On free associative algebras linearly graded by finite groups

TL;DR

This paper studies finite group gradings on free associative algebras, analyzing their structure, Hilbert series, and conditions for finite generation, revealing connections between grading triviality and algebraic properties.

Contribution

It provides an explicit formula for the Hilbert series of the graded subalgebra and characterizes when such subalgebras are finitely generated based on grading triviality.

Findings

Hilbert series is rational and explicitly computed.

Finitely generated subalgebras correspond to trivial gradings.

Provides conditions linking grading triviality to algebraic finiteness.

Abstract

As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding to the identity of the group is a free subalgebra which is graded by the usual degree. We look into its Hilbert series and prove that it is a rational function by giving an explicit formula. As an application, we show that, under suitable conditions, this subalgebra is finitely generated if and only if the grading on the base vector space is trivial.