The localized skein algebra is Frobenius

Abdiel, Nel; Frohman, Charles

doi:10.2140/agt.2017.17.3341

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The localized skein algebra is Frobenius

Nel Abdiel and Charles Frohman

Algebraic & Geometric Topology 17 (2017) 3341–3373

DOI: 10.2140/agt.2017.17.3341

Abstract

When $A$ in the Kauffman bracket skein relation is set equal to a primitive $n^{th}$ root of unity $ζ$ with $n$ not divisible by $4$ , the Kauffman bracket skein algebra $K_{ζ} (F)$ of a finite-type surface $F$ is a ring extension of the ${SL}_{2} ℂ$ –character ring of the fundamental group of $F$ . We localize by inverting the nonzero characters to get an algebra $S^{- 1} K_{ζ} (F)$ over the function field of the corresponding character variety. We prove that if $F$ is noncompact, the algebra $S^{- 1} K_{ζ} (F)$ is a symmetric Frobenius algebra. Along the way we prove $K (F)$ is finitely generated, $K_{ζ} (F)$ 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

Mathematical Subject Classification 2010

Primary: 57M27

References

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

Volume 17, issue 6 (2017)