Automorphisms of simple quotients of the Poisson and universal enveloping algebras of $\mathrm{sl}_2$

TL;DR

This paper characterizes the automorphism groups of certain simple quotients of Poisson and universal enveloping algebras of sl_2, showing they are isomorphic and describing their structure as amalgamated products.

Contribution

It provides a detailed description of the automorphism groups of these simple quotient algebras, extending previous results and establishing their isomorphism.

Findings

Automorphism groups are described as amalgamated products.

Automorphism groups of Poisson and universal enveloping algebra quotients are isomorphic.

Generators of automorphism groups are explicitly identified.

Abstract

Let $P(\mathrm{sl}_2(K))$ be the Poisson enveloping algebra of the Lie algebra $\mathrm{sl}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $ $ $P(\mathrm{sl}_2(K))/(C_P-\lambda)$, where $C_P$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in $P(\mathrm{sl}_2(K))$ and $0\neq \lambda\in K$, are proven to be simple in \cite{UZh}. Using a result by L. Makar-Limanov \cite{ML90}, we describe generators of the automorphism group of $P(\mathrm{sl}_2(K))/(C_P-\lambda)$ and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by J. Dixmier \cite{Dixmier73} and O. Fleury \cite{Fleury} for the quotient algebras $U(\mathrm{sl}_2(K))/(C_U-\lambda)$, where $C_U$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in the universal enveloping algebra $U(\mathrm{sl}_2(K))$, we prove that the automorphism groups of $P(\mathrm{sl}_2(K))/(C_P-\lambda)$ and $U(\mathrm{sl}_2(K))/(C_U-\lambda)$ are isomorphic.