Center of stated $\mathrm{SL}(n)$-skein algebras

TL;DR

This paper investigates the algebraic properties of stated SL(n)-skein algebras, focusing on their centers, finite generation, and PI-degrees, using quantum trace maps and embeddings into quantum tori, with implications for representation theory.

Contribution

It provides new insights into the structure and representation theory of stated SL(n)-skein algebras, including their centers and connections to quantum moduli and cluster algebras.

Findings

Centers of stated SL(n)-skein algebras are finitely generated.

The algebras embed into quantum tori related to higher Teichmüller theory.

Representation theory is accessible via the Unicity theorem.

Abstract

In the paper, we show some properties of (reduced) stated $\mathrm{SL}(n)$-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated $\mathrm{SL}(n)$-skein algebras into quantum tori appearing in higher Teichm\"uller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated $\mathrm{SL}(n)$-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially.