2-representation theory of the Taft algebra

Abstract

We study the $2$-representation theory of the Taft category, the tensor category of finite-dimensional modules over the Taft algebra. This category has two left cells. We first classify suitable transitive modules over its based representation ring. For the upper cell, we recall a construction by Etingof and Ostrik giving an infinite family of pairwise nonequivalent simple $2$-representations. For the bottom cell, we pass to the semisimplification of the Taft category and identify a distinguished fusion subcategory with an adjoint $\mathfrak{sl}_2$ Verlinde category. Using algebra objects associated with $\mathsf{ADET}$-diagrams and categorical Clifford theory, we classify the regular exact indecomposable module categories of this semisimplification. For these regular module categories, the split Grothendieck group is a complete invariant. Finally, we identify the two cell $2$-representations of the Taft category.