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Affine walled Brauer algebras and super Schur-Weyl duality. (English) Zbl 1356.17012
Summary: A new class of associative algebras referred to as affine walled Brauer algebras is introduced. These algebras are free with infinite rank over a commutative ring containing 1. Then level two walled Brauer algebras over \(\mathbb{C}\) are defined, which are some cyclotomic quotients of affine walled Brauer algebras. We establish a super Schur-Weyl duality between affine walled Brauer algebras and general linear Lie superalgebras, and realize level two walled Brauer algebras as endomorphism algebras of tensor modules of Kac modules with mixed tensor products of the natural module and its dual over general linear Lie superalgebras, under some conditions. We also prove the weakly cellularity of level two walled Brauer algebras, and give a classification of their simple modules over \(\mathbb{C}\). This in turn enables us to classify the indecomposable direct summands of the said tensor modules.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
20C08 Hecke algebras and their representations
20G43 Schur and \(q\)-Schur algebras
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