Journal article
Degenerations of skein algebras and quantum traces
Transactions of the American Mathematical Society, Vol.378(9), pp.6049-6108
07/08/2025
DOI: 10.1090/tran/9234
Abstract
We introduce a joint generalization, called LRY skein algebras, of Kauffman bracket skein algebras (of surfaces) that encompasses both Roger-Yang skein algebras and stated skein algebras. We will show that, over an arbitrary ground ring which is a commutative domain, the LRY skein algebras are domains and have degenerations (by filtrations) equal to monomial subalgebras of quantum tori. For surfaces without interior punctures, this integrality generalizes a result of Moon and Wong [Consequences of the compatibility of skein algebra and cluster algebra on surfaces, arXiv:2201.08833, 2022] to the most general ground ring. We also calculate the Gelfand-Kirillov dimension of LRY algebras and show they are Noetherian if the ground ring is. Moreover they are orderly finitely generated. To study the LRY algebras and prove the above-mentioned results, we construct quantum traces, both the so-called XX-version for all surfaces and also an AA-version for a smaller class of surfaces. We also introduce a modified version of Dehn-Thurston coordinates for curves which are more suitable for the study of skein algebras as they pick up the highest degree terms of products in certain natural filtrations.
Details
- Title: Subtitle
- Degenerations of skein algebras and quantum traces
- Creators
- Wade Bloomquist - Department of Mathematics, University of Iowa, IowaHiroaki Karuo - Gakushuin UniversityThang Lê - Georgia Institute of Technology
- Resource Type
- Journal article
- Publication Details
- Transactions of the American Mathematical Society, Vol.378(9), pp.6049-6108
- DOI
- 10.1090/tran/9234
- ISSN
- 0002-9947
- eISSN
- 1088-6850
- Publisher
- American Mathematical Society
- Number of pages
- 60
- Language
- English
- Date published
- 07/08/2025
- Academic Unit
- Mathematics
- Record Identifier
- 9984944720302771
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