[Paper Review] Module traces and Hopf group-coalgebras

Andres F. Fontalvo Orozco, Azat M. Gainutdinov|arXiv (Cornell University)|Sep 4, 2018

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper introduces a generalized notion of module traces for pivotal categories equipped with module endofunctors, extending the concept of modified traces beyond unimodular Hopf algebras. It proves the existence of a non-degenerate module trace on the projective tensor ideal for any finite-dimensional pivotal Hopf algebra over a field, including non-unimodular ones, and generalizes this to Hopf group-coalgebras, with explicit computations for Taft algebras and non-restricted quantum groups at roots of unity.

ABSTRACT

Let H be a finite-dimensional pivotal and unimodular Hopf algebra over a field k. It was shown in [BBGa] that the projective tensor ideal in H-mod admits a unique non-degenerate modified trace, a natural generalisation of the categorical trace. This paper provides an extension of this result to a much more general setting. We first extend the notion of the modified trace to the so-called module trace for a given k-linear module category M over a pivotal category C equipped with a module endofunctor. We provide a non-trivial class of examples of such module traces. In particular, we show that any finite-dimensional pivotal Hopf k-algebra, not necessarily unimodular, admits a non-degenerate module trace on its projective tensor ideal. We also extend this result to pivotal Hopf group-coalgebras of finite type, and we give explicit calculations for the family of Taft Hopf algebras and non-restricted Borel quantum groups at roots of unity.

Motivation & Objective

• To generalize the notion of modified traces beyond semisimple and unimodular settings.
• To define and study module traces in the context of module categories over pivotal categories with module endofunctors.
• To establish the existence of non-degenerate module traces on the projective tensor ideal for any finite-dimensional pivotal Hopf algebra.
• To extend the theory to Hopf group-coalgebras of finite type and compute explicit examples.
• To provide a framework for constructing topological invariants in non-semisimple settings using these generalized traces.

Proposed method

• Introduce the concept of a module trace on a module category over a pivotal category with a module endofunctor.
• Define the pull-back of module traces via module functors to relate different module category structures.
• Use a reduction lemma to simplify the trace condition on tensor ideals.
• Construct the module trace using a generalized integral and the action of the module endofunctor.
• Apply the theory to Hopf group-coalgebras by introducing a G-integral and α-symmetrized G-integral.
• Verify the trace axioms (cyclicity and partial trace) using graphical calculus and algebraic identities in the Hopf algebra setting.

Experimental results

Research questions

• RQ1 Can the modified trace construction be generalized beyond unimodular and semisimple categories?
• RQ2 Does a non-degenerate module trace exist on the projective tensor ideal of any finite-dimensional pivotal Hopf algebra, even if unimodular?
• RQ3 How can the module trace concept be extended to Hopf group-coalgebras of finite type?
• RQ4 What explicit formulas can be derived for the module trace in concrete examples like Taft algebras and non-restricted quantum groups at roots of unity?
• RQ5 How does the module trace interact with duality and tensor structures in non-semisimple settings?

Key findings

• A non-degenerate module trace exists on the projective tensor ideal of any finite-dimensional pivotal Hopf algebra over a field, regardless of unimodularity.
• The construction generalizes the modified trace from [BBGa] to a broader class of algebras, including non-unimodular ones.
• For the family of Taft Hopf algebras, the module trace is explicitly computed and shown to be non-degenerate.
• In the case of non-restricted Borel quantum groups at roots of unity, the module trace is constructed via the α-symmetrized G-integral and yields a non-degenerate pairing.
• The trace satisfies cyclicity and partial trace axioms, ensuring compatibility with topological invariants in non-semisimple TQFTs.
• The module trace on H-pmod is determined by the action of the module endofunctor and the integral, with explicit evaluation given by t_V_s(R_z,x^s) = z^{1+n}/r.