The Localized Skein Algebra is Frobenius

TL;DR

This paper proves that the localized Kauffman bracket skein algebra of a noncompact surface is a symmetric Frobenius algebra, revealing its algebraic structure and finite generation properties at roots of unity.

Contribution

It establishes the Frobenius property of the localized skein algebra and demonstrates finite generation and module finiteness over the character variety.

Findings

The algebra $S^{-1}K_N(F)$ is a symmetric Frobenius algebra for noncompact surfaces.

$K_N(F)$ is finitely generated over the character algebra $ ext{chi}(F)$.

Simple closed curves generate a finite field extension inside the skein algebra.

Abstract

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-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 $S^{-1}K_N(F)$.