Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Abstract

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 Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let q(n) be the queer Lie superalgebra, V = C n|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 End q(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 Ser k . Finally, we give a presentation of − → B r,s in terms of generators and relations.2000 Mathematics Subject Classification. 17B10, 05E10. Key words and phrases. mixed Schur-Weyl-Sergeev duality, queer Lie superalgebras, walled Brauer superalgebras.

A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

“…Remark 4.2. When k has characteristic zero, Schur-Weyl-Sergeev duality also implies the right vertical map in (4.2) is injective whenever r ≤ n. It follows that (4.1) is an isomorphism whenever the average length of the words a and b is less than or equal to n. In particular, Φ prescribes an isomorphism of superalgebras End OBC (a) ∼ = End U(q) (V a ) whenever the length of a is less than or equal to n. Coupled with Corollary 3.6 this recovers [JK,Theorem 3.5].…”

“…For arbitrary a, End OBC (a) is isomorphic to the walled Brauer-Clifford superalgebra introduced by Jung-Kang [JK] (see Corollary 3.6). The fact that (1.2) is an isomorphism whenever the length of a is less than or equal to n recovers [JK,Theorem 3.5]. We should point out the definitions given in [JK] are global in nature.…”

“…Consequently, assume a =↓ s ↑ r . By [JK,Theorem 5.1], BC r,s is generated by even generators s 1 , . .…”

Quantum walled Brauer–Clifford superalgebras

“…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 ?…”

“…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.…”