Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras

We introduce a new family of superalgebras B→r,s for r,s⩾0 such that r+s>0, which we call the walled Brauer superalgebras, and prove the mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let q(n) be the queer Lie superalgebra, V=Cn|n the natural representation of q(n) and W the dual of V. We prove that, if n⩾r+s, the superalgebra B→r,s is isomorphic to the supercentralizer algebra Endq(n)(V⊗r⊗W⊗s)op of the q(n)-action on the mixed tensor space V⊗r⊗W⊗s. As an ingredient for the proof of our main result, we construct a new diagrammatic realization D→k of the Sergeev superalgebra Serk. Finally, we give a presentation of B→r,s in terms of generators and relations.