Exact module categories over $\mathrm{Rep}(u_q(\mathfrak{sl}_2))$

TL;DR

The paper provides a complete explicit classification of indecomposable exact module categories over ${\mathrm{Rep}}(u_q(\mathfrak{sl}_2))$ for odd order $q$ by realizing each as ${}_{\sigma}A$ for a right $u_q$-simple left $u_q$-comodule algebra with trivial coinvariants. Central to the approach are Morita duality between ${}_H\mathfrak{M}$ and ${}^H\mathfrak{M}$, the equivariant Eilenberg-Watts theorem, and a systematic cocycle deformation from the graded model ${\mathrm{gr}}(u_q)$ to $u_q$, enabling an explicit lift-and-deform construction. The authors first classify graded left coideal subalgebras of ${\mathrm{gr}}(u_q)$, then lift them, determine Morita equivalence classes, and finally compute the corresponding cocycle deformations, producing a complete list of left $u_q$-comodule algebras whose representation categories yield indecomposable exact module categories. The result furnishes concrete, computable models ${A}$ with explicit generators, relations, and coactions, advancing understanding of module categories in non-semisimple finite tensor categories and offering a foundation for further study of Serre functors, simple objects, and autoequivalences in this setting.

Abstract

We give a complete list of indecomposable exact module categories over the finite tensor category $\mathrm{Rep}(u_q(\mathfrak{sl}_2))$ of representations of the small quantum group $u_q(\mathfrak{sl}_2)$, where $q$ is a root of unity of odd order. Each of them is given as the category of representations of a left comodule algebra over $u_q(\mathfrak{sl}_2)$ explicitly presented by generators and relations.

Theorems & Definitions (131)

• Definition 2.1
• Definition 2.2
• Definition 2.3: 2014arXiv1406.4204D
• Lemma 2.4: the Morita theorem 2014arXiv1406.4204D
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Definition 2.7
• Lemma 2.8