ON ASSOCIATED VARIETY FOR LIE SUPERALGEBRAS

By M. Duflo, Vera Serganova. 2005

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We define the associated variety XM of a module M over a finite- dimensional superalgebra g, and show how to extract information about M from these geometric data. XM is a subvariety of the cone X of self-commuting odd elements. For finite-dimensional M, XM is invariant under the action of the underlying Lie group G0. For simple superalgebra with invariant symmetric form, X has finitely many G0-orbits; we associate a number (rank) to each such orbit. One can also associate a number (degree of atypicality) to an irreducible finite-dimensional representation. We prove that if M is an irreducible g-module of degree of atypicality k, then XM lies in the closure of all orbits on X of rank k. If g = gl(m|n) we prove that XM coincides with this closure.

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Citations in the corpus

Generalized trace and modified dimension functions on ribbon categories (2010, methodology): “[13, 19 ,34]), and recently Serganova has announced a proof for the classical contragradiant Lie superalgebras using category equivalences, Zuckerman functors, and a character formula of Penkov [51].”
Deligne Categories and the Limit of Categories Rep(GL(m|n)) (2015, methodology): “the tensor product V(λ) ⊗V∗. Since starting from the trivial module we can obtain any V(λ) by tensoring with V and V∗(see Corollary 6.2.2), the statement follows. 7. Homology functor 7.1. Recall from [DS] that for any Lie superalgebra g and an odd element x∈g such that [x,x] = 0, one can deﬁne a fun”