Three Examples of Graded Lie Groups

TL;DR

This paper develops three fundamental examples of graded Lie groups—general linear, orthogonal, and symplectic—extending classical Lie theory to the realm of $ obreak bZ$-graded geometry with explicit constructions and applications.

Contribution

It introduces and constructs explicit examples of graded Lie groups and their Lie algebras, extending classical Lie theory to $ obreak bZ$-graded geometry with geometric and functor-of-points approaches.

Findings

Constructed graded general linear, orthogonal, and symplectic groups.

Showed their Lie algebras are isomorphic to expected subalgebras.

Discussed isomorphisms and potential applications of these graded groups.

Abstract

Lie theory is, beyond any doubt, an absolutely essential part of differential geometry. It is therefore necessary to seek its generalization to $\mathbb{Z}$-graded geometry. In particular, it is vital to construct non-trivial and explicit examples of graded Lie groups and their corresponding graded Lie algebras. Three fundamental families of graded Lie groups are developed in this paper: the general linear group associated with any graded vector space, the graded orthogonal group associated with a graded vector space equipped with a metric, and the graded symplectic group associated with a graded vector space equipped with a symplectic form. We provide both a direct geometric construction and a functor-of-points perspective. It is shown that their corresponding Lie algebras are isomorphic to the anticipated subalgebras of the graded Lie algebra of linear endomorphisms. Isomorphisms of graded Lie groups induced by linear isomorphisms, as well as possible applications, are also discussed.