The Green ring of a restricted enveloping algebra in characteristic 2
Nicolás Andruskiewitsch, Dirceu Bagio, Saradia Della Flora, Daiana Flôres
TL;DR
The paper determines the Green ring of the restricted enveloping algebra $\mathfrak{u}(\mathfrak{m})$ in characteristic $2$ by leveraging the tame classification of indecomposables into simple modules, string modules, and band modules, and then computes all tensor product decompositions to obtain an explicit presentation $r(\mathfrak{u}(\mathfrak{m})) \cong \mathbb{Z}[X]/I$ with generators and relations. It further analyzes the semisimplification of $\operatorname{rep}\,\mathfrak{u}(\mathfrak{m})$, showing that the ser(ie) category is equivalent to the category of $\Gamma$-graded vector spaces for $\Gamma=C_{2}\times\mathbb{Z}$, via a functor sending syzygies to graded simples. These results provide a complete description of the representation ring and its semisimplified version for this tame Hopf algebra, and clarify how indecomposable modules interact under tensor products in characteristic $2$. The work thus advances understanding of Green rings and semisimple quotients for restricted enveloping algebras in low characteristic.
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.