Algebraic & Geometric Topology Volume 23, issue 3 (2023)

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ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

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Finite presentations for stated skein algebras and lattice gauge field theory

Julien Korinman

Algebraic & Geometric Topology 23 (2023) 1249–1302

DOI: 10.2140/agt.2023.23.1249

Bibliography

1	N Abdiel, C Frohman, The localized skein algebra is Frobenius, Algebr. Geom. Topol. 17 (2017) 3341 MR3709648
2	A Y Alekseev, H Grosse, V Schomerus, Combinatorial quantization of the Hamiltonian Chern–Simons theory, I, Comm. Math. Phys. 172 (1995) 317 MR1350411
3	A Y Alekseev, H Grosse, V Schomerus, Combinatorial quantization of the Hamiltonian Chern–Simons theory, II, Comm. Math. Phys. 174 (1996) 561 MR1370081
4	A Y Alekseev, A Z Malkin, Symplectic structure of the moduli space of flat connection on a Riemann surface, Comm. Math. Phys. 169 (1995) 99 MR1328263
5	A Y Alekseev, V Schomerus, Representation theory of Chern–Simons observables, Duke Math. J. 85 (1996) 447 MR1417624
6	M Audin, Lectures on gauge theory and integrable systems, from: "Gauge theory and symplectic geometry" (editors J Hurtubise, F Lalonde, G Sabidussi), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 488, Kluwer (1997) 1 MR1461568
7	J W Barrett, Skein spaces and spin structures, Math. Proc. Cambridge Philos. Soc. 126 (1999) 267 MR1670233
8	S Baseilhac, P Roche, Unrestricted quantum moduli algebras, I : The case of punctured spheres, SIGMA Symmetry Integrability Geom. Methods Appl. 18 (2022) 025 MR4400433
9	D Ben-Zvi, A Brochier, D Jordan, Integrating quantum groups over surfaces, J. Topol. 11 (2018) 874 MR3847209
10	G M Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978) 178 MR506890
11	W Bloomquist, T T Q Lê, The Chebyshev–Frobenius homomorphism for stated skein modules of 3–manifolds, Math. Z. 301 (2022) 1063 MR4405678
12	F Bonahon, H Wong, Quantum traces for representations of surface groups in SL₂(ℂ), Geom. Topol. 15 (2011) 1569 MR2851072
13	F Bonahon, H Wong, Representations of the Kauffman bracket skein algebra, I : Invariants and miraculous cancellations, Invent. Math. 204 (2016) 195 MR3480556
14	K A Brown, K R Goodearl, Lectures on algebraic quantum groups, Birkhäuser (2002) MR1898492
15	E Buffenoir, P Roche, Two-dimensional lattice gauge theory based on a quantum group, Comm. Math. Phys. 170 (1995) 669 MR1337137
16	E Buffenoir, P Roche, Link invariants and combinatorial quantization of Hamiltonian Chern–Simons theory, Comm. Math. Phys. 181 (1996) 331 MR1414836
17	D Bullock, Rings of SL₂(ℂ)–characters and the Kauffman bracket skein module, Comment. Math. Helv. 72 (1997) 521 MR1600138
18	D Bullock, A finite set of generators for the Kauffman bracket skein algebra, Math. Z. 231 (1999) 91 MR1696758
19	D Bullock, C Frohman, J Kania-Bartoszyńska, Topological interpretations of lattice gauge field theory, Comm. Math. Phys. 198 (1998) 47 MR1657365
20	D Bullock, J Kania-Bartoszyńska, C Frohman, Skein quantization and lattice gauge field theory, from: "Knot theory and its applications" (editor C Adams), Elsevier (1998) 811 MR1628758
21	D Bullock, J H Przytycki, Multiplicative structure of Kauffman bracket skein module quantizations, Proc. Amer. Math. Soc. 128 (2000) 923 MR1625701
22	V Chari, A Pressley, A guide to quantum groups, Cambridge Univ. Press (1995) MR1358358
23	L Charles, J Marché, Multicurves and regular functions on the representation variety of a surface in SU(2), Comment. Math. Helv. 87 (2012) 409 MR2914854
24	J Cooke, Excision of skein categories and factorisation homology, Adv. Math. 414 (2023) 108848 MR4536120
25	F Costantino, J Korinman, T T Q Lê, Stated skein algebras for Tannakian ribbon categories, in preparation
26	F Costantino, T T Q Lê, Stated skein algebras of surfaces, J. Eur. Math. Soc. 24 (2022) 4063 MR4493620
27	M Faitg, Holonomy and (stated) skein algebras in combinatorial quantization, preprint (2020) arXiv:2003.08992
28	M Faitg, Projective representations of mapping class groups in combinatorial quantization, Comm. Math. Phys. 377 (2020) 161 MR4107927
29	V V Fock, A A Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and the r–matrix, from: "Moscow Seminar in Mathematical Physics" (editors A Y Morozov, M A Olshanetsky), Amer. Math. Soc. Transl. Ser. 2 191, Amer. Math. Soc. (1999) 67 MR1730456
30	C Frohman, J Kania-Bartoszynska, The structure of the Kauffman bracket skein algebra at roots of unity, Math. Z. 289 (2018) 889 MR3830231
31	S Gunningham, D Jordan, P Safronov, The finiteness conjecture for skein modules, Invent. Math. 232 (2023) 301 MR4557403
32	B Haïoun, Relating stated skein algebras and internal skein algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 18 (2022) 042 MR4437512
33	J Kamnitzer, P Tingley, The crystal commutor and Drinfeld’s unitarized R–matrix, J. Algebraic Combin. 29 (2009) 315 MR2496310
34	A N Kirillov, N Reshetikhin, q–Weyl group and a multiplicative formula for universal R–matrices, Comm. Math. Phys. 134 (1990) 421 MR1081014
35	J Korinman, Quantum groups and braiding operators in quantum Teichmüller theory, preprint (2019) arXiv:1907.01732
36	J Korinman, Triangular decomposition of character varieties, preprint (2019) arXiv:1904.09022
37	J Korinman, Stated skein algebras and their representations, preprint (2021) arXiv:2105.09563
38	J Korinman, Unicity for representations of reduced stated skein algebras, Topology Appl. 293 (2021) 107570 MR4229477
39	J Korinman, A Quesney, Classical shadows of stated skein representations at roots of unity, preprint (2019) arXiv:1905.03441
40	T T Q Lê, Triangular decomposition of skein algebras, Quantum Topol. 9 (2018) 591 MR3827810
41	T T Q Lê, Quantum Teichmüller spaces and quantum trace map, J. Inst. Math. Jussieu 18 (2019) 249 MR3915288
42	J L Loday, B Vallette, Algebraic operads, 346, Springer (2012) MR2954392
43	C Meusburger, D K Wise, Hopf algebra gauge theory on a ribbon graph, Rev. Math. Phys. 33 (2021) 2150016 MR4274370
44	J H Przytycki, A S Sikora, On skein algebras and Sl₂(ℂ)–character varieties, Topology 39 (2000) 115 MR1710996
45	N Reshetikhin, V G Turaev, Invariants of 3–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547 MR1091619
46	R Santharoubane, Algebraic generators of the skein algebra of a surface, preprint (2018) arXiv:1803.09804
47	N Snyder, P Tingley, The half-twist for U_q(𝔤) representations, Algebra Number Theory 3 (2009) 809 MR2579396
48	D P Thurston, Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA 111 (2014) 9725 MR3263305
49	P Tingley, A minus sign that used to annoy me but now I know why it is there (two constructions of the Jones polynomial), from: "Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F R Jones’ 60th birthday" (editors S Morrison, D Penneys), Proc. Centre Math. Appl. Austral. Nat. Univ. 46, Austral. Nat. Univ. (2017) 415 MR3635680
50	V G Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. 24 (1991) 635 MR1142906
51	E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351 MR990772