Quantum traces and embeddings of stated skein algebras into quantum tori
type: Item
instance of: scholarly article
title: Quantum traces and embeddings of stated skein algebras into quantum tori
zbMATH Open document ID: 1503.57015
author: Tao Yu
author: Thang T. Q. Le
published in: Selecta Mathematica. New Series
publication date: 2022-06-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/2012.15272
review text: Skein modules were introduced by Przytycki and independently by Turaev to describe knot and link invariants of the Kauffman bracket kind on general \(3D\) manifolds. The case of \(3D\) manifolds which are thickened surfaces is particularly fruitful, as it developed into almost a separate domain of mathematics, with connections to Teichmüller spaces, classical and quantum, cluster algebras, noncommutative geometry, quantum groups, and others. It is known that there exist quantum trace maps that embed different realizations of the stated skein algebras in certain quantum tori that are defined using various coordinates in certain generalizations of Teichmüller spaces, and introduced by Bonahon and Wong and by one of the authors. The paper focuses on one hand on generalizing the results to possibly most general \(2D\) surfaces (thickened), with boundaries and punctures, and on the other on finding relationships between the different quantum maps and the embeddings, and there are numerous important results proved in the paper that clarify these relationships.
reviewed by: Robert Owczarek
Mathematics Subject Classification ID: 57K20
Mathematics Subject Classification ID: 57K10
zbMATH DE Number: 7542651
zbMATH Keywords: skein algebras
zbMATH Keywords: quantum trace
zbMATH Keywords: quantum tori
zbMATH Keywords: Teichmüller spaces
MaRDI profile type: Publication
OpenAlex ID: W3114145440
arXiv ID: 2012.15272
cites work: The localized skein algebra is Frobenius
cites work: Combinatorial quantization of the Hamiltonian Chern-Simons theory. I
cites work: A uniqueness property for the quantization of Teichmüller spaces
cites work: Integrating quantum groups over surfaces
cites work: Quantum cluster algebras.
cites work: Three-manifold invariants derived from the Kauffman bracket
cites work: The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds
cites work: Quantum traces for representations of surface groups in SL\(_{2}(\mathbb C)\)
cites work: Poisson orders, symplectic reflection algebras and representation theory
cites work: Two dimensional lattice gauge theory based on a quantum group
cites work: Rings of \(SL_2(\mathbb{C})\)-characters and the Kauffman bracket skein module
cites work: UNDERSTANDING THE KAUFFMAN BRACKET SKEIN MODULE
cites work: A quantum Teichmüller space
cites work: Multicurves and regular functions on the representation variety of a surface in \(SU(2)\)
cites work: Q4524254
cites work: Modular group representations in combinatorial quantization with non-semisimple Hopf algebras
cites work: Moduli spaces of local systems and higher Teichmüller theory
cites work: Q3592894
cites work: The quantum dilogarithm and representations of quantum cluster varieties
cites work: Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths
cites work: The A-polynomial from the noncommutative viewpoint
cites work: Cluster algebras and Weil-Petersson forms
cites work: Q3154382
cites work: Quantum traces in quantum Teichmüller theory
cites work: State models and the Jones polynomial
cites work: Global crystal bases of quantum groups
cites work: The hyperbolic volume of knots from the quantum dilogarithm
cites work: Quantization of Teichmüller spaces and the quantum dilogarithm
cites work: Unicity for representations of reduced stated skein algebras
cites work: QUANTUM TEICHMÜLLER SPACES AND QUANTUM TRACE MAP
cites work: Triangular decomposition of skein algebras
cites work: On the AJ conjecture for knots
cites work: Stated skein modules of marked 3-manifolds/surfaces, a survey
cites work: THE QUANTUM TEICHMÜLLER SPACE AS A NONCOMMUTATIVE ALGEBRAIC OBJECT
cites work: On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism
cites work: Algorithmic topology and classification of 3-manifolds
cites work: Q3786586
cites work: Skein and cluster algebras of marked surfaces
cites work: The colored Jones polynomials and the simplicial volume of a knot
cites work: Q4934978
cites work: On skein algebras and \(Sl_2(\mathbb{C})\)-character varieties
cites work: Skein algebras of surfaces
cites work: Gelfand-Kirillov dimension of skew PBW extensions.
cites work: A cluster realization of \(U_q(\mathfrak {sl}_{\mathfrak {n}})\) from quantum character varieties
cites work: Skein quantization of Poisson algebras of loops on surfaces
cites work: Conway and Kauffman modules of a solid torus
cites work: Quantum invariants of knots and 3-manifolds
DOI: 10.1007/S00029-022-00781-3
description: scientific article