Representation theory, topological field theory, and the Andrews-Curtis conjecture

TL;DR

This paper explores a representation-theoretic approach involving Hopf algebras and topological quantum field theories to investigate the longstanding Andrews-Curtis conjecture, aiming to identify potential counterexamples.

Contribution

It introduces a novel connection between representation theory, topological quantum field theory, and the Andrews-Curtis conjecture, proposing a new method to detect counterexamples.

Findings

Proposes a specific representation-theoretic framework for the conjecture

Constructs a topological quantum field theory based on this framework

Suggests this approach could identify counterexamples to the conjecture

Abstract

We pose a representation-theoretic question motivated by an attempt to resolve the Andrews-Curtis conjecture. Roughly, is there a triangular Hopf algebra with a collection of self-dual irreducible representations $V_i$ so that the product of any two decomposes as a sum of copies of the $V_i$, and $\sum (\rank V_i)^2=0$? This data can be used to construct a `topological quantum field theory' on 2-complexes which stands a good chance of detecting counterexamples to the conjecture.