Quantum walled Brauer–Clifford superalgebras

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