Fabio Gavarini

A real form of a complex Lie algebra is the subset of fixed points of some "real structure", that is an antilinear involution; a similar description applies for real forms of complex (Lie or algebraic) groups. For complex Lie superalgebras, the notion of "real structure" extends in two different variants, called standard (a straightforward generalization) and graded (somewhat more sophisticated): the notion of "real form", however, stands problematic in the graded case. I will present the functorial version of "real structure" (standard or graded), and show that the notion of "real form" then properly extends, in both cases; along the same lines, I will introduce real structures and real forms for complex supergroups. Then, basing on a suitable notion of "Hermitian form" on complex superspaces, I will introduce unitary Lie superalgebras and supergroups (again standard or graded); any Lie superalgebra which embeds into a unitary one will then be called "super-compact" - and similary for supergroups. Finally, I will give nice existence/uniqueness results of super-compact real forms for complex Lie superalgebras which are simple of basic (or "contragredient") type, and similarly for their associated connected simply-connected supergroups. This is based on a joint work with Rita Fioresi, cf. Comm. Math. Phys. 397 (2023), no. 2, 937–965.