Quantized Enveloping Superalgebra of Type $ P$

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The new periplectic q-Brauer algebra Bq,ℓ and the new periplectic q-Schur algebra are introduced in the last section, where we prove that Bq,ℓ can be defined equivalently either using generators and relations or as the centralizer of the action of Uq(pn) on the tensor space: see Theorem 5.5.

The proofs of our results require extensive computations: further details for all the computations can be found in [AGG].

Acknowledgements: The second named author is partly supported by the Simons Collaboration Grant 358245. He also would like to thank the Max Planck Institute in Bonn (where part of this work was completed) for the excellent working conditions. The third named author gratefully acknowledges the ﬁnancial support of the Natural Sciences and Engineering Research Council of Canada provided via the Discovery Grant Program. We thank Patrick Conner, Robert Muth and Vidas Regelskis for help with certain computations in the preliminary stages of the present paper. We also thank Nicholas Davidson and Jonathan Kujawa for some useful discussions.

1 The Lie superalgebra of type P

Let C(n n) be the vector superspace Cn Cn spanned by the odd standard basis vectors e ,...,e and | ⊕ −n −1 the even standard basis vectors e1,...,en. Let Mn|n(C) be the vector superspace consisting of matrices A = (aij ) with aij C and with rows and columns labelled using the integers n, . . . , 1, 1,...,n, so i, j 1, 2,..., n∈ . Set p(i)=1 Z if n i 1 and p(i)=0 Z if 1 − i n. The parity ∈ {± ± ± } ∈ 2 − ≤ ≤ − ∈ 2 ≤ ≤ of the elementary matrix Eij is p(i)+ p(j) mod 2. We denote by gln|n the Lie superalgebra over C whose underlying vector space is Mn|n(C) and which is equipped with the Lie superbracket

[E , E ]= δ E ( 1)(p(i)+p(j))(p(k)+p(l))δ E . ij kl jk il − − il kj

st st p(i)(p(j)+1) Recall that the supertranspose ( ) on gln|n is given by the formula (Eij ) = ( 1) Eji. The involution ι on gl which will be relevant· for this paper is given by ι(X)= π(Xst) where− π : gl gl n|n − n|n −→ n|n is the linear map given by π(Eij )= E−i,−j .

Deﬁnition 1.1. The Lie superalgebra pn of type P , which is also called the periplectic Lie superalgebra, is the subspace of ﬁxed points of gl under the involution ι, that is, p = X gl ι(X)= X . n|n n { ∈ n|n | }

A B If X p with and A,B,C,D M (C), then D = At, B = Bt and C = Ct where t denotes ∈ n CD ∈ n − − the transpose with respect to the diagonal i = j. For convenience, we set − E = E + ι(E )= E ( 1)p(i)(p(j)+1)E . ij ij ij ij − − −j,−i

The superbracket on pn is given by

[E , E ] = δ E ( 1)(p(i)+p(j))(p(k)+p(l))δ E ji lk il jk − − jk li δ ( 1)p(l)(p(k)+1)E δ ( 1)p(j)(p(i)+1)E (1) − i,−k − j,−l − −j,l − −i,k

A basis of pn is provided by all the matrices Eij with indices i and j respecting one of the following inequalities: 1 j < i n or 1 i = j n or n i = j 1. ≤ | | | |≤ ≤ ≤ − ≤ − ≤− Note that E = ( 1)p(i)(p(j)+1)E for all i, j 1,..., n , hence E =0 when 1 i n. ij − − −j,−i ∈ {± ± } i,−i ≤ ≤

2 2 Lie bisuperalgebra structure

To construct a Lie bisuperalgebra structure on pn, 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, a1, a2) consists of a Lie superalgebra a equipped with an ad-invariant supersymmetric non-degenerate bilinear form B along with two Lie subsuperalgebras a1, a2 of a which are B-isotropic transversal subspaces of a. Note that such a bilinear form B defines a non-degenerate pairing between a and a and a supercobracket δ : a a⊗2 via 1 2 1 → 1 B⊗2(δ(X), Y Y )= B(X, [Y , Y ]), 1 ⊗ 2 1 2 where X a , Y , Y a . ∈ 1 1 2 ∈ 2 Definition 2.1. The “butterfly” Lie superalgebra bn is the subspace of gln|n spanned by Eij with 1 i < j n and by E + E , E for 1 i n. ≤ | | | |≤ ii −i,−i i,−i ≤ ≤

Note that gln|n = pn bn. It is well-known that the bilinear form B( , ) on gln|n given by the super-trace, B(A, B) = Str(AB), is ad-invariant,⊕ supersymmetric and non-degenrate.· ·

One easily checks that B(X1,X2) = 0 if X1,X2 pn or if X1,X2 bn. Hence we have the following result. ∈ ∈

Proposition 2.2. (gln|n, pn, bn) is a Manin supertriple. Remark 2.3. A similar Manin supertriple is given in [LeSh], §2.2.

The quantum superalgebra that we will deﬁne in the next section will be a quantization of the Lie bisuperalgebra structure given by the Manin supertriple (gln|n, pn, bn).

We extend the form B( , ) to a non-degenerate pairing B⊗2 on gl C gl by setting · · n|n ⊗ n|n B⊗2(X X , Y Y ) = ( 1)|X2||Y1|B(X , Y )B(X , Y ) 1 ⊗ 2 1 ⊗ 2 − 1 1 2 2 |X2||Y1| for all homogeneous elements X1,X2, Y1, Y2 pn. The sign ( 1) is necessary to make this form ad-invariant. ∈ −

Let 1 1 s = ( 1)p(j)E E + E (E + E )+ E E (2) − ij ⊗ ji 2 ii ⊗ ii −i,−i 2 −i,i ⊗ i,−i 1≤|jX|<|i|≤n 1≤Xi≤n 1≤Xi≤n A scalar multiple of this element is called the fake Casimir element in [BDEA+1].

Proposition 2.4. s is a solution of the classical Yang-Baxter equation: [s12, s13] + [s12, s23] + [s13, s23]=0.

The proof of the above proposition follows from the lemma below, which should be well-known among experts.

Lemma 2.5. Let p be a ﬁnite dimensional Lie superalgebra and suppose that (p, p1, p2) is a Manin triple B ′ with respect to a certain supersymmetric, invariant, bilinear form ( , ). Let Xi i∈I , Xi i∈I be bases of B ′ · · { } ′ { } p1 and p2, respectively, dual in the sense that (Xi,Xj )= δij . Set s = i∈I Xi Xi. Then s is a solution of the classical Yang-Baxter equation. ⊗ P

We next compute the supercobracket δ using the identity B(X, [Y , Y ]) = B(δ(X), Y Y ) for all X p 1 2 1 ⊗ 2 ∈ n

3 and all Y , Y b . The formula for δ is (assuming, without loss of generality, that j i ): 1 2 ∈ n | | ≤ | | n δ(E ) = ( 1)p(k)+1 E E ( 1)(p(i)+p(k))(p(j)+p(k))E E ij − ik ⊗ kj − − kj ⊗ ik k=−n |j| 0) + ( 1)p(i)E E + E E 2 − −j,j ⊗ i,−j i,−j ⊗ −j,j 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 1+1 X, s]. (4) ⊗ ⊗

3 Quantized enveloping superalgebra

In this section, we deﬁne the quantized enveloping superalgebra Uqpn 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 the field C(q) of rational functions in the variable q and set C (n n) = q q | C C C(n n). q ⊗ | Definition 3.1. Let S EndC (C (n n)⊗2) be given by the formula: ∈ q q | q q−1 S = 1+ (q 1)E + (q−1 1)E (E + E )+ − E E − ii − −i,−i ⊗ ii −i,−i 2 i,−i ⊗ −i,i 1≤i≤n −n≤i≤−1 X X +(q q−1) ( 1)p(j)E E (5) − − ij ⊗ ji 1≤|jX|<|i|≤n ⊗2 Remark 3.2. If we define S instead as an element of EndC[[~]](C~(n n) ) by the same formula as in definition −1 ~/2 −~/2 ⊗2 | ⊗2 ⊗2 3.1 but with q, q replaced by e ,e and Cq(n n) replaced by C~(n n) , which equals C(n n) [[~]], then S =1+ ~s + O(~2). | | |

Theorem 3.3. S is a solution of the quantum Yang-Baxter equation: S12S13S23 = S23S13S12.

Proof. The proof consists of verifying long computations. To simplify them, we have used the following method. Set f(q) = S12S13S23 S23S13S12. The main idea is to consider f(q) as a Laurent polynomial 3 − ⊗3 i C C ′ i=−3 fiq with coeﬃcients fi in End n|n . Then one shows the eight relations f(a) = 0, f (b) = 0, ′′ Pf (c) =0 for a,b,c = 1 and b = √ 1. (Actually, just seven of those are enough.) We can then deduce that f(q) is a scalar multiple± of (q ± q−−1)3 and we show that the coeﬃcient of q3 in f(q) is zero. − Here are some more details.

Let us set C = (E + E ) (E + E ). ii −i,−i ⊗ ii −i,−i 1≤Xi≤n

4 Then q + q−1 S =1+(q q−1)s + 1 C. − 2 −

For convenience, we introduce the following notation:

[sC]= s12C13 + s12C23 + s13C23 + C12s13 + C12s23 + C13s23 s C s C s C C s C s C s − 23 13 − 23 12 − 13 12 − 23 13 − 23 12 − 13 12 [sCC]= s C C + C s C + C C s s C C C S C C C s 12 13 23 12 13 23 12 13 23 − 23 13 12 − 23 13 12 − 23 13 12 [ssC]= s s C + C s s + s C s s s C C s s s C s 12 13 23 12 13 23 12 13 23 − 23 13 12 − 23 13 12 − 23 13 12

The relations f(a) = 0, f ′(b) = 0, f ′′(c) = 0 for a,b,c = 1 and b = √ 1 follow from the next two lemmas and checking these involves explicit computations. ± ± − Lemma 3.4. [sC] = 2[sCC] Lemma 3.5. [ssC]=0

For instance, f ′( 1) = 0 follows from f ′( 1) = 4[sC] + 8[sCC] and the two lemmas. Furthermore, − − − f ′′( 1) = 4[sC] + 8[sCC] 16[ssC] + 8([s , s ] + [s , s ] + [s , s ]). − − − 12 13 12 23 13 23 Therefore, f ′′( 1) = 0 thanks to Lemmas 2.5, 3.4, and 3.5. Similarly, the two lemmas above imply that − f ′(√ 1)=2√ 1[sC] 4√ 1[sCC] 4[ssC] − − − − − vanishes.

3 The last step in the proof of Theorem 3.3 is to show the vanishing of the coeﬃcient f3 of q . We have 1 1 1 1 f = s s s s s s + [sCC]+ [ssC]+ C C C C C C , 3 12 13 23 − 23 13 12 4 2 8 12 13 23 − 8 23 13 12 which simpliﬁes to 1 s s s s s s + [sCC] (6) 12 13 23 − 23 13 12 4

thanks to Lemma 3.5 and C12C13C23 C23C13C12 = 0. Verifying that (6) vanishes follows by direct and extensive 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 pn is the Z2-graded Cq algebra Uqpn generated by elements t ,t−1 with 1 i j n and i, j 1,..., n which satisfy the− following relations: ij ii ≤ | | ≤ | |≤ ∈ {± ± } t = t , t =0 if i> 0, t =0 if i > j ; (7) ii −i,−i −i,i ij | | | |

T12T13S23 = S23T13T12 (8) ⊗2 where T = t C E and the last equality holds in U p C EndC (C (n n)) . The Z -degree |i|≤|j| ij ⊗ ij q n ⊗ (q) (q) q | 2 of t is p(i)+ p(j). ij P

U p is a Hopf algebra with antipode given by T T −1 and with coproduct given by q n 7→ n ∆(t )= ( 1)(p(i)+p(k))(p(k)+p(j))t t . ij − ik ⊗ kj kX=−n

5 4 Limit when q 1 and quantization 7→

We want to explain how Up can be viewed as the limit when q 1 of U p and how the co-Poisson Hopf n 7→ q n algebra structure on Upn, which is inherited from the cobracket δ on pn, can be recovered from the coproduct on Uqpn.

tij tii−1 −1 Set τ = −1 if i = j and set τ = . Let be the localization of C[q, q ] at the ideal generated ij q−q 6 ii q−1 A by q 1. Let U p be the -subalgebra of U p generated by τ when 1 i j n. − A n A q n ij ≤ | | ≤ | |≤ p(j) Theorem 4.1. The map ψ : Upn UApn/(q 1)UApn given by ψ(Eji) = ( 1) τ ij for i < j , 1 i = j n, and ψ(E )= 2τ −→for 1 i n−, is an associative C-superalgebra− isomorphism.| | | | ≤ ≤ −i,i − i,−i ≤ ≤

Proof. First, we need to write down explicitly the deﬁning relation (8). Comparing coeﬃcients of Eij Ekl on both sides of relation (8), we obtain: ⊗

( 1)(p(i)+p(j))(p(k)+p(l))t t t t + θ(i, j, k) δ δ ǫt t − ij kl − kl ij |j|<|l| − |k|<|i| il kj (p(i)+p(j))(p(k)+p(l)) −1 + ( 1) δj>0(q 1) + δj< 0(q 1) δjl+ δj,−l tij tkl − − − δ (q 1)+ δ (q−1 1) δ + δ t t − i>0 − i<0 − ik i,−k kl ij p(j) + θ(i, j, k)δj>0δj,−lǫti,−j tk,−l ( 1) δi<0δi,−kǫt−k,lt−i,j − − + ( 1)p(j)(p(i)+1)ǫ ( 1)p(i)p(a)θ(i, j, k)δ δ t t + ( 1)p(−j)p(a)δ δ t t − − j,−l |a|<|l| i,−a ka − i,−k |k|<|a| al −a,j −n≤a≤n X =0 (9)

In the identity above, we set

θ(i, j, k) = sgn(sgn(i) + sgn(j) + sgn(k)) and ǫ = q q−1. −

In order to check that ψ([Eji, Ekl]) = [ψ(Eji), ψ(Ekl)], we proceed as follows. We apply ψ on both sides of (1). To show that the resulting right hand side coincides with [ψ(Eji), ψ(Ekl)], we use (9) and pass to the quotient U p /(q 1)U p . This is done via a long case-by-case veriﬁcation for i,j,k,l. A n − A n

From the way UApn 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 Cq(n n) is a representation of Uqpn via the assignment T S, hence also of p by restriction. More explicitly,| 7→ UA n τ ( 1)p(i)E if i < j , and τ E , τ (E q−1E ) if 1 i n. ij 7→ − ji | | | | i,−i 7→ −i,i ii 7→ ii − −i,−i ≤ ≤

Let πℓ be the quotient homomorphism

⊗ℓ ⊗ℓ ⊗ℓ ⊗ℓ End (C (n n) ) End (C (n n) )/(q 1)End (C (n n) ) = EndC(C(n n) ). A A | −→ A A | − A A | ∼ | ⊗ℓ The composite πℓ φℓ descends to a homomorphism πℓ φℓ from UApn/(q 1)UApn to EndC(C(n n) ). The ◦ ◦ − ⊗ℓ | composite πℓ φℓ ψ is the superalgebra homomorphism Upn EndC(C(n n) ) induced by the natural p -module structure◦ ◦ on C(n n)⊗ℓ twisted by the automorphism−→ of p given by| E ( 1)p(i)+p(j)E . n | n ij 7→ − ij

We can combine the homomorphisms πℓ φℓ ψ for all ℓ 1 to obtain a homomorphism pn ∞ C ⊗ℓ ◦ ◦C ≥ U −→ ℓ=1 EndC( (n n) ). This map is injective since (n n) is a faithful representation of pn. It follows that ψ is injective as| well. | Q

6 We next show that a PBW-type theorem holds for Uqpn. For this, we ﬁrst introduce a total order on the set of generators t , 1 i j n, of U p as follows. We declare that t t if ≺ ij ≤ | | ≤ | |≤ q n ij ≺ kl

This order leads to a total lexicographic order on the set of words formed by the generators tij . Namely, if A = A1 Ar and B = B1 Bs are two such words in the sense that each Ak for 1 k r and each Bl for 1 l ···s is equal to some generator··· t , then A B if r~~0 i i ∈ \{ } i diagonal, and ki = 1 if Ai is odd.~~

~~Theorem 4.2. The reduced monomials form a basis of Uqpn over Cq.~~

Proof. We ﬁrst show that the set of reduced monomials spans Uqpn. Note that it is enough to show that all quadratic monomials are in the span of this set. Let tij tkl be a quadratic monomial which is not reduced. We have that either tkl = tij , or i = k, j = l and tij is odd. In the latter case, as shown in Case (21a), 2 6 equation (9) implies tij =0. In the former case, we proceed with a case-by-case reasoning considering seven mutually exclusive subcases:

~~Let’s consider in some details subcase (c). The remaining subcases are handled in a similar manner. In subcase (c), (9) simpliﬁes to:~~

( 1)(p(i)+p(j))(p(k)+p(−j)) δ q + δ q−1 t t t t + θ(i, j, k)δ ǫt t − j>0 j<0 ij k,−j − k,−j ij j>0 i,−j kj p(j)(p(i)+1) p(i)p(a) (10) + ( 1) ǫ ( 1) θ(i, j, k)δ ti,−atka =0 − − |a|<|j| −nX≤a≤n Let us assume that l = j = 1. Then the previous equation reduces to | | | | ( 1)(p(i)+p(j))(p(k)+p(−j)) δ q + δ q−1 t t + θ(i, j, k)δ ǫt t = t t − j>0 j<0 ij k,−j j>0 i,−j kj k,−j ij Replacing j by j leads to the equation − ( 1)(p(i)+p(−j))(p(k)+p(j)) δ q + δ q−1 t t + θ(i, j, k)δ ǫt t = t t − j<0 j>0 i,−j kj − j<0 ij k,−j kj i,−j 7 The monomials tk,−j tij and tkj ti,−j are properly ordered and the previous two equations can be solved to express tij tk,−j and ti,−j tkj in terms of the former.

We then proceed by descending induction on j and show that tij tk,−j can be expressed as a linear combination 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, the − monomials ti,−atka with a < j can be expressed as linear combinations of properly ordered monomials. Moreover, t t and t |t | are| | already correctly ordered. As in the case l = j = 1, we can then solve k,−j ij kj i,−j | | | | those two equations to express tij tk,−j and ti,−j tkj in terms of properly ordered monomials.

It remains to show that the reduced monomials form a linearly independent set. We follow the approach in [Ol]. Let M1,...,Mr be pairwise distinct reduced monomials in the generators τij such that a1M1 + ... + arMr = 0 for some a1,...,ar Cq. Without loss of generality, we can assume that ai . It is suﬃcient to prove that a , ..., a implies∈ a , ..., a (q 1) . ∈ A 1 r ∈ A 1 r ∈ − A

Recall that there is a surjective homomorphism θ : UApn Upn More precisely, θ is the composite of −1 → ψ from Theorem 4.1 and the projection UApn UApn/(q 1)UApn from Theorem 4.1. Let M i = θ(Mi) and denote bya ¯ the image of a in /(q 1) . Since→ M ,...,M− are pairwise distinct reduced monomials, i i A − A 1 r M 1,..., M r are pairwise distinct monomials in Upn. Then using that

~~a¯ M + +¯a M = θ(a M + ... + a M )=0 1 1 ··· r r 1 1 r r and the (classical) PBW Theorem for Upn, we obtaina ¯1 = ... =a ¯r = 0. Hence a1,...,ar (q 1) as needed. ∈ − A~~

As mentioned in Remark 3.2, we may replace C(q) by C((~)), q by e~/2, and by C[[~]], and an analog A of Theorem 4.1 would hold true, implying that UC[[~]]pn is a ﬂat deformation of Upn. Moreover, the next theorem states that UC[[~]]pn is a quantization of the co-Poisson Hopf superalgebra structure on Upn induced by the Lie bisuperalgebra structure deﬁned in Section 2. To be precise, the cobracket δ on pn extends to a ◦ ⊗2 Poisson co-bracket on Up , which we also denote by δ. Let ( ) be the involution on (UC ~ p ) given by n · [[ ]] n A A ( 1)p(A1)p(A2)A A where p(A ) is the Z/2Z-degree of A ,i =1, 2. 1 ⊗ 2 7→ − 2 ⊗ 1 i i For convenience, for A UC ~ p , we denote by A both the image of A in UC ~ p /hUC ~ p and the ∈ [[ ]] n [[ ]] n [[ ]] n corresponding element in Upn via the isomorphism of the ~-analogue of Theorem 4.1. Similarly, we identify the corresponding elements in UC ~ p /hUC ~ p UC ~ p /hUC ~ p and Up Up . [[ ]] n [[ ]] n ⊗ [[ ]] n [[ ]] n n ⊗ n Theorem 4.3. If A UC ~ p, we have ~−1(∆(A) ∆(A)◦)= δ(A). Hence, UC ~ p is a quantization of ∈ [[ ]] n − [[ ]] n the co-Poisson Hopf superalgebra structure on Upn.

Proof. We show that the identity above holds for the generators τij of UC[[~]]g, so let A = τij . We ﬁrst note that the identity is trivially satisﬁed for i = j, as both sides are zero. Assume henceforth that i = j. Then: 6 e~/2 e−~/2 n ~−1 (∆(τ ) ∆(τ )◦)= − ( 1)(p(i)+p(k))(p(j)+p(k))τ τ τ τ ij − ij ~ − ik ⊗ kj − kj ⊗ ik k=−n |i|0 − i,−i ⊗ −i,j −i,j ⊗ i,−i e~/2 e−~/2 + − δ ( 1)p(i)τ τ τ τ ~ j<0 − i,−j ⊗ −j,j − −j,j ⊗ i,−j

~~8 Thus, in UC[[~]]g/~UC[[~]]g, we have:~~

n ~−1 (∆(τ ) ∆(τ )◦)= ( 1)(p(i)+p(k))(p(j)+p(k))τ τ τ τ ij − ij − ik ⊗ kj − kj ⊗ ik k=−n |i|0 −i,j ⊗ i,−i − i,−i ⊗ −i,j + δ ( 1)p(i)τ τ τ τ j<0 − i,−j ⊗ −j,j − −j,j ⊗ i,−j We next compute δ(τ ij ) using the isomorphism of Theorem 4.1 and (3).

δ(τ ) = ( 1)p(j)δ(E ) ij − ji n = ( 1)p(j)+p(k) ( 1)(p(i)+p(k))(p(j)+p(k))E E E E − − ki ⊗ jk − jk ⊗ ki k=−n |i|0 E E − 2 − j,−j ⊗ −j,i 2 −i,i ⊗ j,−i δ δ + j<0 ( 1)p(i)+p(j)E E + i>0 ( 1)p(j)E E 2 − −j,i ⊗ j,−j 2 − j,−i ⊗ −i,i = ~−1 (∆(τ ) ∆(τ )◦) ij − ij as needed.

~~5 Periplectic q-Brauer algebra~~

~~C⊗l In [Mo], D. Moon identiﬁed the centralizer of the action of pn on the tensor space n|n. This centralizer is called the periplectic Brauer algebra in the literature: see [Co, CP, CE1, CE2].~~

We have a representation of Uqpn on Cq(n n) via the assignment T S, and thus we also have a ⊗l | 7→ representation on each tensor power Cq(n n) . In this section, we identify the centralizer of the action of ⊗l | Uqpn on Cq(n n) and call it the periplectic q-Brauer algebra. For the quantum group of type Q, this was done in [Ol] and| the centralizer of its action is called the Hecke-Cliﬀord superalgebra. Quantum analogs of the Brauer algebra were studied in [M] where they appear as centralizers of the action of twisted quantized tw tw enveloping algebras Uq on and Uq spn on tensor representations (here, spn is the symplectic Lie algebra); see also [We].

Definition 5.1. The periplectic q-Brauer algebra Bq,l is the associative C(q)-algebra generated by elements t and c for 1 i l 1 satisfying the following relations: i i ≤ ≤ − (t q)(t + q−1)=0, c2 =0, c t = q−1c , t c = qc for 1 i l 1; (11) i − i i i i − i i i i ≤ ≤ − t t = t t , t c = c t , c c = c c if i j 2; (12) i j j i i j j i i j j i | − |≥ t t t = t t t , c c c = c , c c c = c for 1 i l 2; (13) i j i j i j i+1 i i+1 − i+1 i i+1 i − i ≤ ≤ − t c c = t c + (q q−1)c c , c c t = c t + (q q−1)c c (14) i i+1 i − i+1 i − i+1 i i+1 i i+1 − i+1 i − i+1 i Remark 5.2. Setting q = 1 in this definition yields the algebra Al from Definition 2.2 in [Mo].

Lemma 5.3. We have Uqpn-module homomorphisms ϑ : Cq(n n) Cq(n n) C(q) and ǫ : C(q) C (n n) C (n n) given by ϑ(e e )= δ ( 1)p(a) and ǫ(1) =| n⊗ e | e →. → q | ⊗ q | a ⊗ b a,−b − a=−n a ⊗ −a P 9 Proof. It is enough to check that, for all the generators t of U p and any tensor v C (n n) C (n n), ij q n ∈ q | ⊗ q |

~~ϑ(tij (v)) = tij (ϑ(v)) and ǫ(tij (1)) = tij (ǫ(1)). (15)~~

~~Here is a brief sketch of some of the computations.~~

~~Using the formula for the coproduct, we have:~~

~~n t (e e )= ( 1)(p(i)+p(k))(p(k)+p(j))+(p(k)+p(j))p(a) t (e ) t (e ) (16) ij a ⊗ −a − ik a ⊗ kj −a kX=−n This can be made more explicit using~~

n δbi(1−2p(i))+δb,−i(2p(i)−1) tii(ea) = q Ebb(ea); bX=−n t (e ) = (q q−1)δ E (e ); i,−i a − i>0 −i,i a t (e ) = (q q−1)( 1)p(i)E (e ), if i = j . ij a − − ji a | | 6 | | We obtain, for instance,

δa1,i(1−2p(i))+δa1,−i(2p(i)−1) δa2,i(1−2p(i))+δa2 ,−i(2p(i)−1) t (e 1 e 2 )= q q e 1 e 2 ii a ⊗ a a ⊗ a If a = a = a, this simpliﬁes to e e and this allows us to check (15) quickly for i = j. 2 − 1 − a ⊗ −a Furthermore,

t (e e ) = ( 1)p(a)δ t (e ) t (e )+ δ t (e ) t (e ) i,−i a ⊗ −a − i>0 ii a ⊗ i,−i −a i>0 i,−i a ⊗ −i,−i −a It follows that t n e e = 0, so the identity for ǫ in(15) holds for j = i. i,−i a=−n a ⊗ −a − Suppose now thatPa = a . Then 1 6 − 2 −1 t (e 1 e 2 )= δ δ(a = a = i)(q q )qe e i,−i a ⊗ a i>0 1 2 − i ⊗ −i + δ δ(a = a = i)(q q−1)qe e i>0 1 2 − −i ⊗ i

Observe that ϑ(ei e−i + e−i ei) = 0, so we have shown that ϑ(ti,−i(ea1 ea2 )) = ti,−i(ϑ(ea1 ea2 )) and this proves (15) for⊗ϑ when j =⊗ i. ⊗ ⊗ − Next, we consider the case i = j . To prove the identity for ǫ in (15), we use again (16) and obtain that n | | 6 | | t e e = 0 by considering subcases i = a, j = a, and k = a. To show that (15) holds for ij a ⊗ −a ± ± ± a=−n ! X ϑ we also proceed with case-by-case veriﬁcation. The case a1,a2 i, j is immediate. If a1 i, j , a i, j , anda = a , then 6∈ {± ± } ∈ {± ± } 2 6∈ {± ± } 1 6 − 2 −1 2 (p(i)+p(a2))(p(a2)+p(j))+(p(a2 )+p(j))p(a1) p(i)+p(a2) t (e 1 e 2 ) = (q q ) ( 1) ( 1) E 2 (e 1 ) E 2 (e 2 ) ij a ⊗ a − − − a i a ⊗ ja a −1 p(i) + (q q )( 1) E (e 1 ) E 2 2 (e 2 ). − − ji a ⊗ a a a

~~This shows that ϑ(tij (ea1 ea2 ))=0= tij (ϑ(ea1 ea2 )). Similarly, we obtain the desired identity in the other cases. ⊗ ⊗~~

⊗2 ⊗2 By composing ϑ and ǫ, we obtain a Uqpn-module homomorphism ǫ ϑ : Cq(n n) Cq(n n) . In terms of elementary matrices, this linear map is given by n ( 1)p(a)p◦(b)E |E →, which| we abbreviate a,b=−n − ab ⊗ −a,−b by c. The super-permutation operator P on C (n n)⊗2 is given by P = n ( 1)p(b)E E , so q P| a,b=−n − ab ⊗ ba P 10 (π◦st)2 c = P where (π st)2 stands for the map π st applied to the second tensor in the previous formula for P . ◦ ◦

~~⊗l ⊗l We can extend c to a Uqpn-module homomorphism ci : Cq(n n) Cq(n n) for 1 i l 1 by applying c to the ith and (i + 1)th tensors. | → | ≤ ≤ −~~

⊗l ⊗l The linear map Cq(n n) Cq(n n) given by PiSi,i+1 where Pi is the super-permutation operator th | th→ | acting on the i and (i + 1) tensors is also a Uqpn-module homomorphism: this is a consequence of the fact that S is a solution of the quantum Yang-Baxter relation. Proposition 5.4. The tensor superspace C (n n)⊗l is a module over B if we let t act as P S and c q | q,l i i i,i+1 i act as ci.

Proof. That the linear operators PiSi,i+1 satisfy the braid relation (the first relation in (13)) is a consequence of the fact that S is a solution of the quantum Yang-Baxter relation. The relations (12) for the operators PiSi,i+1 and ci can be easily verified. As for the other relations, they can be checked via direct computations. ⊗2 ⊗3 It is enough to check the relations (11) on Cq(n n) and the relations (14) on Cq(n n) . We briefly sketch some of those computations below. | |

First, note that cP = c and P c = c. Also, we easily obtain the following: − n n c (q 1) E E = c (q−1 1) E E =0, − ii ⊗ ii − −i,−i ⊗ −i,−i i=1 ! i=1 ! n X n Xn c (q 1) E E = (q 1) E E , − ii ⊗ −i,−i − ab ⊗ −a,−b i=1 ! a=−n X X Xb=1 n n −1 c (q−1 1) E E = (q−1 1) ( 1)p(a)E E , − −i,−i ⊗ ii − − ab ⊗ −a,−b i=1 ! a=−n X X bX=−n −1 n n c E E = E E , i,−i ⊗ −i,i − ab ⊗ −a,−b i=−n ! a=−n X X Xb=1 n c ( 1)p(j)E E = ( 1)p(a)(p(i)+1)+p(j)E E =0.  − ij ⊗ ji − a,−i ⊗ −a,i a=−n 1≤|jX|<|i|≤n X 1≤|jX|<|i|≤n   Therefore, we have that c(S 1)=(q−1 1)c, hence cS = q−1c. Now using that c = cP , we obtain the third relation in (11). Similarly,− we prove (S− 1)c = (q 1)c, and then using P c = c, we− obtain the fourth relation in (11). − −

~~For the remaining relations we use the following formula:~~

n n PS = ( 1)p(j)E E + (q 1) (E E ) − ij ⊗ ji − −i,i ⊗ i,−i i,j=−n i=1 X X n n + (q 1) (E E ) (q−1 1) (E E ) − ii ⊗ ii − − i,−i ⊗ −i,i i=1 i=1 X X n −1 (q−1 1) (E E ) + (q q−1) (E E ) − − −i,−i ⊗ −i,−i − −i,−i ⊗ ii i=1 i=−n X X + (q q−1) (E E ) + (q q−1) ( 1)p(i)p(j)E E − jj ⊗ ii − − ji ⊗ −j,−i |jX|<|i| |jX|<|i|

11 As mentioned after the deﬁnition of Bq,l, the module structure given in the previous proposition commutes with the action of U (p ) on C (n n)⊗l. We thus have algebra homomorphisms q n q | ⊗l ⊗l B End (C (n n) ) and U (p ) EndB (C (n n) ). q,l −→ Uq (pn) q | q n −→ q,l q | ⊗l The main theorem of this section states that Bq,l is the full centralizer of the action of Uq(pn) on Cq(n n) when n l. | ≥ Theorem 5.5. The map B End (C (n n)⊗l) is surjective and it is injective when n l. q,l −→ Uq (pn) q | ≥

This is a q-analogue of Theorem 4.5 in [Mo]. The proof follows the lines of the proof of Theorem 3.28 in [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.

−1 −1 Recall that = C[q, q ](q−1) is the localization of C[q, q ] at the ideal generated by q 1. Let Bq,l( ) be the -associativeA algebra which is deﬁned via the same generators and relations as B −. A A q,l Proposition 5.6. The quotient algebra Bq,l( )/(q 1)Bq,l( ) is isomorphic to the algebra Al given in Deﬁnition 2.2 in [Mo]. A − A

Proof. It follows immediately from the deﬁnitions of both A and B ( ) that we have a surjective algebra l q,l A homomorphism Al ։ Bq,l( )/(q 1)Bq,l( ). That it is injective can be proved as in the proof of Proposition 3.21 in [BGJKW] using TheoremA − 4.1 in [Mo].A

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 gln (see for instance [Do]), the following deﬁnition is natural.

~~Deﬁnition 5.7. The q-Schur superalgebra Sq(pn,l) of type P is the centralizer of the action of Bq,l on ⊗l ⊗l C (n n) , that is, S (p ,l) = EndB (C (n n) ). q | q n q,l q |~~

We have an algebra homomorphism Uq(pn) Sq(pn,l): it is an open question whether or not this map −→ C ⊗l is surjective. We also have an algebra homomorphism Bq,l EndSq (pn,l)( q(n n) ) and it is natural to expect that it should be an isomorphism, perhaps under certain−→ conditions on n and| l.

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~~Department of Mathematics, University of Texas at Arlington Arlington, TX 76021, USA [email protected]~~

~~University of Alberta, Department of Mathematical and Statistical Sciences, CAB 632 Edmonton, AB T6G 2G1, Canada [email protected]~~