Cohomology and support varieties for Lie superalgebras

Boe, Brian; Kujawa, Jonathan; Nakano, Daniel

doi:10.1090/S0002-9947-2010-05096-2

Cohomology and support varieties for Lie superalgebras
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by Brian D. Boe, Jonathan R. Kujawa and Daniel K. Nakano PDF

Trans. Amer. Math. Soc. 362 (2010), 6551-6590 Request permission

Abstract:

Unlike Lie algebras, the finite dimensional complex representations of a simple Lie superalgebra are usually not semisimple. As a consequence, despite over thirty years of study, these remain mysterious objects. In this paper we introduce a new tool: the notion of cohomological support varieties for the finite dimensional supermodules for a classical Lie superalgebra $\mathfrak {g} = \mathfrak {g}_{\bar 0} \oplus \mathfrak {g}_{\bar 1}$ which are completely reducible over ${\mathfrak g}_{\bar 0}$. They allow us to provide a new, functorial description of the previously combinatorial notions of defect and atypicality. We also introduce the detecting subalgebra of $\mathfrak {g}$. Its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic. Using invariant theory we prove that there are close connections between the cohomology and support varieties of $\mathfrak {g}$ and the detecting subalgebra.

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