Module categories over affine supergroup schemes (Q2031514)

Let \(k\) be an algebraically closed field of characteristic \(0\) or \(p>2\). Let \(G\) be a finite group scheme over \(k\), \(\mathrm{Coh}(G)\) be the finite tensor category of finite dimensional \({\mathcal O}(G)\)-modules over \(k\), and \(\mathrm{Rep}(G)\) be the finite tensor category of finite dimensional rational representations of \(G\) over \(k\). In the author's previous paper [Quantum Topol. 6, No. 1, 1--37 (2015; Zbl 1338.14018)], the indecomposable exact module categories over \(\mathrm{Rep}(G)\) are classified, which are generalizing the classification of \textit{P. Etingof} and \textit{V. Ostrik} [Mosc. Math. J. 4, No. 3, 627--654, 782--783 (2004; Zbl 1077.18005)] for constant groups \(G\). The goal of the paper under review is to extend the above results to the super case, and to classify finite dimensional triangular Hopf algebras with the Chevalley property over \(k\). Now, let \({\mathcal G}\) be an affine supergroup scheme over \(k\). The author classifies the indecomposable exact module categories over the tensor category \(\mathrm{sCohf} ({\mathcal G})\) of (coherent sheaves of) finite dimensional \({\mathcal O}({\mathcal G})\)- supermodules in terms of \((\mathcal{H}, \Psi)\)-equivariant coherent sheaves on \({\mathcal G}\). From it the classification of indecomposable geometrical module categories over \(\mathrm{sRep}({\mathcal G})\) is deduced. When \({\mathcal G}\) is finite, the result yields the classification of all indecomposable exact module categories over the finite tensor category \(\mathrm{sRep}({\mathcal G})\). In particular, a classification of twists for the supergroup algebra \(k{\mathcal G}\) of a finite supergroup scheme \({\mathcal G}\) is obtained, and then finite dimensional triangular Hopf algebras with the Chevalley property over \(k\) are classified.