Character varieties as a tensor product

TL;DR

This paper presents a novel tensor product perspective on representation and character varieties, enabling new insights into their structure, deformations, and applications in quantum topology.

Contribution

It introduces a tensor product framework for representation and character varieties using PROP of cocommutative Hopf algebras, with applications to deformation theory and quantum representations.

Findings

Provides a new tensor product viewpoint for character varieties.

Derives a functor leading to representation homology.

Explores deformations related to 3-manifold groups and quantum topology.

Abstract

In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting applications. First, it gives a straightforward way of deriving the functor sending a discrete group to the functions on its representation variety, which leads to representation homology. Second, using a suitable deformation of the functors involved in this construction, one can obtain deformations of the representation and character varieties for the fundamental groups of 3-manifolds, and could lead to better understanding of quantum representations of mapping class groups.