$\mathfrak{gl}(1,1)$和$\mathfrak{sl}(1,1)$的不可积超对称及其分类
Non-integrable supersymmetries and their classification for $\mathfrak{gl}(1,1)$ and $\mathfrak{sl}(1,1)$
摘要：
描述了经典李群上的无穷小超对称性，这些超对称性不一定能积分到李群上。它们产生了一个超对称性的概念，这个概念比李群作用的假设不那么严格，但仍然暗示着李群的潜在作用。与李群相反，表现理论李群（RTLSG）是作为与Harish-Chandra超对相关的家族出现的。然而RTLSG的态射直接对应于Harish-Chandra超对的态射。特殊的RTLSG可以从Berezin和Kostant给出的李超群的显式构造中导出。李超代数$\mathfrak{gl}(1,1)$或$\mathfrak{sl}(1,1)$也出现在高维经典李超代数中，提供了RTLSG的有趣的第一个例子。利用参数空间和完备不变量集，给出了实和复$\mathfrak{gl}(1,1)$ -和$\mathfrak{sl}(1,1)$ -Harish-Chandra超对的RTLSG的分类.假设基础李群是连通的，但可能不是单连通的。
英文摘要：
Infinitesimal supersymmetries over classical Lie groups that do not necessarily integrate to Lie supergroups are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish-Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish-Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie supergroups given by Berezin and Kostant. The Lie superalgebras $\mathfrak{gl}(1,1)$ or $\mathfrak{sl}(1,1)$ appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex $\mathfrak{gl}(1,1)$- and $\mathfrak{sl}(1,1)$-Harish-Chandra superpairs is given by parameter spaces and complete sets of invariants. The underlying Lie group is assumed to be connected but possibly not simply connected.
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