Noncommutatively Graded Algebras

TL;DR

This paper introduces noncommutatively graded algebras, generalizing classical graded algebra results and establishing foundational properties and structures like universal enveloping algebras and a graded PBW theorem.

Contribution

It extends the theory of graded algebras to noncommutative settings, including Lie algebras, and proves key structural theorems in this new context.

Findings

Established existence of universal graded enveloping algebras for noncommutatively graded Lie algebras.

Proved a graded version of the Poincaré-Birkhoff-Witt theorem.

Generalized classical graded algebra results to the noncommutative case.

Abstract

Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning identity elements, inverses, existence of limits and colimits and adjointness of certain functors. In the particular instance of noncommutatively graded Lie algebras, we establish the existence of universal graded enveloping algebras and we show a graded version of the Poincar\'e-Birkhoff-Witt theorem.