Cohomology and support varieties for Lie superalgebras II

Abstract

In [2] (Preprint, 2006, arXiv:math.RT/0609363) 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 of Kac supermodules for Type-I Lie superalgebras and of the simple supermodules for g l(m|n). The latter result verifies our earlier conjecture for g l(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.