Hiroaki Karuo

<p>We compute the Azumaya loci of Kauffman-bracket skein algebras of closed surfaces at odd roots of unity and provide partial results for open surfaces as well. As applications, we give an alternative definition of the projective representations of the Torelli groups derived from non-semisimple TQFTs and we strengthen a result by Frohman-Kania Bartoszynska-Lê about the dimensions of some quotients of the skein modules of closed 3-manifolds.</p>

<p> In the paper, we show some properties of (reduced) stated -skein algebras related to their centers for essentially bordered punctured bordered 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 -skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem by Brown and Goodearl [ Lectures on algebraic quantum groups , Birkhäuser Verlag, Basel, 2002] and Frohman, Kania-Bartoszynska, and Lê [Invent. Math. 215 (2019), pp. 609–650], we can understand the representation theory of (reduced) stated -skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially. </p>

We consider a generalization of the Kauffman bracket skein algebra of a surface that is generated by loops and arcs between marked points on the interior or boundary, up to skein relations defined by Muller and Roger-Yang. We compute the center of this Muller-Roger-Yang skein algebra and show that it is almost Azumaya when the quantum parameter $q$ is a primitive $n$-th root of unity with odd $n$. We also discuss the implications on the representation theory of the Muller-Roger-Yang generalized skein algebra.