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We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov–Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.
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Dimension and Trace of the Kauffman Bracket Skein Algebra
Let F F be a finite type surface and ζ \zeta a complex root of unity. The Kauffman bracket skein algebra K ζ ( F ) 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 ζ ( F ) 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 F .
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- Award ID(s):
- 1811114
- PAR ID:
- 10322656
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society, Series B
- Volume:
- 8
- Issue:
- 18
- ISSN:
- 2330-0000
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/btran/69
