Bergman's Centralizer Theorem and quantization

TL;DR

This paper proves Bergman's Centralizer Theorem using generic matrices and Kontsevich's quantization, showing that centralizers in free associative algebras are polynomial rings in one variable, with no higher transcendence commutative subalgebras.

Contribution

It provides a new proof of Bergman's theorem employing quantization techniques and characterizes the structure of centralizers in free associative algebras over fields of positive characteristic.

Findings

Centralizers are polynomial rings in one variable.

No commutative subalgebra with transcendent degree ≥ 2 exists.

The proof uses generic matrices and Kontsevich's quantization method.

Abstract

We prove Bergman's theorem on centralizers by using generic matrices and Kontsevich's quantization method. For any field $\textbf{k} $ of positive characteristics, set $A=\textbf{k} \langle x_1,\dots,x_s\rangle$ be a free associative algebra, then any centralizer $\mathcal{C}(f)$ of nontrivial element $f\in A\backslash \textbf{k}$ is a ring of polynomials on a single variable. We also prove that there is no commutative subalgebra with transcendent degree $\geq 2$ of $A$.