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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2007
- DOI:
- 10.48550/arXiv.0708.3191
- arXiv:
- arXiv:0708.3191
- Bibcode:
- 2007arXiv0708.3191B
- Keywords:
-
- Mathematics - Representation Theory;
- 17B56;
- 17B10
- E-Print:
- 28 pages, the proof of Proposition 4.5.1 was corrected, several other small errors were fixed