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Frobenius algebras derived from the Kauffman bracket skein algebra

Charles Frohman

Department of Mathematics, The University of Iowa, Iowa, USA

E-mail Address: charles-frohman@uiowa.edu

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and

Nel Abdiel

Department of Mathematics, Division of Mathematical Sciences, The University of Iowa, Iowa, USA

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https://doi.org/10.1142/S0218216516500164Cited by:5 (Source: Crossref)

Abstract

The Kauffman bracket skein algebra of a compact oriented surface when the variable $A$ $A$ in the Kauffman bracket is set equal to $e^{π i / N}$ $e^{π i / N}$ , where $N$ $N$ is an odd counting number, is a central extension of the ring of $S L_{2} ℂ$ -characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

Keywords:

AMSC: 57M27

References

1. N. Abdiel and C. Frohman, The localized Skein algebra is Frobenius, preprint (2015), arXiv:1501.02631 [math.GT]. Google Scholar
2. W. Bloomquist and C. Frohman, Multiplying in the Skein algebra of a punctured torus, Comment. Math. Helv. 78 (2003) 1–17. Crossref, Google Scholar
3. F. Bonahon and H. Wong, Representations of the Kauffman skein algebra I: Invariants and miraculous cancellations, preprint (2012), arXiv:1206.1638 [math.GT]. Google Scholar
4. D. Bullock, Rings of $S L_{2} (ℂ)$ -characters and the Kauffman bracket skein module, Comment. Math. Helv. 72(4) (1997) 521–542. Crossref, Web of Science, Google Scholar
5. D. Bullock, C. Frohman and J. Kania-Bartoszynska, Understanding the Kauffman bracket skein module, J. Knot Theory Ramifications 8(3) (1999) 265–277. Link, Web of Science, Google Scholar
6. D. Bullock and J. H. Przytycki, Multiplicative structure of Kauffman bracket Skein module quantizations, Proc. Amer. Math. Soc. 128(3) (1999) 923–931. Crossref, Web of Science, Google Scholar
7. C. Frohman and R. Gelca, Skein modules and the noncommutative torus, Trans. Amer. Math. Soc. 352(10) (2000) 4877–4888. Crossref, Web of Science, Google Scholar
8. C. Frohman and J. Kania-Bartoszynska, The Kauffman bracket skein module of $#_{2} S^{1} \times S^{2}$ at a root of unity, preprint. Google Scholar
9. J. Hoste and J. H. Przytycki, The $(2, \infty)$ -skein module of lens spaces; a generalization of the Jones polynomial, J. Knot Theory Ramifications 2(3) (1993) 321–333. Link, Google Scholar
10. J. Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society Student Texts, Vol. 59 (Cambridge University Press, Cambridge, 2004), xiv+240 pp. Google Scholar
11. P. J. McCarthy, Algebraic Extensions of Fields, Corrected reprint of the 2nd edn. (Dover Publications, New York, 1991), x+166 pp. Google Scholar
12. J. H. Przytycki and A. Sikora, On skein algebras and $S L_{2} (ℂ)$ -character varieties, Topology 39 (2000) 115–148. Crossref, Web of Science, Google Scholar
13. A. S. Sikora and B. W. Westbury, Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007) 439–478. Crossref, Web of Science, Google Scholar