Infinitesimal supersymmetries over Lie groups and their classification for gl(1,1) and sl(1,1)

Infinitesimal supersymmetries over classical Lie groups that are not necessarily induced by a Lie supergroup 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 Kostant and Koszul. The Lie superalgebras $\mathfrak{g}\mathfrak{l}(1,1)$ or $\mathfrak{s}\mathfrak{l}(1,1)$ appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex $\mathfrak{g}\mathfrak{l}(1,1)$- and $\mathfrak{s}\mathfrak{l}(1,1)$-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.