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Proceedings of the London Mathematical Society Advance Access originally published online on May 2, 2008
Proceedings of the London Mathematical Society 2009 98(1):19-44; doi:10.1112/plms/pdn019
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© 2008 London Mathematical Society

Cohomology and support varieties for Lie superalgebras II

Brian D. Boe

Department of Mathematics
University of Georgia
Athens, GA 30602
USA
brian@math.uga.edu

Jonathan R. Kujawa

Department of Mathematics
University of Oklahoma
Norman, OK 73019
USA
kujawa@math.ou.edu

Daniel K. Nakano

Department of Mathematics
University of Georgia
Athens, GA 30602
USA

Received 15 August 2007. Revision received 14 February 2008.

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 Formula (m|n). The latter result verifies our earlier conjecture for Formula (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.

2000 Mathematics Subject Classification 17B56, 17B10 (primary), 13A50 (secondary).

The second author was partially supported by NSF grants DMS-0402916 and DMS-0734226, and the third author was partially supported by NSF grants DMS-0400548 and DMS-0654169.


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