A note on affine representable algebras

TL;DR

This paper demonstrates that affine representable algebras over a field can be embedded into polynomial algebras and confirms that their Gelfand-Kirillov dimension is always an integer.

Contribution

It proves that affine representable algebras can be chosen as polynomial algebras and refines Markov's theorem on their Gelfand-Kirillov dimension.

Findings

Affine representable algebras can be embedded into polynomial algebras.

Gelfand-Kirillov dimension of these algebras is always an integer.

Refinement of Markov's theorem on the dimension.

Abstract

We consider affine representable algebras, that is, finitely generated algebras over a field that can be embedded into some matrix algebra over a commutative algebra. We show that this algebra can in fact be chosen to be a polynomial algebra. We also prove a refined version of a theorem of V.T. Markov stating that the Gelfand-Kirillov dimension of any affine representable algebra is an integer.