Stated skein algebras and their representations

Abstract

This is a survey on stated skein algebras and their representations.

Mapping class group representations derived from stated skein algebras

“…The classification of representations of (reduced) stated skein algebras at roots of unity is closely related to the computation of the equivariant symplectic leaves as detailed in [Kor21a] and briefly reviewed in Section 4.2. In particular, Theorem 1.1 will follow from the fact that X (Σ * g ) contains a single equivariant symplectic leaf.…”

“…The affine variety X admits a well-understood geometric interpretation as a finite cover of the SL 2 (relative) character variety of Σ and it is easy to find points which are stable under the action of the mapping class group or the Torelli group. However computing the Azumaya locus is a quite difficult problem (see [GJSa,Kor21a] for recent developments) and there is no reason to believe, in general, that it contains such fixed points. For instance for the Witten-Reshetikhin-Turaev representations, the induced characters do not belong to the Azumaya locus (see [BW16b]) and it is a highly non trivial fact (which follows from TQFTs properties) that they are Mod(Σ)-stable.…”

Mapping Class Group Representations Derived from Stated Skein Algebras

“…In Section 4, we review the notions of Azumaya loci and Poisson orders which led to the proof of Theorem 1.1. Parts of the exposition of these three sections already appeared in the author's unpublished survey [45]. In Section 5, we construct the representations π : G → PGL(V (O)) and provide some tools to prove that a given mapping class does not belong to the kernel.…”

“…The classification of representations of (reduced) stated skein algebras at roots of unity is closely related to the computation of the equivariant symplectic leaves as detailed in [45] and briefly reviewed in Section 4.2. In particular, Theorem 1.1 will follow from the fact that X (Σ * g ) contains a single equivariant symplectic leaf.…”

“…The affine variety X admits a well-understood geometric interpretation as a finite cover of the SL 2 relative representation variety of Σ and it is easy to find points which are stable under the action of the mapping class group or the Torelli group. However computing the Azumaya locus is a quite difficult problem (see [34,45] for recent developments) and there is no reason to believe, in general, that it contains such fixed points. For instance for the Witten-Reshetikhin-Turaev representations, the induced characters do not belong to the Azumaya locus (see [12]) and it is a highly non trivial fact (which follows from TQFTs properties) that they are Mod(Σ)-stable.…”