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2017 The localized skein algebra is Frobenius
Nel Abdiel, Charles Frohman
Algebr. Geom. Topol. 17(6): 3341-3373 (2017). 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 Kζ(F) of a finite-type surface F is a ring extension of the SL2–character ring of the fundamental group of F. We localize by inverting the nonzero characters to get an algebra S1Kζ(F) over the function field of the corresponding character variety. We prove that if F is noncompact, the algebra S1Kζ(F) is a symmetric Frobenius algebra. Along the way we prove K(F) is finitely generated, Kζ(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.

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Nel Abdiel. Charles Frohman. "The localized skein algebra is Frobenius." Algebr. Geom. Topol. 17 (6) 3341 - 3373, 2017. https://doi.org/10.2140/agt.2017.17.3341

Information

Received: 11 January 2015; Revised: 11 May 2017; Accepted: 27 May 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06791650
MathSciNet: MR3709648
Digital Object Identifier: 10.2140/agt.2017.17.3341

Subjects:
Primary: 57M27

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 6 • 2017
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