Bergman's Centralizer Theorem and quantization

Abstract

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

10 pages

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