G-set Theory and Applications in Lie Theory

TL;DR

This paper introduces new concepts in Lie theory, focusing on G-sets and conjugacy, analyzing their properties and applications from computational and observability perspectives.

Contribution

It develops the theory of G-sets and conjugacy in Lie groups, providing a rigorous framework for their analysis and applications in transformation groups.

Findings

Defined G-sets as G-equivariant maps from Lie groups to manifolds

Analyzed the properties of G-sets in relation to transformation groups

Provided theoretical justification for G-sets in group actions on manifolds

Abstract

This paper is devoted to the development and applications of some (new) basic concepts in Lie theory, both from `computational" and "observability" viewpoint. We specify set of all "G-equivariant" maps from a given Lie group G to the underlying manifold M, namely $G$-set, and also we introduce "conjugacy" in Lie group theory. The next goal of this paper is detailed analysis of the G-sets in connection with underlying transformation groups and providing a rigorous theoretical justification of "G-sets", when a group of transformations G acts on manifold M.