#### Volume 17, issue 6 (2017)

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The localized skein algebra is Frobenius

### Nel Abdiel and Charles Frohman

Algebraic & Geometric Topology 17 (2017) 3341–3373
##### Abstract

When $A$ in the Kauffman bracket skein relation is set equal to a primitive root of unity $\zeta$ with $n$ not divisible by $4$, the Kauffman bracket skein algebra ${K}_{\zeta }\left(F\right)$ of a finite-type surface $F$ is a ring extension of the ${SL}_{2}ℂ$–character ring of the fundamental group of $F\phantom{\rule{0.3em}{0ex}}$. We localize by inverting the nonzero characters to get an algebra ${S}^{-1}{K}_{\zeta }\left(F\right)$ over the function field of the corresponding character variety. We prove that if $F$ is noncompact, the algebra ${S}^{-1}{K}_{\zeta }\left(F\right)$ is a symmetric Frobenius algebra. Along the way we prove $K\left(F\right)$ is finitely generated, ${K}_{\zeta }\left(F\right)$ is a finite-rank module over the coordinate ring of the corresponding character variety, and learn to compute the trace that makes the algebra Frobenius.

##### Keywords
skein algebra, Frobenius
Primary: 57M27
##### Publication
Received: 11 January 2015
Revised: 11 May 2017
Accepted: 27 May 2017
Published: 4 October 2017
##### Authors
 Nel Abdiel Department of Mathematics University of Iowa Iowa City, IA United States Charles Frohman Department of Mathematics University of Iowa Iowa City, IA United States