Unicity for Representations of the Kauffman bracket Skein Algebra

TL;DR

This paper proves the uniqueness of irreducible representations of Kauffman bracket skein algebras for all oriented finite type surfaces at any root of unity, using a general classification theorem based on the algebra's center.

Contribution

It establishes the unicity conjecture for all oriented finite type surfaces at all roots of unity, characterizing the algebra's center and its module structure.

Findings

Center of skein algebra is the coordinate ring of an affine algebraic variety.

Skein algebra is finitely generated as a module over its center.

Irreducible representations are classified by their central characters.

Abstract

This paper resolves the unicity conjecture of Bonahon and Wong for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine $k$-algebra over an algebraically closed field $k$, that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebra is finitely generated as a module over its center. It is shown that for any orientable surface the center of the skein algebra at any root of unity is the coordinate ring of an affine algebraic variety.