Real forms of complex Lie superalgebras and supergroups

TL;DR

This paper explores the concept of real forms of complex Lie superalgebras and supergroups, introducing a functorial approach and a generalized notion of compact real forms, with existence results for certain simple structures.

Contribution

It presents a functorial framework for real forms of Lie superalgebras and supergroups, including a new concept of compact real form and existence proofs for specific simple cases.

Findings

Established a functorial method to obtain real forms as fixed points.

Introduced a generalized notion of compact real form.

Proved existence results for simple contragredient Lie superalgebras.

Abstract

We investigate the notion of real form of complex Lie superalgebras and supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real forms as fixed points, as in the ordinary setting. We also introduce a more general notion of compact real form for Lie superalgebras and supergroups, and we prove some existence results for Lie superalgebras that are simple contragredient and their associated connected simply connected supergroups.