QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS

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1 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS GEORGIA BENKART 1, NICOLAS GUAY 2, JI HYE JUNG 3, SEOK-JIN KANG 4, AND STEWART WILCOX 5 Abstract We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras BC r,s(q) The superalgebra BC r,s(q) is a quantum deformation of the walled Brauer- Clifford superalgebra BC r,s and a super version of the quantum walled Brauer algebra We prove that BC r,s(q) is the centralizer superalgebra of the action of U q(q(n)) on the mixed tensor space Vq r,s = Vq r (Vq) s when n r + s, where V q = C(q) (n n) is the natural representation of the quantum enveloping superalgebra U q(q(n)) and Vq is its dual space We also provide a diagrammatic realization of BC r,s(q) as the (r, s)-bead tangle algebra BT r,s(q) Finally, we define the notion of q-schur superalgebras of type Q and establish their basic properties Introduction Schur-Weyl duality has been one of the most inspiring themes in representation theory, because it reveals many hidden connections between the representation theories of seemingly unrelated algebras By the duality functor, one algebra appears as the centralizer of the other acting on a common representation space Many interesting and important algebras have been constructed as centralizer algebras in this way For example, the group algebra CΣ k of the symmetric group Σ k appears as the centralizer of the gl(n)-action on V k, where V = C n is the natural representation of the general linear Lie algebra gl(n) Similarly, Hecke algebras, Brauer algebras, Birman-Murakami-Wenzl algebras and Hecke- Clifford superalgebras are the centralizer algebras of the action of corresponding Lie (super)algebras or quantum (super)algebras on the tensor powers of their natural representations There are further generalizations of Schur-Weyl duality on mixed tensor powers Let V r,s = V r (V ) s be the mixed tensor space of the natural representation V of gl(n) and its dual space V The centralizer algebra of the gl(n)-action on V r,s is the walled Brauer algebra B r,s (n) The structure and properties of B r,s (n) were first investigated in [1, 12, 20] By replacing gl(n) by the quantum enveloping algebra U q (gl(n)) and V = C n by V q = C(q) n, we obtain as the centralizer algebra the quantum walled Brauer algebra studied in [3, 4, 7, 11, 14] Super versions of the above constructions have been investigated with the following substitutions: Replace gl(n) by gl(m n), C n by C (m n) ; U q (gl(n)) by U q (gl(m n)); and C(q) n by C(q) (m n) as in [15, 16] 2010 Mathematics Subject Classification Primary: 81R50 Secondary: 17B60, 05E10, 57M25, 20G43 Key words and phrases quantum walled Brauer-Clifford superalgebra, queer Lie superalgebra, centralizer algebra, bead tangle algebra, q-schur superalgebra 1 GB acknowledges with gratitude the hospitality of the Banff International Research Station (BIRS), where her collaboration on this project with NG began 2 This work was partly supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada 3 This work was supported by NRF Grant # , NRF and NRF-2013R1A1A This work was partially supported by NRF Grant # and NRF Grant # This work was partly supported by a Postdoctoral Fellowship of the Pacific Institute for the Mathematical Sciences 1

2 2 BENKART ET AL The Lie superalgebra q(n) is commonly referred to as the queer Lie superalgebra because of its unique properties and the fact that it has no non-super counterpart Its natural representation is the superspace (Z 2 -graded vector space) V = C (n n) The corresponding centralizer algebra End q(n) (V r ) was studied by Sergeev in [19], and it is often referred to as the Sergeev algebra Using a modified version of a technique of Fadeev, Reshetikhin and Turaev, Olshanski introduced the quantum queer superalgebra U q (q(n)) and established an analogue of Schur-Weyl duality That is, he showed that there is a surjective algebra homomorphism ρ r n,q : HC r (q) End q(n) (V r ), where HC r (q) is the Hecke-Clifford superalgebra, a quantum version of the Sergeev algebra Moreover, ρ r n,q is an isomorphism when n r On the other hand, in [13] Jung and Kang considered a super version of the walled Brauer algebra For the mixed tensor space V r,s = V r (V ) s, one can ask, What is the centralizer of the q(n)-action on V r,s? In order to answer this question, they introduced two versions of the walled Brauer-Clifford superalgebra, (which is called the walled Brauer superalgebra in [13]) The first is constructed using (r, s)-superdiagrams, and the second is defined by generators and relations The main results of [13] show that these two definitions are equivalent and that there is a surjective algebra homomorphism ρ r,s n : BC r,s End q(n) (V r,s ), which is an isomorphism whenever n r + s The purpose of this paper is to combine the constructions in [18] and [13] to determine the centralizer algebra of the U q (q(n))-action on the mixed tensor space Vq r,s := Vq r (Vq) s We begin by introducing the quantum walled Brauer-Clifford superalgebra BC r,s (q) via generators and relations The superalgebra BC r,s (q) contains the quantum walled Brauer algebra and the Hecke- Clifford superalgebra HC r (q) as subalgebras We then define an action of BC r,s (q) on the mixed tensor space Vq r,s that supercommutes with the action of U q (q(n)) As a result, there is a superalgebra homomorphism ρ r,s n,q : BC r,s (q) End Uq(q(n))(Vq r,s ) Actually, defining such an action is quite subtle and complicated, and we regard this as one of our main results (Theorem 316) We use the fact that BC r,s is the classical limit of BC r,s (q) to show that the homomorphism ρ r,s n,q is surjective and that it is an isomorphism whenever n r + s (Theorem 328) We also give a diagrammatic realization of BC r,s (q) as the (r, s)-bead tangle algebra BT r,s (q) An (r, s)-bead tangle is a portion of a planar knot diagram satisfying the conditions in Definition 41 The algebra BT r,s (q) is a quantum deformation of the (r, s)-bead diagram algebra BD r,s, which is isomorphic to the walled Brauer-Clifford superalgebra BC r,s (Theorem 29) Modifying the arguments in [10], we prove that the algebra BT r,s (q) is isomorphic to BC r,s (q) (Theorem 419), so that BC r,s (q) can be regarded as a diagram algebra In the final section, we introduce q-schur superalgebras of type Q and prove that the classical results for q-schur algebras can be extended to this setting 1 The walled Brauer-Clifford superalgebras To begin, we recall the definition of the Lie superalgebra q(n) and its basic properties Let I = {±i i = 1,, n}, and set Z 2 = Z/2Z The superspace V = C(n n) = C n C n has a standard basis {v i i I} We say that the parity of v i equals i := v i, where v i = 1 if i < 0 and v i = 0 if i > 0 The endomorphism algebra is Z 2 -graded, End C (V) = End C (V) 0 End C (V) 1, and has a basis of matrix units E ij with n i, j n, ij 0, where the parity of E ij is E ij = i + j (mod 2) The general linear Lie superalgebra gl(n n) is obtained from End C (V) by using the supercommutator [X, Y ] = XY ( 1) X Y Y X

3 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 3 for homogeneous elements X,Y The map ι : gl(n n) gl(n n) given by E ij E i, j is an involutive automorphism of gl(n n) Let J = n a=1 (E a, a E a,a ) gl(n n) Definition 11 The queer Lie superalgebra q(n) can be defined equivalently as either the centralizer of J in gl(n n) (under the supercommutator product) or the fixed-point subalgebra of gl(n n) with respect to the involution ι Identifying V with the space of (n n) ( column ) vectors and {v i i I} with the standard basis 0 I for the column vectors, we have J =, and q(n) can be expressed in the matrix form as I 0 {( ) } A B A, B are arbitrary n n complex matrices B A Then q(n) inherits a Z 2 -grading from gl(n n), and a basis for q(n) 0 is given by E 0 ab = E ab + E a, b and for q(n) 1 by E 1 ab = E a, b + E a,b, where 1 a, b n The superalgebra q(n) acts naturally on V by matrix multiplication on the column vectors, and V is an irreducible representation of q(n) The action on V extends to one on V k by k g(v i1 v ik ) = ( 1) ( v i v (12) ij 1 ) g v i1 v ij 1 gv ij v ij+1 v ik, j=1 where g is homogeneous It also extends in a similar fashion to the mixed tensor space V r,s := V r (V ) s, where V is the dual representation of V, and the action on V is given by (gω)(v) := ( 1) g ω ω(gv) for homogeneous elements g q(n), ω V, and v V We assume {ω i i I} is the basis of V dual to the standard basis {v i i I} of V In an effort to construct the centralizer superalgebra End q(n) (V r,s ), Jung and Kang [13] introduced the notion of the walled Brauer-Clifford superalgebra BC r,s The superalgebra BC r,s is generated by even generators s 1,, s r 1, s r+1,, s r+s 1, e r,r+1 and odd generators c 1,, c r+s, which satisfy the following defining relations (for i, j in the allowable range): (13) s 2 i = 1, s i s i+1 s i = s i+1 s i s i+1, s i s j = s j s i ( i j > 1), e 2 r,r+1 = 0, e r,r+1 s j = s j e r,r+1 (j r 1, r + 1), e r,r+1 = e r,r+1 s r 1 e r,r+1 = e r,r+1 s r+1 e r,r+1, s r 1 s r+1 e r,r+1 s r+1 s r 1 e r,r+1 = e r,r+1 s r 1 s r+1 e r,r+1 s r+1 s r 1, c 2 i = 1 (1 i r), c 2 i = 1 (r + 1 i r + s), c i c j = c j c i (i j), s i c i s i = c i+1, s i c j = c j s i (j i, i + 1), c r e r,r+1 = c r+1 e r,r+1, e r,r+1 c r = e r,r+1 c r+1, e r,r+1 c r e r,r+1 = 0, e r,r+1 c j = c j e r,r+1 (j r, r + 1) For 1 j r 1, r + 1 k r + s 1, 1 l r, and r + 1 m r + s, the action of the generators of BC r,s on V r,s (which is on the right) is defined as follows: (v i1 v ir ω ir+1 ω ir+s ) s j = ( 1) i j i j+1 v i1 v ij 1 v ij+1 v ij v ij+2 ω ir+s, (v i1 v ir ω ir+1 ω ir+s ) s k = ( 1) i k i k+1 v i1 ω ik 1 ω ik+1 ω ik w ik+2 ω ir+s, n (v i1 v ir ω ir+1 ω ir+s ) e r,r+1 = δ ir,ir+1 ( 1) ir v i1 v ir 1 v i ω i ω ir+2 ω ir+s, i= n

4 4 BENKART ET AL (v i1 v ir ω ir+1 ω ir+s ) c l = ( 1) i i l 1 v i1 v il 1 Jv il v il+1 ω ir+s, (v i1 v ir w ir+1 w ir+s ) c m = ( 1) i i m 1 v i1 ω im 1 J T ω im ω im+1 ω ir+s, where J T is the supertranspose of J, and the supertranspose is given by E T ij := ( 1)( i + j ) i E ji By direct calculation, it can be verified that this action of the generators gives rise to an action of BC r,s on V r,s Thus, there is an homomorphism of superalgebras, ρ r,s n : BC r,s End C (V r,s ) op The next theorem, due to Jung and Kang [13], identifies a basis for BC r,s In stating it and some subsequent results, we use the following notation: For a nonempty subset A = {a 1 < < a m } of {1,, r + s}, let c A = c a1 c am, and set c = 1 Let e p,q = ϕe r,r+1 ϕ 1, where ϕ = s q 1 s r+1 s p s r 1 for 1 p r and r + 1 q r + s Theorem 14 (i) The actions of BC r,s and q(n) on V r,s supercommute with each other Thus, there is an algebra homomorphism ρ r,s n : BC r,s End q(n) (V r,s ) op (ii) The map ρ r,s n is surjective Moreover, it is an isomorphism if n r + s (iii) The elements c P e p1,q 1 e pa,qa σc Q such that (1) 1 p 1 < < p a r; (2) r + 1 q i r + s, i = 1,, a, are all distinct; (3) σ r s, σ 1 (p 1 ) < < σ 1 (p a ); and (4) P {p 1,, p a }, Q {1,, r} {r + 1,, r + s}\{σ 1 (q 1 ),, σ 1 (q a )} comprise a basis of BC r,s (iv) dim C BC r,s = 2 r+s (r + s)! 2 The (r, s)-bead diagram algebra BD r,s In this section, we give a new diagrammatic realization of the walled Brauer-Clifford superalgebra BC r,s Definition 21 An (r, s)-bead diagram, or simply a bead diagram, is a graph consisting of two rows with r + s vertices in each row such that the following conditions hold: (1) Each vertex is connected by a strand to exactly one other vertex (2) Each strand may (or may not) have finitely many numbered beads The bead numbers on a diagram start with 1 and are distinct consecutive positive integers (3) There is a vertical wall separating the first r vertices from the last s vertices in each row (4) A vertical strand connects a vertex on the top row with one on the bottom row, and it cannot cross the wall A horizontal strand connects vertices in the same row, and it must cross the wall (5) No loops are permitted in an (r, s)-bead diagram The following diagram is an example of a (3, 2)-bead diagram d = 2 3

5 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 5 Beads can slide along a given strand, but they cannot jump to another strand nor can they interchange positions on a given strand For example, the following diagrams are the same as (2, 1)-bead diagrams 2 3 = 2 3 = = 2 3, while the following diagrams are considered to be different 2 2, 2 2 An (r, s)-bead diagram having an even number of beads is regarded as even (resp odd) Let BD r,s be the superspace with basis consisting of the (r, s)-bead diagrams We define a multiplication on BD r,s For (r, s)-bead diagrams d 1, d 2, we put d 1 under d 2 and identify the top vertices of d 1 with the bottom vertices of d 2 If there is a loop in the middle row, we say d 1 d 2 = 0 If there is no loop in the middle row, we add the largest bead number in d 1 to each bead number in d 2, so that if m is the largest bead number in d 1, then a bead numbered i in d 2 is now numbered m + i in d 1 d 2 Then we concatenate the diagrams For example, if d 1 =, 2 3 d 2 =, 2 then d 1 d 2 = = Observe that BD r,s is closed under this product If the number of beads in d 1 (resp d 2 ) is m 1 (resp m 2 ) and d 1 d 2 0, then the number of beads in d 1 d 2 is m 1 + m 2 Hence, the multiplication respects the Z 2 -grading Let d 1, d 2, d 3 BD r,s Note that the connections in (d 1 d 2 )d 3 and d 1 (d 2 d 3 ) are the same The strands where the beads are placed and the bead numbers are also the same in (d 1 d 2 )d 3 and d 1 (d 2 d 3 ) Therefore, (d 1 d 2 )d 3 = d 1 (d 2 d 3 ) The identity element of BD r,s is the diagram with no beads such that each top vertex is connected to the corresponding bottom vertex For 1 i r 1, r + 1 j r + s 1, 1 k r, r + 1 l r + s, let BD r,s be the subalgebra of BD r,s generated by the following diagrams:

6 6 BENKART ET AL s i := i i + 1, s j :=, j j + 1 e r,r+1 :=, 1 r r + 1 r + s c k := k, c l := l Assume L is the (two-sided) ideal of BD r,s generated by the following homogeneous elements, c 2 k + 1 (1 k r), c2 l 1 (r + 1 l r + s), and c ic j + c j c i (1 i j r + s), (22) and let BD r,s be the quotient superalgebra BD r,s/l We say that BD r,s is the (r, s)-bead diagram algebra, or simply the bead diagram algebra For simplicity, we identify cosets in BD r,s with their diagram representatives and make the following identifications in BD r,s : c 2 k = k 2 =, k c 2 l = l 2 =, l and c i c j = i j 2 = = c j c i i j 2 Our aim is to prove that the walled Brauer-Clifford superalgebra BC r,s is isomorphic to BD r,s Towards this purpose, we adopt the following conventions: (i) The vertices on the top row (and on the bottom row) of a bead diagram are labeled 1, 2,, r + s from left to right (ii) The top vertex of a vertical strand or the left vertex of a horizontal strand is the good vertex of the strand (iii) A bead on a horizontal bottom row strand is a bead of type I All other beads are of type II We construct a bead diagram d from the bead diagram d by performing the following steps: (1) Keep the same connections between vertices as in d

7 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 7 (2) If the number of beads along a strand is even, delete all beads on that strand If there is an odd number of beads on a strand, leave only one bead on it Repeat this process for all strands in d (Hence, there is at most one bead on any strand of d) Associate to each remaining bead the good vertex of its strand (3) Renumber (starting with 1) the beads of type I according to the position of its good vertex from left to right (4) Let m be the maximum of bead numbers after Step 3 Renumber (starting with m + 1) the beads of type II according to the position of its good vertex from left to right The resulting diagram is d Since the definition of d depends only on the number of beads along a strand and the good vertices of strands having an odd number of beads, d does not change when we slide beads along a strand, so d is well defined Example 23 If d is as pictured below then d = d = , Next we assign a nonnegative integer γ(d) to the bead diagram d in the following way: (1) Assume the bead numbers of type I in d are 1 η 1 < η 2 < < η p Let a j be the label of the good vertex of the strand in d with the bead η j This determines a sequence a 1,, a p Let l 1 (d) := {(j, k) j < k, a j > a k } (2) Assume the bead numbers of type II in d are 1 ϑ 1 < ϑ 2 < < ϑ q Let b j be the label of the good vertex of the strand in d with the bead ϑ j This determines a sequence b 1,, b q Let l 2 (d) := {(j, k) j < k, b j > b k } r {j {1,, p} aj = i} (3) Let ρ 1 (d) :=, where x denotes the largest integer not 2 i=1 greater than x r {j {1,, q} b j = i} (4) Let ρ 2 (d) = 2 i=1 (5) For each bead η j, its passing number counts the number of beads η k such that η k > η j when η j moves to the good vertex of its strand Let p 1 (d) be the sum of the passing numbers for all η 1,, η p (6) For each bead ϑ j, its passing number counts the number of beads ϑ k with ϑ k < ϑ j when ϑ j moves to the good vertex of its strand Let p 2 (d) be the sum of the passing numbers for all ϑ 1,, ϑ q (7) For each bead ϑ j, count the number of beads η k such that η k > ϑ j ; then let c(d) be the sum of those numbers for all beads ϑ 1,, ϑ q

8 8 BENKART ET AL (8) Let α(d) := r+s i=r+1 {j {1,, q} bj = i} 2 (9) Now set β(d) := l 1 (d) + l 2 (d) + ρ 1 (d) + ρ 2 (d) + p 1 (d) + p 2 (d) + c(d) and γ(d) := β(d) + α(d) Since the definition of γ(d) depends only on the bead numbers, the number of beads on a strand, and the good vertex of a strand having a bead, γ(d) is well defined All values including β(d), α(d), and hence γ(d), are nonnegative integers Example 24 Consider the bead diagram d in Example 23 Three beads 2, 3, 6 are of type I, and the rest are of type II The sequence of labels for the good vertices obtained in Step 1 (resp Step 2) is a 1, a 2, a 3 = 2, 2, 2 (resp b 1,, b 9 = 1, 2, 2, 3, 6, 3, 6, 4, 6) From these sequences we see that l 1 (d) = 0, l 2 (d) = 3, ρ 1 (d) = 1, ρ 2 (d) = 2, and α(d) = 1 When the bead 2 moves to the good vertex on its strand, it must pass the two beads 3, 6 When the beads 3 and 6 move to that same good vertex, they do not have to pass a bead with a larger label Hence p 1 (d) = 2 Similarly p 2 (d) = 2, since only the beads 9 (passing 7 ) and 10 (passing 8 ) contribute to p 2 (d) Only the following beads contribute to c(d): bead with 2, 3, 6 and beads 4 and 5 with 6 Therefore, c(d) = 5 Consequently, β(d) = = 15, α(d) = 1, and γ(d) = 16 The set of (r, s)-bead diagrams without any beads or horizontal strands forms a group under the multiplication defined in BD r,s which is isomorphic to the product Σ r Σ s of symmetric groups In what follows, we identify that group with Σ r Σ s but use boldface when we are regarding an element of Σ r Σ s as a diagram We adopt the following conventions analogous to those for the basis elements of BC r,s, but here we are using the generators for the subalgebra BD r,s of BD r,s : For a nonempty subset A = {a 1 < < a m } of {1,, r} {r+1,, r+s}, set c A := c a1 c am, and let c = 1 Let e p,q := ϕe r,r+1 ϕ 1, where ϕ = s q 1 s r+1 s p s r 1 for 1 p r and r + 1 q r + s Lemma 25 (i) For a bead diagram d, the associated diagram d has an expression of the form c P e p1,q 1 e pa,q a σc Q, where (1) 1 p 1 < < p a r; (2) r + 1 q i r + s, i = 1,, a, are all distinct; (3) σ Σ r Σ s, σ 1 (p 1 ) < < σ 1 (p a ); and (4) P {p 1,, p a }, Q {1,, r} {r + 1,, r + s}\{σ 1 (q 1 ),, σ 1 (q a )} Hence, d BD r,s (ii) γ(d) = 0 if and only if d = d In particular, γ( d) = 0 for all bead diagrams d Proof (i) A diagram without beads can be written as a product e p1,q 1 e pa,q a σ which satisfies conditions (1), (2), and (3) Indeed, a is the number of horizontal strands in d The number p i (resp q i ) is the label of the good vertex (resp right vertex) of the ith horizontal bottom row strand from left to right The number σ 1 (p i ) (resp σ 1 (q i )) is the label of the good vertex (resp right

9 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 9 vertex) of the ith horizontal top row strand from left to right For example, = From Steps 3 and 4 of the construction of d from d, we have d = c P e p1,q 1 e pa,q a σc Q, where P is a set of labels for the good vertices of strands with beads of type I, and Q is a set of labels for the good vertices of strands with beads of type II in d Since we can slide a bead to the good vertex on its strand, condition (4) above can be satisfied (ii) ( ) From the assumption that γ(d) = 0, we have that ρ 1 (d) = ρ 2 (d) = α(d) = 0 Hence, there is at most one bead on each strand in d Since c(d) = 0, the bead numbers of the beads of type I are less than all the bead numbers of the beads of type II From l 1 (d) = l 2 (d) = 0, we deduce that the sequence of good vertices for the stands with beads of the same type are arranged in increasing size That is, the good vertex having the smaller label is connected to the strand with the bead having the smaller bead number From these properties, we determine that nothing is changed when d is constructed from d, so that d = d ( ) It is enough to argue that γ( d) = 0 By Step 2 of the construction of d, we have ρ 1 ( d) = ρ 2 ( d) = α( d) = 0 Also, since there is at most one bead on each strand in d, p 1 ( d) = p 2 ( d) = 0 By Steps 3 and 4, we see that c( d) = l 1 ( d) = l 2 ( d) = 0 As a result, β( d) = α( d) = 0, and hence γ( d) = 0 Example 26 For the diagram d in Example 23, we have p 1 = 2, p 2 = 4, q 1 = 5, q 2 = 6, σ = s 2 s 1 s 5 s 6, σ 1 (2) = 3, σ 1 (4) = 4, σ 1 (5) = 7, σ 1 (6) = 5, P = {2}, and Q = {1, 4, 6} Consequently, d = c 2 e 2,5 e 4,6 s 2 s 1 s 5 s 6 c 1 c 4 c 6 BD 4,3 Lemma 27 The subspace M spanned by {d ( 1) β(d) d d is a bead diagram in BD r,s } contains the two-sided ideal L of BD r,s generated by the elements c 2 k + 1, c2 l 1, and c ic j + c j c i in (22) Proof It suffices to show that ec 2 k f + ef, ec2 l f ef, and ec ic j f + ec j c i f belong to M for any two bead diagrams e, f BD r,s (i) First, we consider ec 2 k f + ef In constructing the diagram d from a diagram d, we delete an even number of beads along each strand Therefore, ẽc2 k f = ẽf If the product c 2 k in ec2 k f creates beads of type I, then ρ 1(ec 2 k f) = ρ 1(ef) + 1 In this case, c(ec 2 k f) c(ef), l 1(ec 2 k f) l 1(ef), and p 1 (ec 2 k f) p 1(ef) mod 2 The other values l 2 (ec 2 k f), ρ 2 (ec 2 k f), p 2(ec 2 k f) are the same as l 2(ef), ρ 2 (ef), p 2 (ef), respectively Thus, β(ec 2 kf) β(ef) + 1 mod 2 If the product c 2 k in ec2 kf creates beads along a vertical strand on the right-hand side of the wall, then ρ 2 (ec 2 k f) = ρ 2(ef) In this case, p 2 (ec 2 k f) p 2(ef) + 1, l 2 (ec 2 k f) l 2(ef), c(ec 2 kf) c(ef) mod 2 The other values l 1, ρ 1, p 1 remain the same for ec 2 k f as for ef Hence, β(ec2 kf) β(ef) + 1 mod 2 The other cases can be checked in a similar manner

10 10 BENKART ET AL As a consequence, (28) ec 2 k f + ef = ec2 k f ( 1)β(ec2 k f)ẽc2 k f + ( 1)β(ec2 k f)ẽc2 k f + ef = ec 2 k f ( 1)β(ec2 k f)ẽc2 k f + ef ( 1)β(ef) ẽf M (ii) To verify ec 2 l f ef M, we can show that ẽc2 l f = ẽf and β(ec2 l f) β(ef) mod 2 as in (i) and then apply a calculation similar to that in (28) (iii) To argue that ec i c j f + ec j c i f M, assume c i creates a bead indexed by a and c j a bead indexed by a + 1 in ec i c j f If we switch the beads containing a and a + 1, we obtain the bead diagram ec j c i f Since the number of beads along each strand does not change, ẽc i c j f = ẽc j c i f We will show that β(ec i c j f) β(ec i c j f)+1 mod 2 Suppose the beads created by c i and c j are of different types, say type I for c i and type II for c j Then c(ec i c j f) = c(ec j c i f) 1 The other values l 1, l 2, ρ 1, ρ 2, p 1, p 2, and α are the same in ec i c j f and ec j c i f Hence, β(ec i c j f) β(ec j c i f)+1 mod 2 The reverse possibility (c i type II and c j type I) can be treated similarly Now assume both c i and c j create beads of type I If the beads are on the same strand, then p 1 (ec i c j f) p 1 (ec j c i f) + 1 mod 2, and the other values do not change If they lie on different strands, then l 1 (ec i c j f) l 1 (ec j c i f)+1 mod 2, and the other values are unchanged Therefore, β(ec i c j f) β(ec j c i f) + 1 mod 2 The case that c i and c j create beads of type II can be handled similarly Applying a computation like the one in (28), we obtain ec i c j f + ec j c i f M We now prove the main theorem of this section Theorem 29 The walled Brauer-Clifford superalgebra BC r,s is isomorphic to the (r, s)-bead diagram algebra BD r,s as associative superalgebras Proof Using the defining relations, we see that the linear map φ r,s : BC r,s BD r,s specified by (210) s i s i, s j s j, e r,r+1 e r,r+1, c k c k, and c l c l, is a well-defined superalgebra epimorphism Recall that BD r,s := BD r,s/l, where L is the two-sided ideal generated by the elements in (22) As L M, by Lemma 27, there is well-defined linear map π r,s : BD r,s BD r,s/m such that π r,s (d + L) = d + M for d BD r,s Since {c P e p1,q 1 e pa,q a σc Q } is a basis of BC r,s by Theorem 14 (iii), we can define a linear map ψ r,s : BC r,s BD r,s/m such that ψ r,s (c P e p1,q 1 e pa,q a σc Q ) = c P e p1,q 1 e pa,q a σc Q + M Moreover, π r,s φ r,s = ψ r,s Therefore, if we can show that ψ r,s is injective, it will follow that φ r,s is injective (hence, an isomorphism) By Lemma 25 (ii), when γ(d) = 0 for a bead diagram d, then d ( 1) β(d) d = 0 Thus, M is spanned by the elements d ( 1) β(d) d with γ(d) 1 Note that γ(c P e p1,q 1 e pa,q a σc Q ) = 0 Therefore, the set {c P e p1,q 1 e pa,qa σc Q +M} of these elements is linearly independent in BD r,s/m, so ψ r,s is indeed injective

11 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 11 Corollary 211 The relation L = M holds In particular, (212) {d ( 1) β(d) d d is a bead diagram in BD r,s, γ(d) 1} is a basis of the two-sided ideal L Proof By Lemma 27 we have that L M Since the mapping φ r,s in (210) is an isomorphism and π r,s = ψ r,s φ 1 r,s, we know that π r,s is injective Thus, for m M, π r,s (m + L) = 0 + M implies that m L From the proof of Theorem 29, we have that the set in (212) spans M (= L) Since γ(d) 1 and γ( d) = 0, it follows that (212) is a linearly independent set 3 The quantum walled Brauer-Clifford superalgebra BC r,s (q) Let q be an indeterminate and C(q) be the field of rational functions in q Set V q = C(q) C V = C(q) C C(n n) Corresponding to any X = k Y k Z k ( End C(q) (V q ) ) 2, let X 12 = k Y k Z k id, X 13 = k Y k id Z k, and X 23 = k id Y k Z k in ( End C(q) (V q ) ) 3, where id = idvq Let ξ = q q 1 and define S = i,j I S ij E ij ( End C(q) (V q ) ) 2 by (31) S = q (δ ij+δ i, j )(1 2 j ) E ii E jj + ξ i,j I i,j I, i<j ( 1) i (E ji + E j, i ) E ij S is known to satisfy the quantum Yang-Baxter equation S 12 S 13 S 23 = S 23 S 13 S 12 In [18], Olshanski constructed a quantization of U(q(n)) of q(n) in terms of S Definition 32 [18] The quantum queer superalgebra U q (q(n)) is the unital associative superalgebra over C(q) generated by elements u ij with i j and i, j I = {±i i = 1,, n}, which satisfy the following relations: (33) u ii u i, i = 1 = u i, i u ii, U 12 U 13 S 23 = S 23 U 13 U 12, where U = i,j I, i j u ij E ij, and the last equality holds in U q (q(n)) C(q) ( EndC(q) (V q ) ) 2 The Z 2 -degree of u ij is i + j By the construction, the assignment u ij S ij is a representation of U q (q(n)) on V q (see [18, Sec 4]) The superalgebra U q (q(n)) is a Hopf superalgebra with coproduct (U) = U 13 U 23 (U q (q(n))) 2 C(q) End C(q) (V q ), or more explicitly, (u ij ) = k I ( 1)( i + k )( k + j ) u ik u kj The counit is given by ε(u) = 1 and the antipode by U U 1 Let Vq r,s := Vq r C(q) (Vq) s be the mixed tensor space of V q and Vq Then Vq r,s is a representation of U q (q(n)) via the coproduct and antipode mappings To describe the structure of the centralizer superalgebra End Uq(q(n))(Vq r,s ), we introduce the quantum walled Brauer-Clifford superalgebra BC r,s (q) Definition 34 The quantum walled Brauer-Clifford superalgebra BC r,s (q) is the associative superalgebra over C(q) generated by even elements t 1, t 2,, t r 1, t 1, t 2,, t s 1, e and odd elements

12 12 BENKART ET AL c 1, c 2,, c r, c 1, c 2,, c s satisfying the following defining relations (for i, j in the allowable range): (35) t 2 i (q q 1 )t i 1 = 0, t i t i+1 t i = t i+1 t i t i+1, (t i ) 2 (q q 1 )t i 1 = 0, t i t i+1t i = t i+1t i t i+1, t i t j = t j t i ( i j > 1), t i t j = t jt i, t i t j = t jt i ( i j > 1), e 2 = 0, et r 1 e = e, et j = t j e (j r 1), et 1e = e, et j = t je (j 1), et 1 r 1 t 1et 1t 1 r 1 = t 1 r 1 t 1et 1t 1 r 1 e, c 2 i = 1, c i c j = c j c i (i j), c i c j = c jc i, (c i ) 2 = 1, c i c j = c jc i (i j), t i c i = c i+1 t i, t i c j = c j t i (j i, i + 1), t i c i = c i+1t i, t i c j = c jt i (j i, i + 1), t i c j = c jt i, c r e = c 1e, c j e = ec j (j r), t i c j = c j t i, ec r = ec 1, c je = ec j (j 1), ec r e = 0 Definition 36 (i) The (finite) Hecke-Clifford superalgebra HC r (q) in [18] is the associative superalgebra over C(q) generated by the even elements t 1, t 2,, t r 1 and the odd elements c 1, c 2,, c r with the following defining relations (for allowable i, j): (37) t 2 i (q q 1 )t i 1 = 0, t i t i+1 t i = t i+1 t i t i+1, t i t j = t j t i ( i j > 1), c 2 i = 1, c i c j = c j c i (i j), t i c i = c i+1 t i, t i c j = c j t i (j i, i + 1) (ii) The quantum walled Brauer algebra H 0 r,s(q) in [11] is the associative algebra over C(q) generated by the elements t 1, t 2,, t r 1, t 1,, t s 1 and e which satisfy the first four lines in (35) Remark 38 The relations in the first three lines in (35) appear in Definition 21 of [11] In line 4 of (35), we have the one relation (39) instead of the following two relations of [11]: (310) et 1 r 1 t 1et 1t 1 r 1 = t 1 r 1 t 1et 1t 1 r 1 e et 1 r 1 t 1et r 1 = et 1 r 1 t 1et 1, t r 1 et 1 r 1 t 1e = t 1et 1 r 1 t 1e The relations in (310) can be derived using (39) and various identities from (35) (especially the fact that t r 1 and t 1 commute) in the following way: et 1 r 1 t 1e = (et 1 r 1e) t 1 r 1 t 1e = e((t 1) 1 t 1) t 1 r 1 et 1 r 1 t 1e = e(t 1) 1 (t 1 t 1 r 1 et 1 r 1 t 1e) = e(t 1) 1 (et 1 t 1 r 1 et 1 r 1 t 1) = (e(t 1) 1 e) t 1t 1 r 1 et 1 r 1 t 1 = et 1t 1 r 1 et 1 r 1 t 1 = et 1 r 1 t 1et 1t 1 r 1 = t 1 r 1 t 1et 1 r 1 t 1e, implying both relations in (310) The following simple expression will be useful in several calculations henceforth Lemma 311 With the conventions ( 1) 0 = 0, j = 1 for any integer j < 0, and j = 0 for j > 0, we have (312) ξ ( 1) j q 2j(1 2 j ) = q (2k 1)(1 2 k ) q (2i+1)(1 2 i ) i<j<k for any nonzero integers i < k, where ξ = q q 1 as above Proof This can be checked by considering the three cases 0 < i < k, i < k < 0 and i < 0 < k

13 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 13 In order to construct an action of BC r,s (q) on the mixed tensor space Vq r,s, we will need a number of U q (q(n))-module homomorphisms Note that C(q) becomes a U q (q(n))-module by sending U to the identity map in End C(q) (C(q) C(q) V q ) Lemma 313 There are U q (q(n))-module homomorphisms : C(q) V q V q and : V q V q C(q) given by (1) = i I v i ω i, (v i ω j ) = ( 1) i q 2i(1 2 i ) (2n+1) δ ij Proof Since is the canonical map C(q) V q Vq, we have (314) (X id) = (id X T ) for any X End C(q) (V q ), where T denotes the supertranspose Since is even, it follows that ( (S 23 ) 1) T 2 ( id) = (S 13 ) 1 ( id) in Hom C(q) (C(q) V q, V q V q V q ), where T 2 indicates taking the supertranspose on the second factor Thus S 13 ( (S 23 ) 1) T 2 ( id) = ( id) The action of U q (q(n)) on V q V q (resp on C(q)) is defined by sending U to S 13 ( (S 23 ) 1) T 2 (resp to id), so this shows that is a U q (q(n))-module homomorphism To check that is a homomorphism, we require an explicit expression for S 13 ( (S 23 ) 1) T 2 We have S 1 = q (δ ij+δ i, j )(1 2 j ) E ii E jj ξ ( 1) i (E ji + E j, i ) E ij i,j I i,j I, i<j If we identify V q with Vq via v i ω i, then (S 1 ) T 1 becomes identified with an endomorphism S of V q V q given by (315) S = q (δ ij+δ i, j )(1 2 j ) E ii E jj ξ i,j I i,j I, i<j ( 1) i j (( 1) i + j E ij + E i, j ) E ij Therefore, identifying V q V q with V q V q, we have that the action on V q V q is defined by sending U to S 13 (S ) 23 = i,j,k I ξ i I + ξ ξ 2 q (δ ij+δ i, j δ jk δ j, k )(1 2 j ) E ii E kk E jj j,k I,j<k j,k I,j<k ( 1) k j q (δ ij+δ i,j )(1 2 j ) E ii ( ( 1) j + k E jk + E j, k ) Ejk ( 1) j q (δ ik+δ i, k )(1 2 k ) ( ) E kj + E k, j Eii E jk i I i,j,k I, j<i<k ( 1) i j + j k + j ( E ij + E i, j ) ( ( 1) k E ik + ( 1) i E i, k ) Ejk

14 14 BENKART ET AL The map can be identified with the map q (2n+1) i I q2i(1 2 i ) ω i ω i, and ω k E ij = δ k,i ω j Moreover, direct calculations show q 2n+1 ( id)s 13 (S ) 23 q 2n+1 ( id) = ξ j<k( 1) k j q (1 2 j ) q 2j(1 2 j )( ( 1) j + k ω j ω k + ω j ω k ) Ejk + ξ j<k( 1) j q (1 2 k ) q 2k(1 2 k )( ( 1) k ( k + j ) ω j ω k + ( 1) ( k +1)( k + j ) ω j ω k ) Ejk ξ 2 ( 1) i j + j k + j q 2i(1 2 i )( ( 1) k + i ( i + j ) ω j ω k j<i<k + ( 1) i +( i +1)( i + j ) ) ω j ω k Ejk = ξ ( q (2k 1)(1 2 k ) q (2j+1)(1 2 j ) ξ ) ( 1) i q 2i(1 2 i ) j<k k>i>j ) (( 1) k j + k + j ω j ω k + ( 1) k j ω j ω k E jk = 0 by (312) Therefore ( id)s 13 (S ) 23 = ( id), so defines a U q (q(n))-module homomorphism Theorem 316 There is an action of BC r,s (q) on Vq r,s which supercommutes with the action of U q (q(n)), such that the action of each generator is given by where id = id Vq, t i id (i 1) P S id (r+s 1 i), c i id (i 1) Φ id (r+s i), t i id (r+i 1) P T S T id (s 1 i), c i id (r+i 1) Φ T id (s i), e id (r 1) id (s 1), (317) P = i,j I( 1) j E ij E ji End(V q V q ), Φ = i I ( 1) i E i, i End(V q ), and T is the supertranspose Explicitly, identifying V q with Vq as above, we have P S = ( 1) i q (δ ij+δ i,j )(1 2 j ) E ji E ij (318) i,j I +ξ i,j I,i<j (E ii E jj ( 1) i + j E i, i E j,j ), P T S T = ( 1) i q (δ ij+δ i,j )(1 2 j ) E ji E ij + ξ i,j I Φ T = i I i,j I,j<i E i, i, = q (2n+1) ( 1) i j q 2j(1 2 j ) E ij E ij i,j I (E ii E jj E i, i E j,j ), Proof The fact that the actions of t i and c i are U q (q(n))-module endomorphisms satisfying the relations of HC r (q) is shown in [18] Consider the linear map given by the cyclic permutation σ : Vq s Vq s, v 1 v s ( 1) i<j v i v j v s v 1 Conjugating by σ, we obtain another action of HC s (q) on Vq s specified by t i id (s 1 i) SP id (i 1), c i id (s i) Φ id (i 1)

15 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 15 These maps are also U q (q(n))-module endomorphisms (even though σ is not) Applying the antiautomorphism of HC s (q) that sends t i to t s i and c i to c s+1 i, we see that satisfy the required relations t i id (i 1) P T S T id (s 1 i), c i id (i 1) Φ T id (s i) Since and are U q (q(n))-module homomorphisms, the same is true of e We have (id Φ T ) = q (2n+1) i I q 2i(1 2 i ) ω i ω i = (Φ id) Thus, (id Φ T ) = (Φ id), so ec r = ec 1 Also (id ΦT ) = (Φ id) by (314), so c r e = c 1 e We have ( )( ) (1) = q (2n+1) q 2j(1 2 j ) ω j ω j v i v i j I i I = q (2n+1) i I ( 1) i q 2i(1 2 i ) = 0 (Φ id) (1) = q (2n+1) ( i I q 2i(1 2 i ) ω i ω i )( j I v j v j ) = 0, so e 2 = ec r e = 0 Now using q 2n+1 (E ij id) (1) = ( 1) i q 2i(1 2 i ) δ ij and identifying V q with V q C(q), we have q 2n+1 (id )(P S id)(id ) = q (2j+1)(1 2 j ) E jj + ξ ( 1) i q 2i(1 2 i ) E jj j I i,j I, j<i = q (2j+1)(1 2 j ) + ξ ( 1) i q 2i(1 2 i ) E jj j I = q 2n+1 id by (312) i,j I, j<i Thus (id )(P S id)(id ) = (id ), so et r 1 e = e Similarly, q 2n+1 ( id)(id P T S T )( id) = j q (2j+1)(1 2 j ) E jj + ξ i>j ( 1) i q 2i(1 2 i ) E jj = q 2n+1 id Thus ( id)(id P T S T )( id) = ( id), so et 1 e = e Identifying V q Vq with V q C(q) Vq, we have (P id V q Vq )(id V q id V q ) (1) = (P id) i,j v j v i ω i ω j = i,j ( 1) i j v i v j ω i ω j The above is the canonical map C(q) V q V q V q V q, so by the same reasoning that led to (314), we know Thus (SP id)(p id)(id id) = (id P T S T )(P id)(id id) (319) (P S id)(id id) = (id P T S T )(id id)

16 16 BENKART ET AL To prove the corresponding expression for, we must explicitly compute the following: q 4n+2 (id id)(p S id) ( = q 4n+2 (id id) ( 1) i q (δ ij+δ i,j )(1 2 j ) E ji E ij id id i,j + ξ ) (E jj E ii ( 1) i + j E j, j E i,i ) id id i>j = i,j +ξ i>j ( 1) i j q (δ ij+δ i,j +2j)(1 2 j )+2i(1 2 i ) ω i ω j ω i ω j q 2i(1 2 i )+2j(1 2 j ) ( ω j ω i ω i ω j + ( 1) j ω j ω i ω i ω j ), Thus, q 4n+2 (id id)(id P T S T ) ( = q 4n+2 (id id) ( 1) i q (δ ij+δ i,j )(1 2 j ) id id E ji E ij i,j + ξ ) id id (E ii E jj E i, i E j,j ) i>j ( = q 2n+1 ( 1) i q (δ ij+δ i,j +2j)(1 2 j ) id ω j ω i E ij + ξ i>j i,j ) q 2i(1 2 i ) (id ω i ω i E jj id ω i ω i E j,j ) = q 4n+2 (id id)(p S id) (320) (id id)(p S id) = (id id)(id P T S T ) Finally, using S 1 P = i,j (E ii E jj + ( 1) i + j E i,i E j, j ) ( 1) j q (δ ij+δ i,j )(1 2 j ) E ij E ji ξ i>j and we have (with the help of (312)) q 2n+1 (E ij E kl ) = δ ik δ jl ( 1) i j + i + j q 2i(1 2 i ), q 2n+1 (id id)(s 1 P P T S T )(id id) = q 2j(1 2 j ) ( 1) i j E ij E ij i,j + ξ ) (q (2i 1)(1 2 i ) q (2j+1)(1 2 j ) ) (E ii E jj + ( 1) i E i,i E j,j i>j ξ 2 ( ) ( 1) k q 2k(1 2 k ) E ii E jj + ( 1) i E i,i E j,j i>k>j = q 2n+1

17 Combining this with (320) gives QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 17 (id id)(s 1 P P T S T )(id id)(p S id) = (id id) (id id)(p S id) = (id id)(s 1 P P T S T )(id id)(id P T S T ) Thus, et 1 r 1 t 1 et r 1 = et 1 r 1 t 1 et 1 Similarly combining with (319) shows that (P S id)(id id)(s 1 P P T S T )(id id) = (P S id)(id id) (id id) = (id P T S T )(id id)(s 1 P P T S T )(id id) Hence, t r 1 et 1 r 1 t 1 e = t 1 et 1 r 1 t 1e, and it follows that et 1 r 1 t 1 et 1 t 1 r 1 = t 1 r 1 t 1 et 1 r 1 t 1 e Proposition 321 The walled Brauer-Clifford superalgebra BC r,s is the classical limit of the quantum walled Brauer-Clifford superalgebra BC r,s (q) Proof To see this, let R = C[q, q 1 ] (q 1) be the localization of C[q, q 1 ] at the ideal generated by q 1 Let BC r,s (R) be the R-subalgebra of BC r,s (q) generated by t 1,, t r 1, c 1,, c r, t 1,, t s 1, c 1,, c s, e Let V R = R C V and set V r,s R = R C V r,s It follows from [13, Thm 51] that there is a natural epimorphism from the walled Brauer-Clifford superalgebra BC r,s onto (R/(q 1)R) R BC r,s (R) = BC r,s (R)/(q 1)BC r,s (R) and hence a natural epimorphism π : BC r,s BC r,s (R)/(q 1)BC r,s (R) We want to argue that π is an isomorphism Let ρ r,s n : BC r,s End C (V r,s ) op be the representation given just before Theorem 14 The action of BC r,s (q) on Vq r,s defined in Theorem 316 restricts to a representation ρ r,s n,r : BC r,s(r) End R (V r,s R ) Let ρr,s n,r be the homomorphism BC r,s (R)/(q 1)BC r,s (R) End R (V r,s R )/(q 1)End R(V r,s R ) Since V r,s R is a free R-module, the algebra End R(V r,s R ) is also free; thus, it is possible to identify End R (V r,s R )/(q 1)End R(V r,s R ) with End C(V r,s ) Let j be the anti-involution of BC r,s which fixes each generator Upon the previous identification, the composite ρ r,s n,r π is equal to ρr,s n j, as can be checked from the action of the generators on the mixed tensor space given in Theorem 316 (Setting q = 1 in the formula for S in (31) gives the identity map) When n r + s, the map ρ r,s n is known to be injective by [13, Thm 45] Therefore, if n r + s, then ρ r,s n,r π must also be injective, hence so is π In conclusion, π is an isomorphism In the proof of Theorem 51 in [13], a vector space basis of the walled Brauer-Clifford superalgebra BC r,s is constructed In this section, we obtain a basis of BC r,s (q) which specializes to the one in [13] when q 1 (in a suitable sense) Our basis is inspired by the basis of the quantum walled Brauer algebra H n r,s(q) constructed in Section 2 of [11] (see Corollary 326 below) Definition 322 [11] A monomial n in normal form in the generators t 1, t 2,, t r 1 is a product of the form n = p 1 p 2 p r 1, where p i = t 1 i t 1 i 1 t 1 j for some j with 1 j i + 1 (If j = i + 1, then p i = 1) A monomial n in normal form in the generators t 1, t 2,, t s 1 is a product of the form n = p 1 p 2 p s 1, where p i = t i t i 1 t j for some j with 1 j i + 1 (If j = i + 1, then p i = 1)

18 18 BENKART ET AL Definition 323 Suppose that I = (i 1,, i a ) with 1 i 1 < < i a r, J = (j 1,, j a ) with 0 j k s 1 for k = 1,, a, and if k 1 k 2, then j k1 j k2 Let Ĩ I and J {1, 2,, r+s}\j A monomial m in normal form in BC r,s (q) is one of the form where m = cĩ k=1,,a t j k t j k 1 t 1ti 1 k t 1 r 2 t 1 r 1 et 1 r 1 t 1 r 2 t 1 i k t 1 t j k 1t j k c Jnn, (1) n is a monomial in normal form in the generators t 1, t 2,, t r 1 ; (2) n is a monomial in normal form in the generators t 1, t 2,, t s 1 ; (3) the product is arranged from k = 1 to k = a from left to right; (4) cĩ is the product of c i over i Ĩ in increasing order, and c is defined similarly J (5) Moreover, it is required that if n = p 1 p 2 p r 1 and p i = t 1 i t 1 i 1 t 1 j, then σ 1 (i 1 ) < < σ 1 (i a ) where σ = σ 1 σ 2 σ r 1 and σ i is the cycle (i + 1 i i 1 j + 1 j) Theorem 324 The set B of monomials m in normal form is a basis of BC r,s (q) over C(q) Proof As in Step 1 of [13, Thm 51], there are relations analogous to those labeled (1)-(8), except that words of strictly smaller length need to be added on one side of each equality By using induction on the length of words, we can verify that the set B spans BC r,s (q) over C(q) Since BC r,s (R) is a finitely generated torsion-free R-module, it is free over R Now by a standard argument in abstract algebra (cf [9, Chap 4, Thm 511]), it follows that B is linearly independent over C(q) Corollary 325 The dimension of BC r,s (q) over C(q) is (r + s)! 2 r+s Proof This follows from Theorem 324, Step 2 in the proof [13, Thm 51], and [11, Lem 17] Corollary 326 The subalgebra of BC r,s (q) generated by t 1,, t r 1, c 1,, c r (resp by t 1,, t s 1 and c 1,, c s) is isomorphic to the finite Hecke-Clifford superalgebra HC r (q) (resp to HC s (q)) The subalgebra generated by t 1,, t r 1, t 1,, t s 1, and e is isomorphic to the quantum walled Brauer algebra H 0 r,s(q) in [11] Proof The first assertion follows from the fact that the set {cĩn}, as Ĩ ranges over the subsets of I = {1,, r} and n ranges over the monomials in normal form in t 1, t 2,, t r 1, is a basis of HC r (q) over C(q) For the second assertion, one can show that the set B consisting of the elements k=1,,a t j k t j k 1 t 1t 1 i k t 1 r 2 t 1 r 1 et 1 r 1 t 1 r 2 t 1 i k t 1 t j k 1t j k nn, where n is a monomial in normal form in t 1, t 2,, t r 1 satisfying the condition (5) in Definition 323, and n is a monomial in normal form in t 1, t 2,, t s 1, spans H0 r,s(q) over C(q) Since dim C(q) H 0 r,s(q) = (r + s)! and B (r + s)!, the set B is a basis of H 0 r,s(q) over C(q) We will frequently deduce properties of BC r,s (q) from the corresponding properties of BC r,s using the following well-known facts about specialization, which we prove here for convenience Lemma 327 Suppose R is a Noetherian local integral domain whose maximal ideal is generated by a single element x R Let ψ : A B be a homomorphism of finitely generated R-modules, and consider the corresponding induced homomorphism ψ : A/xA B/xB, ψ(a + xa) = ψ(a) + xb

19 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 19 (i) If ψ is surjective, then ψ is surjective (ii) If B is torsion free and ψ is injective, then ψ is injective, and its cokernel is also torsion free Proof (i) Let C be the cokernel of ψ The right exact sequence A ψ B C induces a right exact sequence A/xA ψ B/xB C/xC By assumption, ψ is surjective, so C/xC = 0 Thus C = 0 by Nakayama s lemma (ii) Let K be the kernel of ψ If k K, then ψ(k) + xa is in the kernel of ψ, which is zero by assumption Thus k xa, so k = xa for some a A Thus xψ(a) = ψ(k) = 0, so ψ(a) = 0 since B is torsion free Thus, a K, so K = xk Again by Nakayama s lemma, K = 0, so ψ is injective Finally, choose an (R/xR)-basis X of A/xA and extend it to a basis X Y of B/xB (here we are identifying ψ(x) with X by injectivity) By lifting these basis elements arbitrarily to A and B, we obtain a commutative diagram R X R X R Y A B B/A where the top two modules are free over R, and the vertical maps induce isomorphisms of (R/xR)- vector spaces By what we ve shown so far, R X A is surjective and R X R Y B is an isomorphism Therefore B/A = R Y is free, and in particular, is torsion free We now show that BC r,s (q) gives the centralizer of the action of U q (q(n)) on Vq r,s We deduce this from the corresponding result in the classical case, which is proven in [13] Theorem 328 Let ρ r,s n,q : BC r,s (q) End Uq(q(n))(Vq r,s ) be the representation of BC r,s (q) coming from Theorem 316 Then ρ r,s n,q is surjective, and when n r + s, it is an isomorphism Proof Let π : BC r,s (R/(q 1)R) R BC r,s (R) be the isomorphism established in Proposition 321 Consider the following elements of U q (q(n)) for i, j I = {±i i = 1,, n} with i j: { ũ ij = (q 1) 1 u ij 1 if i = j, u ij if i j Let Ũ = i j ũ ij E ij U q (q(n)) C End C (V) By (31), the action of ũ ij on V q lies in End R (V R ) End C(q) (V q ) Moreover, under the surjection End R (V R ) End C (V) given by evaluation at q = 1, the action of ũ ij maps to the action of the following element of q(n): { ( 1) i E 0 ii if i = j, u ij = ( 1) i 2E 1 δ i, j ( 1) j j,( 1) i i if i < j Similarly, by (315), the action of ũ ij on V q lies in End R (V R ) and maps to the action of u ij on V by evaluation at q = 1 Finally, since the coproduct on U q (q(n)) sends (Ũ) = Ũ 13 + Ũ 23 + (q 1)Ũ 13 Ũ 23, the corresponding statements extend to the action of ũ ij on V r,s q

20 20 BENKART ET AL Now let EndŨ(V r,s R ) denote the space of endomorphisms in End R(V r,s R ) which supercommute with the action of ũ ij for all i j We will show that the R-module homomorphism ψ : BC r,s (R) EndŨ(V r,s R ) is surjective, and an isomorphism if n r + s Note that if X End R (V r,s R ) is such that (q 1)X supercommutes with u ij, then X also supercommutes with u ij Therefore, the induced homomorphism (R/(q 1)R) R EndŨ(V r,s R ) End C(V r,s ) is injective Moreover since the elements {u ij i j} generate q(n), this map factors through End q(n) (V r,s ) We obtain the following diagram π BC r,s (R/(q 1)R) R BC r,s (R) End q(n) (V r,s ) id ψ (R/(q 1)R) R EndŨ(V r,s End C (V r,s ) R ) Now Theorem 35 of [13] shows that the homomorphism ρ r,s n : BC r,s End q(n) (V r,s ) op given by the BC r,s -module action is surjective, and also injective for n r + s It follows that (R/(q 1)R) R EndŨ(V r,s R ) End q(n)(v r,s ) is an isomorphism for all n, so (R/(q 1)R) R ψ is surjective for all n and injective for n r + s Since EndŨ(V r,s R ) is torsion free, we conclude by Lemma 327 that BC r,s (R) EndŨ(V r,s R ) is surjective for all n and injective for n r + s Finally, since the ũ ij generate U q (q(n)), we have C(q) R EndŨ(V r,s ) = End Uq(q(n))(V r,s q ) Therefore, tensoring by C(q), we obtain the desired result Remark 329 The following question is left open: Does U q (q(n)) surject onto End BCr,s(q)(V r,s q )? 4 The (r, s)-bead tangle algebras BT r,s (q) In this section, we introduce a diagrammatic realization of the quantum walled Brauer-Clifford superalgebra BC r,s (q) given in Definition 34 Definition 41 An (r, s)-bead tangle is a portion of a planar knot diagram in a rectangle R with the following conditions: (1) The top and bottom boundaries of R each have r + s vertices in some standard position (2) There is a vertical wall that separates the first r vertices from the last s vertices on the top and bottom boundaries (3) Each vertex must be connected to exactly one other vertex by an arc (4) Each arc may (or may not) have finitely many numbered beads The bead numbers in the tangle start with 1 and are distinct consecutive positive integers (5) A vertical arc connects a vertex on the top boundary to a vertex on the bottom boundary of R, and it cannot cross the wall A horizontal arc connects two vertices on the same boundary of R, and it must cross the wall (6) An (r, s)-bead tangle may have finitely many loops

21 QUANTUM WALLED BRAUER-CLIFFORD SUPERALGEBRAS 21 The following is an example of (3, 2)-bead tangle 2 3 We want to stress that an (r, s)-bead tangle is in the plane, not in 3-dimensional space We consider a bead as a point on the arc Two (r, s)-bead tangles are regularly isotopic if they are related by a finite sequence of the Reidemeister moves II, III together with isotopies fixing the boundaries of R Reidemeister move II : < > < > Reidemeister move III: < > We observe that there are isotopies fixing the boundaries of the rectangles between the following tangles: * * < > * * < > * *, * * < > * * < > * * Therefore, moving a bead along a non-crossing arc or an over-crossing arc gives tangles that are regularly isotopic We want to emphasize that the following are not regularly isotopic: * * and * *, * * and * * We identify an (r, s)-bead tangle with its regular isotopy class, and denote by BT r,s the set of (r, s)-bead tangles (up to regular isotopy) The (r, s)-bead tangle in which there are even (resp odd) number of beads is regarded as even (resp odd)

22 22 BENKART ET AL Now we define a multiplication on BT r,s For (r, s)-bead tangles d 1, d 2, we place d 1 under d 2 and identify the top row of d 1 with the bottom row of d 2 We add the largest bead number in d 1 to each bead number in d 2, as we did for (r, s)-bead diagrams, and then concatenate the tangles For example, if d 1 =, 2 3 d 2 =, 2 then, d 1 d 2 = = We observe BT r,s is closed under this product, and it is Z 2 -graded The shape of the arcs are the same in (d 1 d 2 )d 3 and d 1 (d 2 d 3 ) Moreover, the locations of the beads and the bead numbers are also the same in (d 1 d 2 )d 3 and d 1 (d 2 d 3 ) It follows that the multiplication on BT r,s is associative Hence BT r,s is a monoid with identity element For 1 i r 1, 1 j s 1, 1 k r, 1 l s, we define the following (r, s)-bead tangles: σ i := i i + 1, σ j :=, j j + 1 h :=, 1 r r + 1 r + s c k := k, c l := l From Reidemeister move II, we obtain the following elements in BT r,s :