Cohomology and Support Varieties for Lie Superalgebras II

Abstract

In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

28 pages, the proof of Proposition 4.5.1 was corrected, several other small errors were fixed

References in corpus (4)

• On associated variety for Lie superalgebras
• On representations of the Lie superalgebra p(n)
• Crystal structures arising from representations of $GL(m|n)$
• Cohomology and Support Varieties for Lie Superalgebras

Cited by in corpus (8)

• On support varieties for Lie superalgebras and finite supergroup schemes
• Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)
• Model structures, categorial quotients and representations of super commutative Hopf algebras II, The case Gl(m,n)
• Support varieties and modules of finite projective dimension for modular Lie superalgebras (with an appendix on homological dimensions over Noether Algebras by Luchezar L. Avramov and Srikanth B. Iyengar)
• On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n|n)$
• Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and Connections
• Pieri type rules and $GL(2|2)$ tensor products
• Endotrivial modules for the general linear Lie superalgebra