Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 025, 78 pages      arXiv:1912.02440

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Stéphane Baseilhac and Philippe Roche
IMAG, Univ Montpellier, CNRS, Montpellier, France

Received April 09, 2021, in final form March 07, 2022; Published online March 29, 2022

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, 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 $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the ''daisy'' graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. 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 $\mathcal{O}(G^n)$. 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 $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$. This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on $K_{\zeta}(\Sigma)$.

Key words: quantum groups; invariant theory; character varieties; skein algebras.

pdf (1062 kb)   tex (121 kb)  


  1. Alekseev A.Yu., Integrability in the Hamiltonian Chern-Simons theory, St Petersburg Math. J. 6 (1995), 241-253, arXiv:hep-th/9311074.
  2. Alekseev A.Yu., Grosse H., Schomerus V., Combinatorial quantization of the Hamiltonian Chern-Simons theory. I, Comm. Math. Phys. 172 (1995), 317-358, arXiv:hep-th/9403066.
  3. Alekseev A.Yu., Grosse H., Schomerus V., Combinatorial quantization of the Hamiltonian Chern-Simons theory. II, Comm. Math. Phys. 174 (1996), 561-604, arXiv:hep-th/9408097.
  4. Alekseev A.Yu., Malkin A.Z., Symplectic structure of the moduli space of flat connection[s] on a Riemann surface, Comm. Math. Phys. 169 (1995), 99-119, arXiv:hep-th/9312004.
  5. Alekseev A.Yu., Malkin A.Z., Meinrenken E., Lie group valued moment maps, J. Differential Geom. 48 (1998), 445-495, arXiv:dg-ga/9707021.
  6. Alekseev A.Yu., Schomerus V., Representation theory of Chern-Simons observables, Duke Math. J. 85 (1996), 447-510, arXiv:q-alg/9503016.
  7. Audin M., Lectures on gauge theory and integrable systems, in Gauge Theory and Symplectic Geometry (Montreal, PQ, 1995), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 488, Kluwer Acad. Publ., Dordrecht, 1997, 1-48.
  8. Baseilhac S., Benedetti R., Analytic families of quantum hyperbolic invariants, Algebr. Geom. Topol. 15 (2015), 1983-2063, arXiv:1212.4261.
  9. Baseilhac S., Benedetti R., Non ambiguous structures on 3-manifolds and quantum symmetry defects, Quantum Topol. 8 (2017), 749-846, arXiv:1506.01174.
  10. Baseilhac S., Benedetti R., On the quantum Teichmüller invariants of fibred cusped 3-manifolds, Geom. Dedicata 197 (2018), 1-32, arXiv:1704.05667.
  11. Baseilhac S., Roche P., Unrestricted quantum moduli algebras, II: Noetherianity and simple fractions rings at roots of $1$, arXiv:2106.04136.
  12. Baumann P., On the center of quantized enveloping algebras, J. Algebra 203 (1998), 244-260.
  13. Baumann P., Another proof of Joseph and Letzter's separation of variables theorem for quantum groups, Transform. Groups 5 (2000), 3-20.
  14. Ben-Zvi D., Brochier A., Jordan D., Quantum character varieties and braided module categories, Selecta Math. (N.S.) 24 (2018), 4711-4748, arXiv:1606.04769.
  15. Bonahon F., Miraculous cancellations for quantum ${\rm SL}_2$, Ann. Fac. Sci. Toulouse Math. 28 (2019), 523-557, arXiv:1708.07617.
  16. Bonahon F., Wong H., Quantum traces for representations of surface groups in ${\rm SL}_2(\mathbb C)$, Geom. Topol. 15 (2011), 1569-1615, arXiv:1003.5250.
  17. Bonahon F., Wong H., Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations, Invent. Math. 204 (2016), 195-243, arXiv:1206.1638.
  18. Bonahon F., Wong H., Representations of the Kauffman bracket skein algebra II: Punctured surfaces, Algebr. Geom. Topol. 17 (2017), 3399-3434, arXiv:1206.1639.
  19. Buffenoir E., Roche P., Two-dimensional lattice gauge theory based on a quantum group, Comm. Math. Phys. 170 (1995), 669-698, arXiv:hep-th/9405126.
  20. Buffenoir E., Roche P., Link invariants and combinatorial quantization of Hamiltonian Chern-Simons theory, Comm. Math. Phys. 181 (1996), 331-365, arXiv:q-alg/9507001.
  21. Buffenoir E., Roche P., Terras V., Quantum dynamical coboundary equation for finite dimensional simple Lie algebras, Adv. Math. 214 (2007), 181-229, arXiv:math.QA/0512500.
  22. Bullock D., Frohman C., Kania-Bartoszyńska J., Topological interpretations of lattice gauge field theory, Comm. Math. Phys. 198 (1998), 47-81, arXiv:q-alg/9710003.
  23. Bullock D., Frohman C., Kania-Bartoszyńska J., The Kauffman bracket skein as an algebra of observables, Proc. Amer. Math. Soc. 130 (2002), 2479-2485, arXiv:math.GT/0010330.
  24. Caldero P., Éléments ad-finis de certains groupes quantiques, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 327-329.
  25. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  26. Costantino F., Lê T.T.Q., Stated skein algebras of surfaces, J. Eur. Math. Soc., to appear, arXiv:1907.11400.
  27. De Concini C., Kac V.G., Representations of quantum groups at roots of $1$, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., Vol. 92, Birkhäuser Boston, Boston, MA, 1990, 471-506.
  28. De Concini C., Kac V.G., Procesi C., Quantum coadjoint action, J. Amer. Math. Soc. 5 (1992), 151-189.
  29. De Concini C., Lyubashenko V.V., Quantum function algebra at roots of $1$, Adv. Math. 108 (1994), 205-262.
  30. De Concini C., Procesi C., Quantum groups, in $D$-Modules, Representation Theory, and Quantum Groups (Venice, 1992), Lecture Notes in Math., Vol. 1565, Springer, Berlin, 1993, 31-140.
  31. Dixmier J., Enveloping algebras, Graduate Studies in Mathematics, Vol. 11, Amer. Math. Soc., Providence, RI, 1996.
  32. Donin J., Kulish P.P., Mudrov A.I., On a universal solution to the reflection equation, Lett. Math. Phys. 63 (2003), 179-194, arXiv:math.QA/0210242.
  33. Donin J., Mudrov A., Reflection equation, twist, and equivariant quantization, Israel J. Math. 136 (2003), 11-28, arXiv:math.QA/0204295.
  34. Drinfeld V.G., Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, in Problems of Modern Quantum Field Theory (Alushta, 1989), Editors A.A. Belavin, A.U. Klimyk, A.B. Zamolodchikov, Res. Rep. Phys., Springer, Berlin, 1989, 1-13.
  35. Drinfeld V.G., On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342.
  36. Exel R., Partial group actions, Lectures notes at the ICMAT, 2013.
  37. Exel R., Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, Vol. 224, Amer. Math. Soc., Providence, RI, 2017, arXiv:1511.04565.
  38. Faitg M., Modular group representations in combinatorial quantization with non-semisimple Hopf algebras, SIGMA 15 (2019), 077, 39 pages, arXiv:1805.00924.
  39. Faitg M., Mapping class groups and skein algebras in combinatorial quantization, Ph.D. Thesis, Université de Montpellier, 2019, arXiv:1910.04110.
  40. Faitg M., Projective representations of mapping class groups in combinatorial quantization, Comm. Math. Phys. 377 (2020), 161-198, arXiv:1812.00446.
  41. Faitg M., Holonomy and (stated) skein algebras in combinatorial quantization, Quant. Topol., to appear, arXiv:2003.08992.
  42. Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Comm. Math. Phys. 265 (2006), 47-93, arXiv:hep-th/0504093.
  43. Fock V.V., Rosly A.A., Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix, in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999, 67-86, arXiv:math.QA/9802054.
  44. Frohman C., Kania-Bartoszynska J., A matrix model for quantum $\rm SL_2$, in Diagrammatic Morphisms and Applications (San Francisco, CA, 2000), Contemp. Math., Vol. 318, Amer. Math. Soc., Providence, RI, 2003, 107-112, arXiv:math.QA/0010328.
  45. Frohman C., Kania-Bartoszynska J., Lê T., Unicity for representations of the Kauffman bracket skein algebra, Invent. Math. 215 (2019), 609-650, arXiv:1707.09234.
  46. Frohman C., Kania-Bartoszynska J., Lê T., Dimension and trace of the Kauffman bracket skein algebra, Trans. Amer. Math. Soc. Ser. B 8 (2021), 510-547, arXiv:1902.02002.
  47. Frolov S.A., The centre of the graph and moduli algebras at roots of $1$, Comm. Math. Phys. 200 (1999), 599-619, arXiv:q-alg/9602036.
  48. Ganev I., Jordan D., Safronov P., The quantum Frobenius for character varieties and multiplicative quiver varieties, arXiv:1901.11450.
  49. Goodearl K.R., Warfield Jr. R.B., An introduction to noncommutative Noetherian rings, 2nd ed., London Mathematical Society Student Texts, Vol. 61, Cambridge University Press, Cambridge, 2004.
  50. Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, Vol. 6, Amer. Math. Soc., Providence, RI, 1996.
  51. Joseph A., Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 29, Springer-Verlag, Berlin, 1995.
  52. Joseph A., Letzter G., Local finiteness of the adjoint action for quantized enveloping algebras, J. Algebra 153 (1992), 289-318.
  53. Joseph A., Letzter G., Separation of variables for quantized enveloping algebras, Amer. J. Math. 116 (1994), 127-177.
  54. Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  55. Kauffman L.H., Lins S.L., Temperley-Lieb recoupling theory and invariants of $3$-manifolds, Annals of Mathematics Studies, Vol. 134, Princeton University Press, Princeton, NJ, 1994.
  56. Kirby R., Melvin P., The $3$-manifold invariants of Witten and Reshetikhin-Turaev for ${\rm sl}(2,{\mathbb C})$, Invent. Math. 105 (1991), 473-545.
  57. Knop F., A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. 140 (1994), 253-288.
  58. Kolb S., Lorenz M., Nguyen B., Yammine R., On the adjoint representation of a Hopf algebra, Proc. Edinb. Math. Soc. 63 (2020), 1092-1099, arXiv:1905.03020.
  59. Kraft H., Procesi C., Classical invariant theory - a primer, Notes from courses given in Basel, available at, 1996.
  60. Kulish P.P., Sklyanin E.K., Algebraic structures related to reflection equations, J. Phys. A: Math. Gen. 25 (1992), 5963-5975, arXiv:hep-th/9209054.
  61. Lê T.T.Q., Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000), 273-306, arXiv:math.QA/0408358.
  62. Lê T.T.Q., Triangular decomposition of skein algebras, Quantum Topol. 9 (2018), 591-632, arXiv:1609.04987.
  63. Lickorish W.B.R., An introduction to knot theory, Graduate Texts in Mathematics, Vol. 175, Springer-Verlag, New York, 1997.
  64. Lusztig G., Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237-249.
  65. Lusztig G., Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), 89-113.
  66. Lusztig G., Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
  67. Lyubashenko V.V., Invariants of $3$-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172 (1995), 467-516, arXiv:hep-th/9405167.
  68. Lyubashenko V.V., Majid S., Braided groups and quantum Fourier transform, J. Algebra 166 (1994), 506-528.
  69. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  70. Meusburger C., Wise D.K., Hopf algebra gauge theory on a ribbon graph, Rev. Math. Phys. 33 (2021), 2150016, 93 pages, arXiv:1512.03966.
  71. Parshall B., Wang J.P., Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), vi+157 pages.
  72. Przytycki J.H., Sikora A.S., Skein algebras of surfaces, Trans. Amer. Math. Soc. 371 (2019), 1309-1332, arXiv:1602.07402.
  73. Reshetikhin N.Yu., Semenov-Tian-Shansky M.A., Quantum $R$-matrices and factorization problems, J. Geom. Phys. 5 (1988), 533-550.
  74. Reshetikhin N.Yu., Takhtadzhyan L.A., Faddeev L.D., Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.
  75. Roche P., Arnaudon D., Irreducible representations of the quantum analogue of ${\rm SU}(2)$, Lett. Math. Phys. 17 (1989), 295-300.
  76. Rybnikov L., Cactus group and monodromy of Bethe vectors, Int. Math. Res. Not. 2018 (2018), 202-235, arXiv:1409.0131.
  77. Semenov-Tian-Shansky M.A., Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), 1237-1260.
  78. Turaev V.G., Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter & Co., Berlin, 1994.
  79. Van Daele A., Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), 917-932.
  80. Varadarajan V.S., Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, Vol. 102, Springer-Verlag, New York, 1984.

Previous article  Next article  Contents of Volume 18 (2022)