗ E jk − E jk ⊗ E ki (cid:17) −
12 ( − p ( j ) (cid:16) ( − p ( j ) E jj − ( − p ( i ) E ii (cid:17) ⊗ E ji + 12 ( − p ( j ) E ji ⊗ (cid:16) ( − p ( j ) E jj − ( − p ( i ) E ii (cid:17) − δ j< − p ( j ) E j, − j ⊗ E − j,i + δ i> E − i,i ⊗ E j, − i + δ j< − p ( i )+ p ( j ) E − j,i ⊗ E j, − j + δ i> − p ( j ) E j, − i ⊗ E − i,i = ~ − (∆( τ ij ) − ∆( τ ij ) ◦ )as needed. q -Brauer algebra In [Mo], D. Moon identified the centralizer of the action of p n on the tensor space C ⊗ ln | n . This centralizer iscalled the periplectic Brauer algebra in the literature: see [Co, CP, CE1, CE2].We have a representation of U q p n on C q ( n | n ) via the assignment T S , and thus we also have arepresentation on each tensor power C q ( n | n ) ⊗ l . In this section, we identify the centralizer of the action of U q p n on C q ( n | n ) ⊗ l and call it the periplectic q -Brauer algebra. For the quantum group of type Q , this wasdone in [Ol] and the centralizer of its action is called the Hecke-Clifford superalgebra. Quantum analogs ofthe Brauer algebra were studied in [M] where they appear as centralizers of the action of twisted quantizedenveloping algebras U twq o n and U twq sp n on tensor representations (here, sp n is the symplectic Lie algebra);see also [We]. Definition 5.1.
The periplectic q -Brauer algebra B q,l is the associative C ( q ) -algebra generated by elements t i and c i for ≤ i ≤ l − satisfying the following relations: ( t i − q )( t i + q − ) = 0 , c i = 0 , c i t i = − q − c i , t i c i = q c i for ≤ i ≤ l −
1; (11) t i t j = t j t i , t i c j = c j t i , c i c j = c j c i if | i − j | ≥
2; (12) t i t j t i = t j t i t j , c i +1 c i c i +1 = − c i +1 , c i c i +1 c i = − c i for ≤ i ≤ l −
2; (13) t i c i +1 c i = − t i +1 c i + ( q − q − ) c i +1 c i , c i +1 c i t i +1 = − c i +1 t i + ( q − q − ) c i +1 c i (14) Remark . Setting q = 1 in this definition yields the algebra A l from Definition 2.2 in [Mo]. Lemma 5.3.
We have U q p n -module homomorphisms ϑ : C q ( n | n ) ⊗ C q ( n | n ) → C ( q ) and ǫ : C ( q ) → C q ( n | n ) ⊗ C q ( n | n ) given by ϑ ( e a ⊗ e b ) = δ a, − b ( − p ( a ) and ǫ (1) = P na = − n e a ⊗ e − a . roof. It is enough to check that, for all the generators t ij of U q p n and any tensor v ∈ C q ( n | n ) ⊗ C q ( n | n ), ϑ ( t ij ( v )) = t ij ( ϑ ( v )) and ǫ ( t ij (1)) = t ij ( ǫ (1)) . (15)Here is a brief sketch of some of the computations.Using the formula for the coproduct, we have: t ij ( e a ⊗ e − a ) = n X k = − n ( − ( p ( i )+ p ( k ))( p ( k )+ p ( j ))+( p ( k )+ p ( j )) p ( a ) t ik ( e a ) ⊗ t kj ( e − a ) (16)This can be made more explicit using t ii ( e a ) = n X b = − n q δ bi (1 − p ( i ))+ δ b, − i (2 p ( i ) − E bb ( e a ); t i, − i ( e a ) = ( q − q − ) δ i> E − i,i ( e a ); t ij ( e a ) = ( q − q − )( − p ( i ) E ji ( e a ) , if | i | 6 = | j | . We obtain, for instance, t ii ( e a ⊗ e a ) = q δ a ,i (1 − p ( i ))+ δ a , − i (2 p ( i ) − q δ a ,i (1 − p ( i ))+ δ a , − i (2 p ( i ) − e a ⊗ e a If a = − a = − a , this simplifies to e a ⊗ e − a and this allows us to check (15) quickly for i = j .Furthermore, t i, − i ( e a ⊗ e − a ) = ( − p ( a ) δ i> t ii ( e a ) ⊗ t i, − i ( e − a ) + δ i> t i, − i ( e a ) ⊗ t − i, − i ( e − a )It follows that t i, − i (cid:0)P na = − n e a ⊗ e − a (cid:1) = 0, so the identity for ǫ in(15) holds for j = − i .Suppose now that a = − a . Then t i, − i ( e a ⊗ e a ) = δ i> δ ( a = a = i )( q − q − ) qe i ⊗ e − i + δ i> δ ( a = a = i )( q − q − ) qe − i ⊗ e i Observe that ϑ ( e i ⊗ e − i + e − i ⊗ e i ) = 0, so we have shown that ϑ ( t i, − i ( e a ⊗ e a )) = t i, − i ( ϑ ( e a ⊗ e a )) andthis proves (15) for ϑ when j = − i .Next, we consider the case | i | 6 = | j | . To prove the identity for ǫ in (15), we use again (16) and obtain that t ij n X a = − n e a ⊗ e − a ! = 0 by considering subcases i = ± a , j = ± a , and k = ± a . To show that (15) holds for ϑ we also proceed with case-by-case verification. The case a , a
6∈ {± i, ± j } is immediate. If a ∈ {± i, ± j } , a
6∈ {± i, ± j } , and a = − a , then t ij ( e a ⊗ e a ) = ( q − q − ) ( − ( p ( i )+ p ( a ))( p ( a )+ p ( j ))+( p ( a )+ p ( j )) p ( a ) ( − p ( i )+ p ( a ) E a i ( e a ) ⊗ E ja ( e a )+ ( q − q − )( − p ( i ) E ji ( e a ) ⊗ E a a ( e a ) . This shows that ϑ ( t ij ( e a ⊗ e a )) = 0 = t ij ( ϑ ( e a ⊗ e a )). Similarly, we obtain the desired identity in theother cases.By composing ϑ and ǫ , we obtain a U q p n -module homomorphism ǫ ◦ ϑ : C q ( n | n ) ⊗ → C q ( n | n ) ⊗ . In termsof elementary matrices, this linear map is given by P na,b = − n ( − p ( a ) p ( b ) E ab ⊗ E − a, − b , which we abbreviateby c . The super-permutation operator P on C q ( n | n ) ⊗ is given by P = P na,b = − n ( − p ( b ) E ab ⊗ E ba , so10 = P ( π ◦ st) where ( π ◦ st) stands for the map π ◦ st applied to the second tensor in the previous formulafor P .We can extend c to a U q p n -module homomorphism c i : C q ( n | n ) ⊗ l → C q ( n | n ) ⊗ l for 1 ≤ i ≤ l − c to the i th and ( i + 1) th tensors.The linear map C q ( n | n ) ⊗ l → C q ( n | n ) ⊗ l given by P i S i,i +1 where P i is the super-permutation operatoracting on the i th and ( i + 1) th tensors is also a U q p n -module homomorphism: this is a consequence of thefact that S is a solution of the quantum Yang-Baxter relation. Proposition 5.4.
The tensor superspace C q ( n | n ) ⊗ l is a module over B q,l if we let t i act as P i S i,i +1 and c i act as c i .Proof. That the linear operators P i S i,i +1 satisfy the braid relation (the first relation in (13)) is a consequenceof the fact that S is a solution of the quantum Yang-Baxter relation. The relations (12) for the operators P i S i,i +1 and c i can be easily verified. As for the other relations, they can be checked via direct computations.It is enough to check the relations (11) on C q ( n | n ) ⊗ and the relations (14) on C q ( n | n ) ⊗ . We briefly sketchsome of those computations below.First, note that c P = − c and P c = c . Also, we easily obtain the following: c ( q − n X i =1 E ii ⊗ E ii ! = c ( q − − n X i =1 E − i, − i ⊗ E − i, − i ! = 0 , c ( q − n X i =1 E ii ⊗ E − i, − i ! = ( q − n X a = − n n X b =1 E ab ⊗ E − a, − b , c ( q − − n X i =1 E − i, − i ⊗ E ii ! = ( q − − n X a = − n − X b = − n ( − p ( a ) E ab ⊗ E − a, − b , c − X i = − n E i, − i ⊗ E − i,i ! = − n X a = − n n X b =1 E ab ⊗ E − a, − b , c X ≤| j | < | i |≤ n ( − p ( j ) E ij ⊗ E ji = n X a = − n X ≤| j | < | i |≤ n ( − p ( a )( p ( i )+1)+ p ( j ) E a, − i ⊗ E − a,i = 0 . Therefore, we have that c ( S −
1) = ( q − − c , hence c S = q − c . Now using that c = − c P , we obtain thethird relation in (11). Similarly, we prove ( S − c = ( q − c , and then using P c = c , we obtain the fourthrelation in (11).For the remaining relations we use the following formula: P S = n X i,j = − n ( − p ( j ) E ij ⊗ E ji + ( q − n X i =1 ( E − i,i ⊗ E i, − i )+ ( q − n X i =1 ( E ii ⊗ E ii ) − ( q − − n X i =1 ( E i, − i ⊗ E − i,i ) − ( q − − n X i =1 ( E − i, − i ⊗ E − i, − i ) + ( q − q − ) − X i = − n ( E − i, − i ⊗ E ii )+ ( q − q − ) X | j | < | i | ( E jj ⊗ E ii ) + ( q − q − ) X | j | < | i | (cid:16) ( − p ( i ) p ( j ) E ji ⊗ E − j, − i (cid:17)
11s mentioned after the definition of B q,l , the module structure given in the previous proposition commuteswith the action of U q ( p n ) on C q ( n | n ) ⊗ l . We thus have algebra homomorphisms B q,l −→ End U q ( p n ) ( C q ( n | n ) ⊗ l ) and U q ( p n ) −→ End B q,l ( C q ( n | n ) ⊗ l ) . The main theorem of this section states that B q,l is the full centralizer of the action of U q ( p n ) on C q ( n | n ) ⊗ l when n ≥ l . Theorem 5.5.
The map B q,l −→ End U q ( p n ) ( C q ( n | n ) ⊗ l ) is surjective and it is injective when n ≥ l . This is a q -analogue of Theorem 4.5 in [Mo]. The proof follows the lines of the proof of Theorem 3.28in [BGJKW], using Proposition 5.6 below and Theorem 4.5 in [Mo] along with Lemma 3.27 in [BGJKW],which can be applied in the present situation.Recall that A = C [ q, q − ] ( q − is the localization of C [ q, q − ] at the ideal generated by q −
1. Let B q,l ( A )be the A -associative algebra which is defined via the same generators and relations as B q,l . Proposition 5.6.
The quotient algebra B q,l ( A ) / ( q − B q,l ( A ) is isomorphic to the algebra A l given inDefinition 2.2 in [Mo].Proof. It follows immediately from the definitions of both A l and B q,l ( A ) that we have a surjective algebrahomomorphism A l ։ B q,l ( A ) / ( q − B q,l ( A ). That it is injective can be proved as in the proof of Proposition3.21 in [BGJKW] using Theorem 4.1 in [Mo].The q -Schur superalgebras of type Q were introduced in [BGJKW] and [DuWa1, DuWa2]. Considering loc. cit. and the earlier work on q -Schur algebras for gl n (see for instance [Do]), the following definition isnatural. Definition 5.7.
The q -Schur superalgebra S q ( p n , l ) of type P is the centralizer of the action of B q,l on C q ( n | n ) ⊗ l , that is, S q ( p n , l ) = End B q,l ( C q ( n | n ) ⊗ l ) . We have an algebra homomorphism U q ( p n ) −→ S q ( p n , l ): it is an open question whether or not this mapis surjective. We also have an algebra homomorphism B q,l −→ End S q ( p n ,l ) ( C q ( n | n ) ⊗ l ) and it is natural toexpect that it should be an isomorphism, perhaps under certain conditions on n and l . References [AGG] S. Ahmed, D. Grantcharov, N. Guay,
Quantized enveloping superalgebra of type P , preprint.[BDEA +
1] M. Balagovic, Z. Daugherty, I. Entova-Aizenbud, I. Halacheva, J. Hennig, M. S. Im, G. Letzter,E. Norton, V. Serganova, C. Stroppel,
The affine
V W supercategory , Selecta Math. (N.S.) (2020),no. 2, Paper No. 20, 42 pp., arXiv:1801.04178.[BDEA +
2] M. Balagovic, Z. Daugherty, I. Entova-Aizenbud, I. Halacheva, J. Hennig, M. S. Im, G. Letzter, E.Norton, V. Serganova, C. Stroppel,
Translation functors and decomposition numbers for the periplecticLie superalgebra p ( n ), Math. Res. Lett. (2019), no. 3, 643–710, arXiv:1610.08470.[BGJKW] G. Benkart, N. Guay, J. H. Jung, S.-J. Kang, S. Wilcox, Quantum walled Brauer-Clifford super-algebras , J. Algebra (2016), 433–474.[CE1] K. Coulembier, M. Ehrig,
The periplectic Brauer algebra II: Decomposition multiplicities , J. Comb.Algebra (2018), no. 1, 19–46. 12CE2] K. Coulembier, M. Ehrig, The periplectic Brauer algebra III: The Deligne category , Algebr RepresentTheory (2020). https://doi.org/10.1007/s10468-020-09976-8.[ChGu] H. Chen, N. Guay,
Twisted affine Lie superalgebra of type Q and quantization of its envelopingsuperalgebra , Math. Z., (2012), no.1, 317–347.[Co] K. Coulembier, The periplectic Brauer algebra , Proc. Lond. Math. Soc. (3) (2018), no. 3, 441–482.[CP] C. Chen, Y. Peng,
Affine periplectic Brauer algebras , J. Algebra (2018), 345–372.[DHIN] Z. Daugherty, I. Halacheva, M.S. Im, and E. Norton,
On calibrated representations of the degenerateaffine periplectic Brauer algebra , arXiv:1905.05148.[Do] S. Donkin,
The q -Schur Algebra , London Mathematical Society Lecture Note Series, , CambridgeUniversity Press, Cambridge, 1998.[DuWa1] J. Du, J. Wan, Presenting queer Schur superalgebras , Int. Math. Res. Not. IMRN 2015, no. 8,2210–2272.[DuWa2] J. Du, J. Wan,
The queer q -Schur superalgebra , J. Aust. Math. Soc. (2018), , no.3, 316–346.[EAS1] Inna Entova-Aizenbud, Vera Serganova, Deligne categories and the periplectic Lie superalgebra ,arXiv:1807.09478.[EAS2] Inna Entova-Aizenbud, Vera Serganova,
Kac-Wakimoto conjecture for the periplectic Lie superalge-bra , arXiv:1905.04712.[FRT] L. Faddeev, N. Reshetikhin, L. Takhtajan,
Quantization of Lie groups and Lie algebras (Russian),Algebra i Analiz (1989), no. 1, 178–206; translation in Leningrad Math. J. (1990), no. 1, 193–225.[FSS] L. Frappa, P. Sorba, A. Sciarrino, Deformation of the strange superalgebra ˜ P ( n ), J. Phys. A: Math.Gen. (1993) 661–665.[GJKK] D. Grantcharov, J.H. Jung., S.-J. Kang, M. Kim, Highest weight modules over quantun queer su-peralgebra U q ( q ( n )), Comm. Math. Phys. (2010), no. 3, 827–860.[GJKKK] D. Grantcharov, J.H. Jung., S.-J. Kang, M. Kashiwara, M. Kim, Quantum Queer Superalgebraand Crystal Bases , Proc. Japan Acad. Ser. A Math. Sci. (2010), no. 10, 177–182, arXiv:1007.4105.[K] V. Kac, Lie superalgebras , Adv. Math. (1977), no. 1, 8–96.[KT] J. Kujawa, B. Tharp, The marked Brauer category , J. Lond. Math. Soc. (2017), no. 2, 393–413.[LeSh] D. Leites, A. Shapovalov, Manin-Olshansky triples for Lie superalgebras , J. Nonlinear Math. Phys. (2000), no. 2, 120–125.[M] A. Molev, A new quantum analog of the Brauer algebra , Quantum groups and integrable systems,Czechoslovak J. Phys. (2003), no. 11, 1073–1078.[Mo] D. Moon, Tensor product representations of the Lie superalgebra p ( n ) and their centralizers , Comm.Algebra (2003), no. 5, 2095–2140.[Na1] M. Nazarov, Yangians of the ”strange” Lie superalgebras , Quantum groups (Leningrad, 1990), LectureNotes in Math. , pp. 90-–97, Springer, Berlin, 1992.[Na2] M. Nazarov,
Yangian of the queer Lie superalgebra , Comm. Math. Phys. (1999), no. 1, 195–223.[Ol] G. Olshanski,
Quantized universal enveloping superalgebra of type Q and a super-extension of the Heckealgebra , Lett. Math. Phys. (1992), no. 2, 93–102.[Ser] V. Serganova, On representations of the Lie superalgebra p ( n ), J. Algebra (2002), 615–630.13We] H. Wenzl, A q -Brauer algebra , J. Algebra358