The Green ring of a restricted enveloping algebra in characteristic 2

Abstract

Let $\mathbb{k}$ 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 $\mathbb{k}$. 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.