Various instances of Harish-Chandra pairs

TL;DR

This paper generalizes the integration of graded Lie algebras to Z-graded cases, explores algebraic structures like groupoids and algebroids, and applies Harish-Chandra pairs to these contexts.

Contribution

It extends the integration techniques from N-graded to Z-graded Lie algebras and develops applications involving Harish-Chandra pairs in algebroid settings.

Findings

Successful generalization of integration from N-graded to Z-graded Lie algebras

Construction of examples using Harish-Chandra pairs

Extension of techniques to algebroids, including action Lie algebroids

Abstract

In this paper we address several algebraic constructions in the context of groupoids, algebroids and $\mathbb Z$-graded manifolds. We generalize the results of integration of $\mathbb N$-graded Lie algebras to the honest $\mathbb Z$-graded case and provide some examples of application of the technique based on Harish-Chandra pairs. We extend the construction to the algebroids setting, the main example being the action Lie algebroid.