Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1

Stéphane Baseilhac ; Philippe Roche

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

Détail

Keywords: quantum groups, invariant theory, character varieties, skein algebras, TQFT.

Bibliographie

[1] Alekseev A.Yu., “Integrability in the Hamiltonian Chern–Simons theory”, St. Petersburg Math. J., 6 (1995), 241–253, arXiv: hep-th/9311074 | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[4] Alekseev A.Yu., Schomerus V., “Representation theory of Chern–Simons observables”, Duke Math. J., 85 (1996), 447–510, arXiv: q-alg/9503016 | DOI | MR | Zbl

[5] Andersen H.H., Polo P., Wen K.X., “Representations of quantum algebras”, Invent. Math., 104 (1991), 1–59 | DOI | MR | Zbl

[6] Andruskiewitsch N., García G.A., “Quantum subgroups of a simple quantum group at roots of $1$”, Compos. Math., 145 (2009), 476–500, arXiv: 0707.0070 | DOI | MR | Zbl

[7] Atiyah M.F., Macdonald I.G., Introduction to commutative algebra, Addison-Wesley series in mathematics, 64, Addison-Wesley Publishing Co., Reading, Mass., 1969 | DOI | MR

[8] Baseilhac S., “Quantum coadjoint action and the $6j$-symbols of $U_q{\rm sl}_2$”, Interactions between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemp. Math., 541, American Mathematical Society, Providence, RI, 2011, 103–143, arXiv: 1101.3440 | DOI | MR | Zbl

[9] Baseilhac S., Benedetti R., “Quantum hyperbolic invariants of 3-manifolds with ${\rm PSL}(2,\mathbb C)$-characters”, Topology, 43 (2004), 1373–1423, arXiv: math.GT/0306280 | DOI | MR | Zbl

[10] Baseilhac S., Benedetti R., “Classical and quantum dilogarithmic invariants of flat ${\rm PSL}(2,\mathbb C)$-bundles over 3-manifolds”, Geom. Topol., 9 (2005), 493–569, arXiv: math.GT/0306283 | DOI | MR | Zbl

[11] Baseilhac S., Benedetti R., “Quantum hyperbolic geometry”, Algebr. Geom. Topol., 7 (2007), 845–917, arXiv: math.GT/0611504 | DOI | MR | Zbl

[12] Baseilhac S., Benedetti R., “The Kashaev and quantum hyperbolic link invariants”, J. Gökova Geom. Topol. GGT, 5 (2011), 31–85, arXiv: 1101.1851 | MR | Zbl

[13] Baseilhac S., Benedetti R., “Analytic families of quantum hyperbolic invariants”, Algebr. Geom. Topol., 15 (2015), 1983–2063, arXiv: 1212.4261 | DOI | MR | Zbl

[14] Baseilhac S., Benedetti R., “Non ambiguous structures on 3-manifolds and quantum symmetry defects”, Quantum Topol., 8 (2017), 749–846, arXiv: 1506.01174 | DOI | MR | Zbl

[15] Baseilhac S., Benedetti R., “On the quantum Teichmüller invariants of fibred cusped 3-manifolds”, Geom. Dedicata, 197 (2018), 1–32, arXiv: 1704.05667 | DOI | MR | Zbl

[16] Baseilhac S., Faitg M., Roche P., Unrestricted quantum moduli algebras, III: surfaces of arbitrary genus and skein algebras, arXiv: 2302.00396

[17] Baseilhac S., Faitg M., Roche P., Structure and representations of quantum moduli and ${\mathfrak g}$-skein algebras at roots of unity, in preparation

[18] Baseilhac S., Roche P., “Unrestricted quantum moduli algebras. I The case of punctured spheres”, SIGMA, 18 (2022), 025, 78 pp., arXiv: 1912.02440 | DOI | MR | Zbl

[19] Bass H., Algebraic $K$-theory, Math. Lect. Note Ser., W.A. Benjamin, Inc., New York, 1968 | MR | Zbl

[20] Baumann P., “Another proof of Joseph and Letzter's separation of variables theorem for quantum groups”, Transform. Groups, 5 (2000), 3–20 | DOI | MR

[21] Beliakova A., Blanchet C., Geer N., “Logarithmic Hennings invariants for restricted quantum $\mathfrak{sl}(2)$”, Algebr. Geom. Topol., 18 (2018), 4329–4358, arXiv: 1705.03083 | DOI | MR | Zbl

[22] 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 | DOI | MR | Zbl

[23] 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 | DOI | MR | Zbl

[24] 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 | DOI | MR | Zbl

[25] Brown K.A., Couto M., “Affine commutative-by-finite Hopf algebras”, J. Algebra, 573 (2021), 56–94, arXiv: 1907.10527 | DOI | MR | Zbl

[26] Brown K.A., Goodearl K.R., Lectures on algebraic quantum groups, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2002 | DOI | MR | Zbl

[27] Brown K.A., Gordon I., “The ramifications of the centres: quantised function algebras at roots of unity”, Proc. London Math. Soc., 84 (2002), 147–178, arXiv: math.RT/9912042 | DOI | MR | Zbl

[28] Brown K.A., Gordon I., Stafford J.T., $\mathcal{O}_\varepsilon[G]$ is a free module over $\mathcal{O}[G]$, arXiv: math.QA/0007179 | MR

[29] 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 | DOI | MR | Zbl

[30] 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 | DOI | MR | Zbl

[31] 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 | DOI | MR | Zbl

[32] Bullock D., “A finite set of generators for the Kauffman bracket skein algebra”, Math. Z., 231 (1999), 91–101 | DOI | MR | Zbl

[33] 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 | DOI | MR | Zbl

[34] Caldero P., “Éléments ad-finis de certains groupes quantiques”, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 327–329 | MR | Zbl

[35] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[36] Costantino F., Lê T.T.Q., “Stated skein algebras of surfaces”, J. Eur. Math. Soc., 24 (2022), 4063–4142, arXiv: 1907.11400 | DOI | MR | Zbl

[37] Cui W., “Canonical bases of modified quantum algebras for type $A_2$”, J. Algebra Appl., 17 (2018), 1850113, 27 pp., arXiv: 1208.5531 | DOI | MR | Zbl

[38] Dabrowski L., Reina C., Zampa A., “$A({\rm SL}_q(2))$ at roots of unity is a free module over $A({\rm SL}(2))$”, Lett. Math. Phys., 52 (2000), 339–342, arXiv: math.QA/0004092 | DOI | MR | Zbl

[39] De Concini C., Kac V.G., “Representations of quantum groups at roots of $1$”, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., 92, Birkhäuser, Boston, MA, 1990, 471–506 | MR

[40] De Concini C., Kac V.G., Procesi C., “Quantum coadjoint action”, J. Amer. Math. Soc., 5 (1992), 151–189 | DOI | MR | Zbl

[41] De Concini C., “Lyubashenko V., Quantum function algebra at roots of $1$”, Adv. Math., 108 (1994), 205–262 | DOI | MR | Zbl

[42] De Concini C., Procesi C., “Quantum groups”, $D$-modules, Representation Theory, and Quantum Qroups (Venice, 1992), Lecture Notes in Math., 1565, Springer, Berlin, 1993, 31–140 | DOI | MR | Zbl

[43] de Graaf W.A., “Constructing canonical bases of quantized enveloping algebras”, Experiment. Math., 11 (2002), 161–170 | DOI | MR | Zbl

[44] De Renzi M., Gainutdinov A.M., Geer N., Patureau-Mirand B., Runkel I., “$3$-dimensional {TQFT}s from non-semisimple modular categories”, Selecta Math. (N.S., 28 (2022), 42, 60 pp., arXiv: 1912.02063 | DOI | MR | Zbl

[45] De Renzi M., Geer N., Patureau-Mirand B., “Renormalized Hennings Invariants and $2+1$-TQFTs”, Comm. Math. Phys., 362 (2020), 855–907, arXiv: 1707.08044 | DOI | MR

[46] Dieudonné J.A., Carrell J.B., “Invariant theory, old and new”, Adv. Math., 4 (1970), 1–80 | DOI | MR | Zbl

[47] Domokos M., Lenagan T.H., “Quantized trace rings”, Q. J. Math., 56 (2005), 507–523, arXiv: math.QA/0407053 | DOI | MR | Zbl

[48] Drinfeld V.G., “On almost cocommutative Hopf algebras”, Leningrad Math. J., 1 (1990), 321–342 | MR | Zbl

[49] Du J., “Global IC bases for quantum linear groups”, J. Pure Appl. Algebra, 114 (1996), 25–37 | DOI | MR | Zbl

[50] Enriquez B., “Le centre des algébres de coordonnées des groupes quantiques aux racines $p^\alpha$-iémes de l'unité”, Bull. Soc. Math. France, 122 (1994), 443–485 | DOI | MR | Zbl

[51] Etingof P., Golberg O., Hensel S., Liu T., Schwendner A., Vaintrob D., Yudovina E., Introduction to representation theory, Stud. Math. Libr., 59, American Mathematical Society, Providence, RI, 2011, arXiv: 0901.0827 | DOI | MR | Zbl

[52] Faitg M., Mapping class groups, skein algebras and combinatorial quantization, Ph.D. Thesis, Montpellier de Université, 2019, arXiv: 1910.04110

[53] Faitg M., “Projective representations of mapping class groups in combinatorial quantization”, Comm. Math. Phys., 377 (2020), 161–198, arXiv: 1812.00446 | DOI | MR | Zbl

[54] Faitg M., “Holonomy and (stated) skein algebras in combinatorial quantization”, Quantum Topol. (to appear) , arXiv: 2003.08992

[55] Fock V.V., Rosly A.A., “Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix”, Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser., 191, American Mathematical Society, Providence, RI, 1999, 67–86, arXiv: math.QA/9802054 | DOI | MR | Zbl

[56] Frenkel I.B., Khovanov M.G., “Canonical bases in tensor products and graphical calculus for $U_q(\mathfrak{sl}_2)$”, Duke Math. J., 87 (1997), 409–480 | DOI | MR | Zbl

[57] 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 | DOI | MR | Zbl

[58] Ganev I., Jordan D., Safronov P., “The quantum Frobenius for character varieties and multiplicative quiver varieties”, J. Eur. Math. Soc. (to appear) , arXiv: 1901.11450 | DOI

[59] Humphreys J.E., Linear algebraic groups, Grad. Texts Math., 21, Springer, New York, 1975 | DOI | MR | Zbl

[60] Jordan D., White N., “The center of the reflection equation algebra via quantum minors”, J. Algebra, 542 (2020), 308–342, arXiv: 1709.09149 | DOI | MR | Zbl

[61] Joseph A., Quantum groups and their primitive ideals, Ergeb. Math. Grenzgeb. (3), 29, Springer, Berlin, 1995 | DOI | MR | Zbl

[62] Joseph A., Letzter G., “Local finiteness of the adjoint action for quantized enveloping algebras”, J. Algebra, 153 (1992), 289–318 | DOI | MR | Zbl

[63] Joseph A., Letzter G., “Separation of variables for quantized enveloping algebras”, Amer. J. Math., 116 (1994), 127–177 | DOI | MR | Zbl

[64] Karuo H., Korinman J., Classification of semi-weight representations of reduced stated skein algebras, arXiv: 2303.09433

[65] Kashiwara M., “On crystal bases of the $Q$-analogue of universal enveloping algebras”, Duke Math. J., 63 (1991), 465–516 | DOI | MR | Zbl

[66] Kashiwara M., “Global crystal bases of quantum groups”, Duke Math. J., 69 (1993), 455–485 | DOI | MR | Zbl

[67] Kashiwara M., “Crystal bases of modified quantized enveloping algebra”, Duke Math. J., 73 (1994), 383–413 | DOI | MR | Zbl

[68] Kashiwara M., “On crystal bases”, Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., 16, American Mathematical Society, Providence, RI, 1995, 155–197 | MR | Zbl

[69] Khoroshkin S.M., Tolstoy V.N., “Universal $R$-matrix for quantized (super)algebras”, Comm. Math. Phys., 141 (1991), 599–617 | DOI | MR | Zbl

[70] Kirillov A.N., Reshetikhin N., “$q$-Weyl group and a multiplicative formula for universal $R$-matrices”, Comm. Math. Phys., 134 (1990), 421–431 | DOI | MR | Zbl

[71] Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts Monogr. Phys., Springer, Berlin, 1997 | DOI | MR | Zbl

[72] 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 | DOI | MR | Zbl

[73] Korinman J., “Unicity for representations of reduced stated skein algebras”, Topology Appl., 293 (2021), 107570, 28 pp., arXiv: 2001.00969 | DOI | MR | Zbl

[74] Korinman J., “Finite presentations for stated skein algebras and lattice gauge field theory”, Algebr. Geom. Topol., 23 (2023), 1249–1302, arXiv: 2012.03237 | DOI | MR

[75] Korinman J., Quesney A., Classical shadows of stated skein representations at odd roots of unity, arXiv: 1905.03441

[76] Kostant B., “Groups over $Z$”, Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), Proc. Sympos. Pure Math., American Mathematical Society, Providence, RI, 1966, 90–98 | DOI | MR

[77] Lang S., Algebraic structures, Addison-Wesley Publishing Co., Reading, Mass., 1967 | MR | Zbl

[78] Lê T.T.Q., “Triangular decomposition of skein algebras”, Quantum Topol., 9 (2018), 591–632, arXiv: 1609.04987 | DOI | MR | Zbl

[79] Lê T.T.Q., Sikora A.S., Stated ${\rm SL}(n)$-skein modules and algebras, arXiv: 2201.00045

[80] Lê T.T.Q., Yu T., “Quantum traces and embeddings of stated skein algebras into quantum tori”, Selecta Math. (N.S.), 28 (2022), 66, 48 pp., arXiv: 2012.15272 | DOI | MR

[81] Levendorskii S.Z., Soibelman Y.S., “Some applications of the quantum Weyl groups”, J. Geom. Phys., 7 (1990), 241–254 | DOI | MR | Zbl

[82] Lusztig G., “Quantum groups at roots of $1$”, Geom. Dedicata, 35 (1990), 89–113 | DOI | MR

[83] Lusztig G., Introduction to quantum groups, Mod. Birkhäuser Class., 110, Birkhäuser, Boston, MA, 1993 | DOI | MR | Zbl

[84] Lusztig G., “Study of a $\mathbb{Z}$-form of the coordinate ring of a reductive group”, J. Amer. Math. Soc., 22 (2009), 739–769 | DOI | MR | Zbl

[85] Lyubashenko V., Majid S., “Braided groups and quantum Fourier transform”, J. Algebra, 166 (1994), 506–528 | DOI | MR | Zbl

[86] Majid S., “Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group”, Comm. Math. Phys., 156 (1993), 607–638, arXiv: hep-th/9208008 | DOI | MR | Zbl

[87] Marlin R., “Anneaux de Grothendieck des variétés de drapeaux”, Bull. Soc. Math. France, 104 (1976), 337–348 | DOI | MR | Zbl

[88] McConnell J.C., Robson J.C., Noncommutative Noetherian rings, Grad. Stud. Math., 30, American Mathematical Society, Providence, RI, 2001 | DOI | MR | Zbl

[89] Meusburger C., “Kitaev lattice models as a Hopf algebra gauge theory”, Comm. Math. Phys., 353 (2017), 413–468, arXiv: 1607.01144 | DOI | MR | Zbl

[90] Murakami J., “Generalized Kashaev invariants for knots in three manifolds”, Quantum Topol., 8 (2017), 35–73, arXiv: 1312.0330 | DOI | MR | Zbl

[91] Paradowski J., “Filtrations of modules over the quantum algebra”, Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., 56, American Mathematical Society, Providence, RI, 1994, 93–108 | DOI | MR | Zbl

[92] Parshall B., Wang J.P., Quantum linear groups, Mem. Amer. Math. Soc., 89, 1991, vi+157 pp. | DOI | MR

[93] Przytycki J.H., Sikora A.S., “Skein algebras of surfaces”, Trans. Amer. Math. Soc., 371 (2019), 1309–1332, arXiv: 1602.07402 | DOI | MR | Zbl

[94] Reshetikhin N.Yu., Semenov-Tian-Shansky M.A., “Quantum $R$-matrices and factorization problems”, J. Geom. Phys., 5 (1988 (1989)), 533–550 | DOI | MR | Zbl

[95] Reshetikhin N.Yu., Turaev V.G., “Invariants of $3$-manifolds via link polynomials and quantum groups”, Invent. Math., 103 (1991), 547–597 | DOI | MR | Zbl

[96] Rowen L.H., Ring theory, Academic Press, Inc., Boston, MA, 1991 | MR | Zbl

[97] Saito Y., “PBW basis of quantized universal enveloping algebras”, Publ. Res. Inst. Math. Sci., 30 (1994), 209–232 | DOI | MR | Zbl

[98] Soibelman Y.S., “Algebra of functions on a compact quantum group and its representations”, Leningrad Math. J., 2 (1990), 161–268 | MR

[99] Springer T.A., Invariant theory, Lect. Notes in Math., 585, Springer, Berlin, 1977 | DOI | MR | Zbl

[100] The Stacks Project Authors, Commutative Alg, Chapter 10, The Stack Project https://stacks.math.columbia.edu

[101] The Stacks Project Authors, Brauer groups, Chapter 11, The Stack Project https://stacks.math.columbia.edu

[102] Vaksman L.L., Soibelman Y.S., “Algebra of functions on the quantum group ${\rm SU}(2)$”, Funct. Anal. Appl., 22 (1988), 170–181 | DOI | MR | Zbl

[103] Varagnolo M., Vasserot E., “Double affine Hecke algebras at roots of unity”, Represent. Theory, 14 (2010), 510–600 | DOI | MR | Zbl

[104] Voigt C., Yuncken R., Complex semisimple quantum groups and representation theory, Lect. Notes in Math., 2264, Springer, Cham, 2020 | DOI | MR | Zbl

[105] Wang Z., On stated ${\rm SL}(n)$-skein modules, arXiv: 2307.10288

[106] Witten E., “Topological quantum field theory”, Comm. Math. Phys., 117 (1988), 353–386 | DOI | MR | Zbl

[107] Xi N.H., “Root vectors in quantum groups”, Comment. Math. Helv., 69 (1994), 612–639 | DOI | MR | Zbl