Quantum Walled Brauer-clifford Superalgebras

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 q = V ⊗r q ⊗ (V∗ q) 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. 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 CΣk of the symmetric group Σk appears as the centralizer of the gl(n)-action on V⊗k, where V = Cn is the natural representation of the general linear Lie algebra gl(n). Similarly, Hecke algebras, Brauer algebras, Birman-Murakami-Wenzl algebras and HeckeClifford 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 Vr,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 Vr,s is the walled Brauer algebra Br,s(n). The structure and properties of Br,s(n) were first investigated in [1, 12, 20]. By replacing gl(n) by the quantum enveloping algebra Uq(gl(n)) and V = Cn by Vq = 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), Cn by C(m|n); Uq(gl(n)) by Uq(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.