Centralizers in PBW Extensions

TL;DR

This paper characterizes the centralizer of the coefficient ring within skew PBW extensions, providing explicit descriptions in specific cases and necessary conditions in the general case, advancing understanding of their algebraic structure.

Contribution

It offers a detailed description of the centralizer in skew PBW extensions, including explicit cases and conditions, enhancing the algebraic theory of these structures.

Findings

Explicit description of the centralizer in the quasi-commutative case

Necessary conditions for the centralizer in the general case

Description of the center of the skew PBW extension of functions

Abstract

In this article we give a description for the centralizer of the coefficient ring $R$ in the skew PBW extension $\sigma(R)<x_1,x_2,\cdots,x_n>.$ We give an explicit description in the quasi-commutative case and state a necessary condition in the general case. We also consider the PBW extension $\sigma(\mathcal{A})<x_1,x_2,\cdots,x_n>$ of the algebra of functions with finite support on a countable set, describing the centralizer of $\mathcal{A}$ and the center of the skew PBW extension.