Non-integrable supersymmetries and their classification for $\mathfrak{gl}(1,1)$ and $\mathfrak{sl}(1,1)$

TL;DR

This paper explores non-integrable infinitesimal supersymmetries over classical Lie groups, focusing on their classification for the Lie superalgebras gl(1,1) and sl(1,1), revealing new structures called RTLSGs.

Contribution

It provides a classification of real and complex RTLSGs associated with gl(1,1) and sl(1,1) superpairs, including parameter spaces and invariants.

Findings

Classified RTLSGs for gl(1,1) and sl(1,1) superpairs.

Identified invariants and parameter spaces for these RTLSGs.

Connected but not necessarily simply connected underlying Lie groups.

Abstract

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.