Universal enveloping of a graded Lie algebra

TL;DR

This paper extends the concept of universal enveloping algebras to G-graded Lie algebras with non-abelian grading groups, establishing existence, uniqueness, and key properties including graded versions of classical theorems.

Contribution

It introduces a construction for graded universal enveloping algebras for non-abelian groups and proves their fundamental properties, generalizing classical results.

Findings

Constructed graded universal enveloping algebra for non-abelian G-graded Lie algebras.

Proved graded versions of Witt's Theorem and Ado's Theorem.

Explored conditions under which a Lie grading is equivalent to an abelian grading.

Abstract

In this paper we construct a graded universal enveloping algebra of a $G$-graded Lie algebra, where $G$ is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading.