Almost complex structures on real Lie supergroups

TL;DR

This paper characterizes when a real Lie supergroup admits an integrable almost complex structure, linking it to complex Lie supergroups, and introduces a universal complexification, offering a new classification approach.

Contribution

It provides necessary and sufficient conditions for integrable almost complex structures on real Lie supergroups and introduces a universal complexification, connecting real and complex supergroup classifications.

Findings

Derived conditions for integrable almost complex structures.

Classified real Lie supergroups with such structures.

Constructed a universal complexification.

Abstract

A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complex Harish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed.