The Green ring of a restricted enveloping algebra in characteristic 2

TL;DR

This paper computes the Green ring of a specific restricted enveloping algebra in characteristic 2 and describes its semisimplified representation category, providing new insights into its module structure.

Contribution

It explicitly calculates the Green ring of the restricted enveloping algebra of the minimal 2-envelope of sl(2) in characteristic 2, a novel result in this context.

Findings

Green ring of the algebra is explicitly determined.

Semisimplification of the module category is described.

Classification of indecomposable modules is utilized in analysis.

Abstract

Let $\Bbbk$ be an algebraically closed field of characteristic $2$ and let $\mathfrak{fsl}(2)$ be the unique, up to isomorphism, $3$-dimensional simple Lie algebra over $\Bbbk$. Denote by $\mathfrak{m}$ the minimal $2$-envelope of $\mathfrak{fsl}(2)$ and by $\mathfrak{u}(\mathfrak{m})$ its corresponding restricted enveloping algebra. The non-isomorphic finite-dimensional indecomposable $\mathfrak{u}(\mathfrak{m})$-modules were classified in \cite{ABDF}. In this paper, the Green ring (or representation ring) for $\mathfrak{u}(\mathfrak{m})$ is calculated. Also, the semisimplification of the representation category of $\mathfrak{u}(\mathfrak{m})$ is determined.