Adventures of Harish-Chandra in $\mathbb Z_2 \times \mathbb Z_2$-graded world
Olga Chekeres, Alexei Kotov, Vladimir Salnikov
TL;DR
The work addresses the structure and integration of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebras, clarifying their relation to Lie superalgebras via an involutive automorphism. It develops the algebraic framework, including bi-graded brackets, Jacobi identities, and the universal enveloping algebra ${\mathscr{U}}({\mathfrak g})$, and establishes an explicit integration scheme to a group object in the category of bi-graded manifolds using Harish-Chandra pairs with the maximal subalgebra ${\mathfrak g}_+ = {\mathfrak g}_{00} \oplus {\mathfrak g}_{11}$. A key contribution is the demonstration of an equivalence between bi-graded Lie algebras and Lie superalgebras equipped with an involution, together with a careful treatment of morphisms and the implications for global (smooth) geometry, where naive sign-trick translations may fail. The results provide a concrete pathway to realize the Lie group–algebra correspondence in multigraded settings and illuminate how these structures behave under integration and involutive symmetries, with potential applications in graded differential geometry and representation theory.
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.