Cohomology and support varieties for Lie superalgebras II
Cohomology and support varieties for Lie superalgebras II
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DOI：
10.1112/plms/pdn019
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发表时间：
2007-08
影响因子：
1.8
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
在[2](Preprint，2006，arxiv：Math.RT/0609363)中，作者利用上同调方法开始了对经典李超代数表示理论的研究。构造了检测子代数，发展了支持簇理论。检测子代数的维度与李超代数的缺陷一致，并猜想简单超模的支撑簇的维度等于超模的非典型性。本文计算了第I类李超代数的Kac超模和gl(m|n)的简单超模的支撑簇。后一个结果验证了我们先前对gl(m|n)的猜想。在我们的研究中，我们还刻画了I型和II型经典李超代数之间的几个主要区别。最后，利用Clifford超代数的支撑簇理论和表示，使非典型性、亏量和超维之间的联系更加精确。
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 gl (m|n). The latter result verifies our earlier conjecture for 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.