Hiroaki Karuo, Zhihao Wang
arXiv 2024年8月22日
In the paper, we show some properties of (reduced) stated 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 SL($n$)-skein algebras into quantum tori appearing in higher
Teichmüller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can
understand the representation theory of (reduced) stated 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.