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arXiv · 1708.04802

Bergman's Centralizer Theorem and quantization

Alexei Kanel Belov·Farrokh Razavinia·Wenchao Zhang
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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 $A$.

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Alexei Kanel Belov, Farrokh Razavinia, Wenchao Zhang. 2017-08-16. Bergman's Centralizer Theorem and quantization. https://doi.org/10.1080/00927872.2017.1372462

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