Dimension and Trace of the Kauffman Bracket Skein Algebra
HTML articles powered by AMS MathViewer
- by Charles Frohman, Joanna Kania-Bartoszynska and Thang Lê;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 510-547
- DOI: https://doi.org/10.1090/btran/69
- Published electronically: July 7, 2021
- HTML | PDF
Abstract:
Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The Kauffman bracket skein algebra $K_\zeta (F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_\zeta (F)$ over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of $F$.References
- Charles Frohman and Nel Abdiel, Frobenius algebras derived from the Kauffman bracket skein algebra, J. Knot Theory Ramifications 25 (2016), no. 4, 1650016, 25. MR 3482494, DOI 10.1142/S0218216516500164
- Nel Abdiel and Charles Frohman, The localized skein algebra is Frobenius, Algebr. Geom. Topol. 17 (2017), no. 6, 3341–3373. MR 3709648, DOI 10.2140/agt.2017.17.3341
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 242802
- Francis Bonahon and Helen Wong, Quantum traces for representations of surface groups in $\textrm {SL}_2(\Bbb C)$, Geom. Topol. 15 (2011), no. 3, 1569–1615. MR 2851072, DOI 10.2140/gt.2011.15.1569
- Francis Bonahon and Helen Wong, Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations, Invent. Math. 204 (2016), no. 1, 195–243. MR 3480556, DOI 10.1007/s00222-015-0611-y
- Francis Bonahon and Helen Wong, Representations of the Kauffman bracket skein algebra II: Punctured surfaces, Algebr. Geom. Topol. 17 (2017), no. 6, 3399–3434. MR 3709650, DOI 10.2140/agt.2017.17.3399
- Laurent Charles and Julien Marché, Multicurves and regular functions on the representation variety of a surface in SU(2), Comment. Math. Helv. 87 (2012), no. 2, 409–431. MR 2914854, DOI 10.4171/CMH/258
- P. M. Cohn, Algebra. Vol. 3, 2nd ed., John Wiley & Sons, Ltd., Chichester, 1991. MR 1098018
- Albert Fathi, François Laudenbach, and Valentin Poénaru, Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR 3053012, DOI 10.1515/9781400839032
- Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
- Charles Frohman and Răzvan Gelca, Skein modules and the noncommutative torus, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4877–4888. MR 1675190, DOI 10.1090/S0002-9947-00-02512-5
- Charles Frohman and Joanna Kania-Bartoszynska, The structure of the Kauffman bracket skein algebra at roots of unity, Math. Z. 289 (2018), no. 3-4, 889–920. MR 3830231, DOI 10.1007/s00209-017-1980-2
- Charles Frohman, Joanna Kania-Bartoszynska, and Thang Lê, Unicity for representations of the Kauffman bracket skein algebra, Invent. Math. 215 (2019), no. 2, 609–650. MR 3910071, DOI 10.1007/s00222-018-0833-x
- Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787, DOI 10.1090/mmono/115
- T. Y. Lam, A first course in noncommutative rings, 2nd ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. MR 1838439, DOI 10.1007/978-1-4419-8616-0
- Thang T. Q. Lê, On Kauffman bracket skein modules at roots of unity, Algebr. Geom. Topol. 15 (2015), no. 2, 1093–1117. MR 3342686, DOI 10.2140/agt.2015.15.1093
- Thang T. Q. Lê, On positivity of Kauffman bracket skein algebras of surfaces, Int. Math. Res. Not. IMRN 5 (2018), 1314–1328. MR 3801463, DOI 10.1093/imrn/rnw280
- Thang T. Q. Lê, Triangular decomposition of skein algebras, Quantum Topol. 9 (2018), no. 3, 591–632. MR 3827810, DOI 10.4171/QT/115
- Thang T. Q. Lê and Jonathan Paprocki, On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism, Algebr. Geom. Topol. 19 (2019), no. 7, 3453–3509. MR 4045358, DOI 10.2140/agt.2019.19.3453
- Feng Luo, Some applications of a multiplicative structure on simple loops in surfaces, Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman (New York, 1998) AMS/IP Stud. Adv. Math., vol. 24, Amer. Math. Soc., Providence, RI, 2001, pp. 123–129. MR 1873113, DOI 10.1090/amsip/024/10
- Feng Luo, Simple loops on surfaces and their intersection numbers, J. Differential Geom. 85 (2010), no. 1, 73–115. MR 2719409
- Feng Luo and Richard Stong, Dehn-Thurston coordinates for curves on surfaces, Comm. Anal. Geom. 12 (2004), no. 1-2, 1–41. MR 2074869
- Paul J. McCarthy, Algebraic extensions of fields, 2nd ed., Dover Publications, Inc., New York, 1991. MR 1105534
- Greg Muller, Skein and cluster algebras of marked surfaces, Quantum Topol. 7 (2016), no. 3, 435–503. MR 3551171, DOI 10.4171/QT/79
- Robert C. Penner, The action of the mapping class group on curves in surfaces, Enseign. Math. (2) 30 (1984), no. 1-2, 39–55. MR 743669
- Józef H. Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999), no. 1, 45–66. MR 1723531
- Józef H. Przytycki and Adam S. Sikora, On skein algebras and $\textrm {Sl}_2(\textbf {C})$-character varieties, Topology 39 (2000), no. 1, 115–148. MR 1710996, DOI 10.1016/S0040-9383(98)00062-7
- Józef H. Przytycki and Adam S. Sikora, Skein algebras of surfaces, Trans. Amer. Math. Soc. 371 (2019), no. 2, 1309–1332. MR 3885180, DOI 10.1090/tran/7298
- Adam S. Sikora and Bruce W. Westbury, Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007), 439–478. MR 2308953, DOI 10.2140/agt.2007.7.439
- Dylan Paul Thurston, Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9725–9732. MR 3263305, DOI 10.1073/pnas.1313070111
- Dylan Paul Thurston, Geometric intersection of curves on surfaces, preprint, available at http://pages.iu.edu/~dpthurst/DehnCoordinates.pdf.
- V. G. Turaev, The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 79–89, 190 (Russian, with English summary); English transl., J. Soviet Math. 52 (1990), no. 1, 2799–2805. MR 964255, DOI 10.1007/BF01099241
- Vladimir G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 635–704. MR 1142906, DOI 10.24033/asens.1639
Bibliographic Information
- Charles Frohman
- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa
- MR Author ID: 234056
- ORCID: 0000-0003-0202-5351
- Email: charles-frohman@uiowa.edu
- Joanna Kania-Bartoszynska
- Affiliation: Division of Mathematical Sciences, The National Science Foundation, Alexandria, Virginia
- MR Author ID: 239347
- Email: jkaniaba@nsf.gov
- Thang Lê
- Affiliation: Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia
- ORCID: 0000-0003-4551-0285
- Email: letu@math.gatech.edu
- Received by editor(s): February 5, 2019
- Received by editor(s) in revised form: January 15, 2021
- Published electronically: July 7, 2021
- Additional Notes: This material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
- © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 510-547
- MSC (2020): Primary 57K31
- DOI: https://doi.org/10.1090/btran/69
- MathSciNet review: 4282692