Quantum walled Brauer–Clifford superalgebras

Abstract

Abstract. We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras BCr,s(q). The superalgebra BCr,s(q) is a quantum deformation of the walled BrauerClifford superalgebra BCr,s and a super version of the quantum walled Brauer algebra. We prove that BCr,s(q) is the centralizer superalgebra of the action of Uq(q(n)) on the mixed tensor space V r,s q⊗s when n ≥ r + s, where Vq = C(q) (n|n) is the natural representation of the quantum enveloping superalgebra Uq(q(n)) and V * q is its dual space. We also provide a diagrammatic realization of BCr,s(q) as the (r, s)-bead tangle algebra BTr,s(q). Finally, we define the notion of q-Schur superalgebras of type Q and establish their basic properties.

The marked Brauer category

“…Another natural question is if there exists a marked version of the Birman–Murakami–Wenzl (BMW) algebra. Preliminary results of a forthcoming paper by Grantcharov and Guay suggest the existence of a quantum group corresponding to the Lie superalgebra , and it is natural to expect a deformation of the marked Brauer algebra that plays the role of the BMW algebra in this setting (see for similar results for ).…”

Quantum walled Brauer algebra: commuting families, Baxterization, and representations

“…Also, if a (skew-shaped) tableau S contains 1 1 , then we let S n 1 1 denote the result of removing the box containing 1 1 . 12 The general solution of (5.4) for ηpxqζ pxq is ηpxqζ pxq "´αβ x Q Fpxq with any Q.…”

“…The quantum version of the algebra was introduced in [6,7,8,9], and its role as the centralizer of U q pgℓ N q on the mixed tensor product X˚b m b X bn was elucidated in [10,11]. We also note a recent "super" extension of quantum walled Brauer algebras in [12].…”

Webs of type Q

“…In [12] Chang-Wang introduce a quantum Howe duality of type Q. This is done by defining an action of quantized enveloping algebras of type Q on a quantized symmetric algebra which also appeared in [2]. In [4] the authors and Davidson show this Howe duality can be used to obtain quantum analogues of the results presented here.…”