Centers and representations of ${\rm SL}_n$ quantum Teichm\"uller spaces

TL;DR

This paper computes the center and classifies irreducible representations of the ${ m SL}_n$ quantum Teichmüller space algebra at roots of unity, linking them to the ${ m SL}_n$ character variety and showing independence from triangulation choices.

Contribution

It provides the first detailed computation of the center and irreducible representations of the ${ m SL}_n$ quantum Teichmüller space algebra at roots of unity, connecting algebraic and geometric structures.

Findings

Computed the center and its rank over the center at roots of unity.

Classified irreducible representations of the balanced Fock-Goncharov algebra.

Linked irreducible representations to points in the ${ m SL}_n$ character variety.

Abstract

In this paper, we compute the center of the balanced Fock-Goncharov algebra and determine its rank over the center when the quantum parameter is a root of unity. These results have potential applications to the study of the center and rank of the ${\rm SL}_n$-skein algebra. Building on this computation, we classify the irreducible representations of the balanced Fock-Goncharov algebra. Due to the Frobenius homomorphism, every irreducible representation of the (projected) ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$ determines a point in the ${\rm SL}_n$ character variety of $\mathfrak{S}$, known as the classical shadow of the representation. By pulling back the irreducible representations of the balanced Fock-Goncharov algebra via the quantum trace map, we show that there exists a ``large'' subset of the ${\rm SL}_n$ character variety such that, for any point in this subset, there exists an irreducible representation of the (projected) ${\rm SL}_n$-skein algebra whose classical shadow is this point. Finally, we prove that, under mild conditions, the representations of the ${\rm SL}_n$-skein algebra obtained in this way are independent of the choice of ideal triangulation.