Representation theory of non-factorizable ribbon Hopf algebras

TL;DR

This paper investigates the representation theory of new non-factorizable ribbon Hopf algebras, classifying modules, analyzing their structure, and exploring implications for topological quantum field theories, revealing complex properties like non-semisimple M"uger centers.

Contribution

It provides a detailed classification of modules and fusion rules for these new ribbon Hopf algebras, extending understanding of their representation categories and properties.

Findings

Classification of indecomposable projective and simple modules

Krull-Schmidt decomposition of the adjoint representation

Non-semisimple M"uger centers in their categories

Abstract

In arXiv:2503.19532 new examples of ribbon Hopf algebras based on the construction due to Nenciu were presented. This piece serves as a sequel where we study the representation theory of these new examples of ribbon Hopf algebras. We classify indecomposable projective and simple modules, find the Krull-Schmidt decomposition of the adjoint representation of Nenciu algebras, and prove fusion rules between its components. We also comment on the properties of M\"uger centres of their representation categories, in particular that they can be non-semisimple. Finally, we consider a new family of ribbon Hopf algebras over fields of prime characteristic $p>2$ in the context of 4-dimensional TQFTs presented in arXiv:2306.03225 that constitute an improvement over examples given therein, although still seemingly falling short of producing powerful invariants of 4-manifolds.