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Cohomology and support varieties for Lie superalgebras


Authors: Brian D. Boe, Jonathan R. Kujawa and Daniel K. Nakano
Journal: Trans. Amer. Math. Soc. 362 (2010), 6551-6590
MSC (2010): Primary 17B56, 17B10; Secondary 13A50
DOI: https://doi.org/10.1090/S0002-9947-2010-05096-2
Published electronically: July 13, 2010
MathSciNet review: 2678986
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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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Additional Information

Brian D. Boe
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: brian@math.uga.edu

Jonathan R. Kujawa
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kujawa@math.ou.edu

Daniel K. Nakano
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: nakano@math.uga.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05096-2
Received by editor(s): April 13, 2009
Published electronically: July 13, 2010
Additional Notes: The research of the first author was partially supported by NSA grant H98230-04-1-0103
The research of the second author was partially supported by NSF grants DMS-0402916 and DMS-0734226
The research of the third author was partially supported by NSF grants DMS-0400548 and DMS-0654169
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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