Current superalgebras and unitary representations
Current superalgebras and unitary representations
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DOI：
10.1016/j.jpaa.2017.12.009
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发表时间：
2017-07
影响因子：
0.8
通讯作者：
K. Neeb;M. Yousofzadeh
中科院分区：
文献类型：
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作者：
K. Neeb;M. Yousofzadeh
在本文中，我们确定了有限维李超群的射影酉表示，其底层李超代数为 g= A⊗ k，其中 k 是一个紧简单李超代数，A 是一个超交换结合（超）代数；关键情况是当 A= Λ s (R) 是格拉斯曼代数时。由于我们对投影表示感兴趣，因此第一步包括确定定义相应中心扩展的余循环。我们的第二个主要结果断言，如果 k 是一个简单的紧李超代数，且 k 1≠{0}，则 Λ s (R)⊗ 的每个（投影）酉表示通过 k 本身的（投影）酉表示来分解 k 个因子，并且这些通过 Jakobsen 分类可知。如果 k 1={0}，那么我们同样将分类问题简化为紧李群 K 与 Clifford-Lie 超群的半直积，Carmeli、Cassinelli、Toigo 和 Varadarajan 已研究过该超群。
In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is g= A⊗ k, where k is a compact simple Lie superalgebra and A is a supercommutative associative (super) algebra; the crucial case is when A= Λ s (R) is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if k is a simple compact Lie superalgebra with k 1≠{0}, then each (projective) unitary representation of Λ s (R)⊗ k factors through a (projective) unitary representation of k itself, and these are known by Jakobsen's classification. If k 1={0}, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.