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Representation theory of non-factorizable ribbon Hopf algebras

Maksymilian Manko

TL;DR

The paper analyzes the representation theory of new non-factorizable ribbon Hopf algebras constructed via Nenciu-type methods, focusing on indecomposable projectives, simple modules, and the adjoint action. It provides explicit idempotent systems, decomposes the regular and adjoint representations, and derives fusion rules and braiding properties for key families of modules, showing that Müger centres can be non-semisimple. It extends these analyses to semidirect biproducts with $u_q\mathfrak{sl}_2$ and symplectic-fermion-type algebras, highlighting how the constituent data control category-theoretic features such as transparency and central structure. The work then connects these algebraic constructions to 4d TQFT contexts, distinguishing cases that are chromatic non-degenerate but not chromatic compact, and finally presents a finite-characteristic construction yielding a new example with non-semisimple Müger centres and chromatic non-degeneracy, while discussing limitations due to stabilization phenomena. Overall, the results broaden the catalog of concrete, non-semisimple ribbon categories and provide concrete tools for investigating 4-manifold invariants within non-factorizable, unimodular frameworks.

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 Müger 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.

Representation theory of non-factorizable ribbon Hopf algebras

TL;DR

The paper analyzes the representation theory of new non-factorizable ribbon Hopf algebras constructed via Nenciu-type methods, focusing on indecomposable projectives, simple modules, and the adjoint action. It provides explicit idempotent systems, decomposes the regular and adjoint representations, and derives fusion rules and braiding properties for key families of modules, showing that Müger centres can be non-semisimple. It extends these analyses to semidirect biproducts with and symplectic-fermion-type algebras, highlighting how the constituent data control category-theoretic features such as transparency and central structure. The work then connects these algebraic constructions to 4d TQFT contexts, distinguishing cases that are chromatic non-degenerate but not chromatic compact, and finally presents a finite-characteristic construction yielding a new example with non-semisimple Müger centres and chromatic non-degeneracy, while discussing limitations due to stabilization phenomena. Overall, the results broaden the catalog of concrete, non-semisimple ribbon categories and provide concrete tools for investigating 4-manifold invariants within non-factorizable, unimodular frameworks.

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 Müger 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 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.

Paper Structure

This paper contains 31 sections, 56 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Preliminaries
  5. Notations for Hopf algebras
  6. Integrals and unimodularity
  7. Quasitriangular and ribbon structures
  8. Algebra generators and tuple notation
  9. The semi-direct biproduct structure
  10. Basic representation theory
  11. The Nenciu construction
  12. Hopf algebra structure
  13. Quasitriangular structure
  14. Examples
  15. Non-factorizable biproducts of $u_q \mathfrak{sl}_2$
  16. ...and 16 more sections

Key Result

Theorem 2.9

Let $H$ be a Hopf algebra. Any element $R \in H \otimes H$ satisfying (QT1)-(QT4) induces a Hopf algebra map where $H^{cop}$ is the coopposite Hopf algebra of $H$, defined above.

Theorems & Definitions (174)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: radford_2012 Definition 10.2.3
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: radford_1994, Section 2.1
  • Proposition 2.10: shimizu_2016 Theorem 1.1, etingof_gelaki_nikshych_ostrik_2016 Exercise 8.6.4
  • ...and 164 more