The Localized Skein Algebra is Frobenius

Abstract

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\ge 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $S{L}_2\mathbb{C}$-characters $\chi (F)$ of the fundamental group of $F$. We localize by inverting the nonzero characters to get an algebra $S^{-1}{K}_N(F)$ over the function field of the character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}{K}_N(F)$ is a symmetric Frobenius algebra. Along the way we prove $K(F)$ is finitely generated, $K_N(F)$ is a finite rank module over $\chi (F)$, and the simple closed curves that make up any simple diagram on $F$ generate a finite field extension of $S^{-1}\chi (F)$ inside $S^{-1}{K}_N(F)$.

29 pages, 10 figures

References in corpus (3)

• A positive basis for surface skein algebras
• On Kauffman Bracket Skein Modules at Root of Unity
• Frobenius Algebras Derived from the Kauffman Bracket Skein Algebra

Cited by in corpus (3)

• Finite presentations for stated skein algebras and lattice gauge field theory
• Algebraic generators of the skein algebra of a surface
• Center of generalized skein algebras