Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres
Let Σ be a finite type surface, and a complex algebraic simple Lie group with Lie algebra . The quantum moduli algebra of (Σ, ) is a quantization of the ring of functions of (Σ), the variety of -characters of π₁(Σ), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. I...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211520 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres. Stéphane Baseilhac and Philippe Roche. SIGMA 18 (2022), 025, 78 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862731352171872256 |
|---|---|
| author | Baseilhac, Stéphane Roche, Philippe |
| author_facet | Baseilhac, Stéphane Roche, Philippe |
| citation_txt | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres. Stéphane Baseilhac and Philippe Roche. SIGMA 18 (2022), 025, 78 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let Σ be a finite type surface, and a complex algebraic simple Lie group with Lie algebra . The quantum moduli algebra of (Σ, ) is a quantization of the ring of functions of (Σ), the variety of -characters of π₁(Σ), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are q()-module-algebras associated to graphs on Σ, where q() is the quantum group corresponding to . We study the structure of the quantum moduli algebra in the case where Σ is a sphere with + 1 open disks removed, ≥ 1, using the graph algebra of the ''daisy'' graph on Σ to make computations easier. We provide new results that hold for arbitrary and generic , and develop the theory in the case where = ϵ, a primitive root of unity of odd order, and =SL(2, ℂ). In such a situation, we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring (ⁿ). We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on (ⁿ). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ℂ[(Σ)] endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators, we identify the Kauffman bracket skein algebra ζ(Σ) at ζ := iϵ¹/² with this quantum moduli algebra specialized at = ϵ. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on ζ(Σ).
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| first_indexed | 2026-04-17T15:17:28Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211520 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T15:17:28Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Baseilhac, Stéphane Roche, Philippe 2026-01-05T12:23:44Z 2022 Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres. Stéphane Baseilhac and Philippe Roche. SIGMA 18 (2022), 025, 78 pages 1815-0659 2020 Mathematics Subject Classification: 16R30; 17B37; 20G42; 57M27; 57R56; 81R50 arXiv:1912.02440 https://nasplib.isofts.kiev.ua/handle/123456789/211520 https://doi.org/10.3842/SIGMA.2022.025 Let Σ be a finite type surface, and a complex algebraic simple Lie group with Lie algebra . The quantum moduli algebra of (Σ, ) is a quantization of the ring of functions of (Σ), the variety of -characters of π₁(Σ), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are q()-module-algebras associated to graphs on Σ, where q() is the quantum group corresponding to . We study the structure of the quantum moduli algebra in the case where Σ is a sphere with + 1 open disks removed, ≥ 1, using the graph algebra of the ''daisy'' graph on Σ to make computations easier. We provide new results that hold for arbitrary and generic , and develop the theory in the case where = ϵ, a primitive root of unity of odd order, and =SL(2, ℂ). In such a situation, we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring (ⁿ). We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on (ⁿ). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ℂ[(Σ)] endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators, we identify the Kauffman bracket skein algebra ζ(Σ) at ζ := iϵ¹/² with this quantum moduli algebra specialized at = ϵ. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on ζ(Σ). We thank our colleagues of the work group on moduli spaces at IMAG for discussions, especially Paul-Emile Paradan and Damien Calaque. We also thank Pavel Etingof, Matthieu Faitg, Charles Frohman, and Catherine Meusburger for their interest and exchanges related to the present work. Finally, we also thank the referees for their suggestions, which greatly improved the exposition of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres Article published earlier |
| spellingShingle | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres Baseilhac, Stéphane Roche, Philippe |
| title | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres |
| title_full | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres |
| title_fullStr | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres |
| title_full_unstemmed | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres |
| title_short | Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres |
| title_sort | unrestricted quantum moduli algebras. i. the case of punctured spheres |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211520 |
| work_keys_str_mv | AT baseilhacstephane unrestrictedquantummodulialgebrasithecaseofpuncturedspheres AT rochephilippe unrestrictedquantummodulialgebrasithecaseofpuncturedspheres |