Adventures of Harish-Chandra in $\mathbb Z_2 \times \mathbb Z_2$-graded world

TL;DR

This paper explores the structure and properties of $ ext{Z}_2 imes ext{Z}_2$-graded Lie algebras, relating them to Lie superalgebras, and investigates the Lie group--algebra correspondence using Harish-Chandra pairs with practical examples.

Contribution

It introduces the theory of $ ext{Z}_2 imes ext{Z}_2$-graded Lie algebras, connecting them to superalgebras and extending the Lie group--algebra correspondence framework to this bi-graded context.

Findings

Characterization of $ ext{Z}_2 imes ext{Z}_2$-graded Lie algebras

Relation to Lie superalgebras with compatible structures

Examples demonstrating the Lie group--algebra correspondence in the bi-graded setting

Abstract

We study $\mathbb Z_2\times\mathbb Z_2$ bi-graded Lie algebras. We describe their properties in relation to Lie superalgebras with some compatible structures. Then we focus on the approach to the Lie group--algebra correspondence based on Harish-Chandra pairs and provide some examples of application of it in the bi-graded setting.