Centralizers in Domains of Finite Gelfand-Kirillov Dimension

TL;DR

This paper investigates the structure of centralizers in finitely generated noetherian domains, establishing bounds on their Gelfand-Kirillov dimension and showing they satisfy polynomial identities under certain conditions.

Contribution

It generalizes previous results by proving bounds on the Gelfand-Kirillov dimension of centralizers and demonstrating polynomial identities in specific cases.

Findings

Centralizers have Gelfand-Kirillov dimension at most one less than the domain.

In GK dimension 3 domains over complex numbers, centralizers satisfy polynomial identities.

Abstract

We study centralizers of elements in domains. We generalize a result of the author and Small, showing that if $A$ is a finitely generated noetherian domain and $a\in A$ is not algebraic over the extended centre of $A$, then the centralizer of $a$ has Gelfand-Kirillov dimension at most one less than the Gelfand-Kirillov dimension of $A$. In the case that $A$ is a finitely generated noetherian domain of GK dimension 3 over the complex numbers, we show that the centralizer of an element a $A$ that is not algebraic over the extended centre of $A$ satisfies a polynomial identity.