Rationality of the Hilbert series of Hopf-invariants of free algebras

TL;DR

This paper proves that the Hilbert series of invariants of free associative algebras under semisimple Hopf algebra actions are rational functions in characteristic zero.

Contribution

It establishes the rationality of the Hilbert series for invariants of free algebras under semisimple Hopf actions, extending previous results to a broader algebraic context.

Findings

Hilbert series of invariants are rational functions

Results hold over fields of characteristic zero

Applicable to free associative algebras under semisimple Hopf actions

Abstract

It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.