Derived representations of quantum character varieties

TL;DR

This paper constructs new representations of quantum character variety algebras on cohomology spaces, linking quantum topology, surface mapping class groups, and Ext functors, with applications to knot invariants and skein algebra representations.

Contribution

It introduces a method to represent quantum moduli algebras on Ext spaces, generalizing previous projective representations of surface mapping class groups.

Findings

Constructed Ext space representations of quantum moduli algebras.

Recovered known projective representations from Lyubashenko theory.

Provided new topological invariants for knots and skein algebras.

Abstract

Quantum moduli algebras $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ were introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the context of quantization of character varieties of surfaces and exist for any quasitriangular Hopf algebra $H$. In this paper we construct representations of $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ on cohomology spaces $\mathrm{Ext}_H^m(X,M)$ for all $m \geq 0$, where $X$ is any $H$-module and $M$ is any $\mathcal{L}_{g,n}(H)$-module endowed with a compatible $H$-module structure. As a corollary and under suitable assumptions on $H$, we obtain projective representations of mapping class groups of surfaces on such Ext spaces. This recovers the projective representations constructed by Lentner-Mierach-Schweigert-Sommerh\"auser from Lyubashenko theory, when the category $\mathcal{C} = H\text{-}\mathrm{mod}$ is used in their construction. Other topological applications are matrix-valued invariants of knots in thickened surfaces and representations of skein algebras on Ext spaces.