The Green ring of a restricted enveloping algebra in characteristic 2

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.

Categories: math.RT math.QA