The Lie superalgebra P(n) and Brauer algebras with signs

Speaker: Vera Serganova, UC Berkeley.

Abstract:

The “strange” Lie superalgebra P(n) is the algebra of endomorphisms of an (n|n)-dimensional vector space V equipped with a non-degenerate odd symmetric form. Representations of P(n) in tensor powers of V are not completely reducible. The centralizer of the P(n)-action in the k-th tensor power of V is given by a certain analogue of the Brauer algebra.Using this algebra one can construct a pseudo-abelian tensor category P-rep, which is a natural analogue of the Deligne categories GL(t)-rep and SO(t)-rep.

Then we construct an abelian tensor category C which satisfy certain universal properties with respect to the categories of representations of P(n) for all n. We discuss combinatorial properties of C and its relationship with P-rep.

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