Cohomology and Support Varieties for Lie Superalgebras
Cohomology and Support Varieties for Lie Superalgebras
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DOI：
10.1090/s0002-9947-2010-05096-2
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发表时间：
2006-09
影响因子：
1.3
通讯作者：
Brian D. Boe;J. Kujawa;D. Nakano
中科院分区：
文献类型：
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作者：
Brian D. Boe;J. Kujawa;D. Nakano
与李代数不同，单李超代数的有限维复表示通常不是半单的。因此，尽管经过了30多年的研究，这些物体仍然是神秘的。在本文中，我们引入了一个新的工具：上同调支持簇的概念，有限维超模的经典李超代数g = g 0 <$g1是完全可约的g 0。它们使我们能够提供一个新的，函子描述以前的组合概念的缺陷和pancicality。我们还引入了g的检测子代数。它的作用类似于正特征有限群理论中的亏子群。利用不变量理论证明了g的上同调簇和支撑簇与检测子代数之间存在着密切的联系。
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 g = g 0 ⊕ g 1 which are completely reducible over g 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 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 g and the detecting subalgebra.