Journal article
The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds
Mathematische Zeitschrift, Vol.301(1), pp.1063-1105
05/01/2022
DOI: 10.1007/s00209-021-02904-6
Abstract
We study the stated skein modules of marked 3-manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of 3-manifolds. We show that there exists a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we showthat the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev-Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of O q2 (SL(2)) through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig's Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev-Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.
Details
- Title: Subtitle
- The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds
- Creators
- Wade Bloomquist - Georgia Institute of TechnologyThang T. Q. Le - Georgia Institute of Technology
- Resource Type
- Journal article
- Publication Details
- Mathematische Zeitschrift, Vol.301(1), pp.1063-1105
- DOI
- 10.1007/s00209-021-02904-6
- ISSN
- 0025-5874
- eISSN
- 1432-1823
- Publisher
- Springer Nature
- Number of pages
- 43
- Grant note
- DMS-1745583; DMS 1811114 / NSF; National Science Foundation (NSF)
- Language
- English
- Date published
- 05/01/2022
- Academic Unit
- Mathematics
- Record Identifier
- 9984936508002771
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