Degenerations Of Skein Algebras And Quantum Traces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916292343103488 |
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| author | Bloomquist, Wade Karuo, Hiroaki Lê, Thang |
| author_facet | Bloomquist, Wade Karuo, Hiroaki Lê, Thang |
| contents | 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 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 X-version for all surfaces and also an A-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_16702 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Degenerations Of Skein Algebras And Quantum Traces Bloomquist, Wade Karuo, Hiroaki Lê, Thang Geometric Topology Quantum Algebra 57K31 (Primary), 57K20, 16W70 (Secondary) 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 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 X-version for all surfaces and also an A-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. |
| title | Degenerations Of Skein Algebras And Quantum Traces |
| topic | Geometric Topology Quantum Algebra 57K31 (Primary), 57K20, 16W70 (Secondary) |
| url | https://arxiv.org/abs/2308.16702 |