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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 SL2ℂ–character ring of the fundamental group of F. We localize by inverting the nonzero characters to get an algebra S−1Kζ(F) over the function field of the corresponding character variety. We prove that if F is noncompact, the algebra S−1Kζ(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.
Keywords: skein algebra, Frobenius
Abdiel, Nel 1 ; Frohman, Charles 1
@article{10_2140_agt_2017_17_3341,
author = {Abdiel, Nel and Frohman, Charles},
title = {The localized skein algebra is {Frobenius}},
journal = {Algebraic and Geometric Topology},
pages = {3341--3373},
publisher = {mathdoc},
volume = {17},
number = {6},
year = {2017},
doi = {10.2140/agt.2017.17.3341},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3341/}
}
TY - JOUR AU - Abdiel, Nel AU - Frohman, Charles TI - The localized skein algebra is Frobenius JO - Algebraic and Geometric Topology PY - 2017 SP - 3341 EP - 3373 VL - 17 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3341/ DO - 10.2140/agt.2017.17.3341 ID - 10_2140_agt_2017_17_3341 ER -
Abdiel, Nel; Frohman, Charles. The localized skein algebra is Frobenius. Algebraic and Geometric Topology, Tome 17 (2017) no. 6, pp. 3341-3373. doi: 10.2140/agt.2017.17.3341
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