Mini-workshop: Bridging number theory and Nichols algebras via deformations. Abstracts from the mini-workshop held January 28 – February 2, 2024. (English) Zbl 07921239
Summary: Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.
MSC:
| 16-06 | Proceedings, conferences, collections, etc. pertaining to associative rings and algebras |
| 00B05 | Collections of abstracts of lectures |
| 00B25 | Proceedings of conferences of miscellaneous specific interest |
| 16G30 | Representations of orders, lattices, algebras over commutative rings |
| 16T05 | Hopf algebras and their applications |
| 11G20 | Curves over finite and local fields |
| 20F36 | Braid groups; Artin groups |
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
References:
| [1] | N. Andruskiewitsch, On infinite-dimensional Hopf algebras. arXiv:2308.13120. |
| [2] | N. Andruskiewitsch, I. Angiono, J. Pevtsova, S. Witherspoon. Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups. Res. Math. Sci. 9:12 (2022). · Zbl 1510.16005 |
| [3] | N. Andruskiewitsch, F. Fantino, M. Graña, L. Vendramin, Finite-dimensional pointed Hopf algebras with alternating groups are trivial, Ann. Mat. Pura Appl. 190, 225-245 (2011). · Zbl 1234.16019 |
| [4] | N. Andruskiewitsch, M. Graña, From racks to pointed Hopf algebras, Adv. Math. 178, 177-243 (2003). · Zbl 1032.16028 |
| [5] | N. Andruskiewitsch and H.-J. Schneider. Finite quantum groups and Cartan matrices, Adv. Math. 154, 1-45 (2000). · Zbl 1007.16027 |
| [6] | N. Andruskiewitsch, H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. 171, 375-417 (2010). · Zbl 1208.16028 |
| [7] | I. Angiono, On Nichols algebras of diagonal type. J. Reine Angew. Math. 683, 189-251 (2013). · Zbl 1331.16023 |
| [8] | I. Angiono and A. García Iglesias. Liftings of Nichols algebras of diagonal type II: all liftings are cocycle deformations. Selecta Math. (N.S.) 25 (2019), Paper No. 5, 95 pp. · Zbl 1406.16031 |
| [9] | I. Heckenberger, E. Meir, and L. Vendramin, Simple Yetter-Drinfeld Modules Over Groups With Prime Dimension and a Finite-Dimensional Nichols Algebra, arXiv:2306.02989. |
| [10] | I. Heckenberger and H.-J. Schneider, Hopf algebras and root systems. Providence, RI: Amer. Math. Soc. (AMS) (2020). · Zbl 1473.16026 |
| [11] | D. Joyce, Simple quandles. J. Algebra 79, 307-318 (1982). · Zbl 0514.20018 |
| [12] | K. Karai. In what sense is the classification of all finite groups “impossible”?, https://mathoverflow.net/q/180355 |
| [13] | E. Meir Geometric perspective on Nichols algebras, J. Algebra 601, 390-422 (2022). · Zbl 1502.16035 |
| [14] | V V. Sergejchuk. On the classification of metabelian p-groups (Russian), in Matrix prob-lems, Inst. Mat. Ukrain. Akad. Nauk. (Kiev), 150-161 (1977). References · Zbl 0444.20018 |
| [15] | N. Andruskiewitsch. An Introduction to Nichols Algebras. In Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. A. Cardona, P. Morales, H. Ocampo, S. Paycha, A. Reyes, eds., pp. 135-195, Springer (2017). · Zbl 1383.81112 |
| [16] | N. Andruskiewitsch, I. Angiono, I. Heckenberger. On finite GK-dimensional Nichols algebras over abelian groups. Mem. Amer. Math. Soc. Volume 271, Number 1329 (2021). · Zbl 1507.16002 |
| [17] | N. Andruskiewitsch, I. Angiono, I. Heckenberger. On finite GK-dimensional Nichols algebras of diagonal type. Contemp. Math. 728 1-23 (2019). · Zbl 1446.17025 |
| [18] | N. Andruskiewitsch, I. Angiono, M. Moya Giusti, Rank 4 Nichols algebras of pale braidings. SIGMA 19 (2023), 021, 41 pages. · Zbl 07707703 |
| [19] | N. Andruskiewitsch, G. Sanmarco, Finite GK-dimensional pre-Nichols algebras of quantum linear spaces and of Cartan type. Trans. Amer. Math. Soc. Ser. B 8, 296-329 (2021). · Zbl 1484.16041 |
| [20] | N. Andruskiewitsch, H.-J. Schneider, Pointed Hopf algebras, New directions in Hopf alge-bras, MSRI series Cambridge Univ. Press; 1-68 (2002). · Zbl 1011.16025 |
| [21] | I. Angiono, E. Campagnolo, G. Sanmarco. Finite GK-dimensional pre-Nichols algebras of super and standard type. J. Pure Appl. Alg., to appear. · Zbl 07740048 |
| [22] | I. Angiono, E. Campagnolo, G. Sanmarco. Finite GK-dimensional pre-Nichols algebras of (super)modular and unidentified type. J. Noncommut. Geom. 17 (2023), 499-525. · Zbl 1528.16030 |
| [23] | I. Angiono, A. García Iglesias. Finite GK-dimensional Nichols algebras of diagonal type and finite root systems, arXiv:2212.08169. |
| [24] | The GAP Group, GAP -Groups, Algorithms, and Programming, Version 4.11.1; 2021, https://www.gap-system.org. |
| [25] | I. Heckenberger. Classification of arithmetic root systems. Adv. Math. 220 59-124 (2009). · Zbl 1176.17011 |
| [26] | I. Heckenberger, H. Yamane. A generalization of Coxeter groups, root systems, and Mat-sumoto’s theorem. Math. Z. 259 255-276 (2008). · Zbl 1198.20036 |
| [27] | V. Kharchenko, A quantum analog of the Poincare-Birkhoff-Witt theorem, Algebra and Logic 38, (1999), 259-276. References · Zbl 0936.16034 |
| [28] | M. Bhargava. Higher composition laws. III. The parametrization of quartic rings. Ann. of Math. (2), 159(3):1329-1360, 2004. · Zbl 1169.11045 |
| [29] | M. Bhargava. Higher composition laws. IV. The parametrization of quintic rings. Ann. of Math. (2), 167(1):53-94, 2008. · Zbl 1173.11058 |
| [30] | K. Chang. Hurwitz spaces, Nichols algebras, and Igusa zeta functions. arXiv preprint arXiv:2306.10446, 2023. |
| [31] | B. Delone and D. Faddeev. The Theory of Irrationalities of the Third Degree, volume Vol. 10 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1964. · Zbl 0133.30202 |
| [32] | M. Kapranov and V. Schechtman. Shuffle algebras and perverse sheaves. Pure Appl. Math. Q., 16(3):573-657, 2020. · Zbl 1464.14022 |
| [33] | A. Landesman, R. Vakil, and M. Wood. Low degree Hurwitz stacks in the Grothendieck ring. arXiv preprint arXiv:2203.01840, 2022. |
| [34] | M. Sato and T. Kimura. A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J., 65:1-155, 1977. · Zbl 0321.14030 |
| [35] | M. Wood. Parametrizing quartic algebras over an arbitrary base. Algebra Number Theory, 5(8):1069-1094, 2011. References · Zbl 1271.11043 |
| [36] | S. Fomin, K. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Progr. Math.172, Birkhäuser, (1999), 146-182. · Zbl 0940.05070 |
| [37] | A. Milinski, H.-J. Schneider, Pointed Indecomposable Hopf Algebras over Coxeter Groups, Contemp. Math. 267, (2000), 215-236. · Zbl 1093.16504 |
| [38] | M. Kapranov, V. Schechtman, Shuffle algebras and perverse sheaves, Pure and Applied Mathematics Quarterly, Special Issue in Honor of Prof. Kyoji Saito’ s 75th Birthday vol 16, 573-657 (2020), International Press. References · Zbl 1464.14022 |
| [39] | Michael Barr. On duality of topological abelian groups. (preprint). |
| [40] | Serge Bouc. The Burnside dimension of projective Mackey functors. RIMS Kôkyûroku, 1440:107-120, 2005. |
| [41] | Jürgen Fuchs, Gregor Schaumann, Christoph Schweigert, and Simon Wood. Grothendieck-verdier duality in categories of bimodules and weak module functors, 2023. |
| [42] | Sebastian Halbig and Tony Zorman. Duality in Monoidal Categories. arXiv e-prints, 2024. |
| [43] | Harald Lindner. A remark on Mackey-functors. Manuscr. Math., 18:273-278, 1976. · Zbl 0321.18002 |
| [44] | Lukas Müller and Lukas Woike. The distinguished invertible object as ribbon dualizing object in the Drinfeld center. arXiv e-prints, 2023. |
| [45] | K.-H. Ulbrich. On Hopf algebras and rigid monoidal categories. Isr. J. Math., 72(1-2):252-256, 1990. References · Zbl 0727.16029 |
| [46] | S. Lentner: New large-rank Nichols algebras over nonabelian groups with commutator subgroup Z 2 , Journal of Algebra 419 (2014). · Zbl 1306.16029 |
| [47] | I. Heckenberger, L. Vendramin: A classification of Nichols algebras of semi-simple Yetter-Drinfeld modules over non-abelian groups, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 2, 299-356. · Zbl 1396.16025 |
| [48] | N. Andruskiewitsch, I. Heckenberger, H.-J. Schneider: The Nichols algebra of a semisimple Yetter-Drinfeld module. American Journal of Mathematics 132 (2008). |
| [49] | I. Angiono, S. Lentner, G. Sanmarco: Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings (2023), Proceedings of the London Mathematical Society 127(9). · Zbl 1540.16025 |
| [50] | T. Creutzig, S. Lentner, M. Rupert: An algebraic theory for logarithmic Kazhdan-Lusztig correspondences, Preprint (2023), arXiv:2306.11492. |
| [51] | P. Etingof, D. Nikshych, V. OStrik, E. Meir: Fusion categories and homotopy theory, Preprint (2009) arXiv:0909.3140. References |
| [52] | G. Carnovale, J. Cuadra, and E. Masut, Non-existence of integral Hopf orders for twists of several simple groups of Lie type. Publicacions Matemàtiques 68 (2024), 73-101. · Zbl 1543.16027 |
| [53] | J. Cuadra and E. Meir, On the existence of orders in semisimple Hopf algebras. Trans. Amer. Math. Soc. 368 (2016), no. 4, 2547-2562. · Zbl 1342.16026 |
| [54] | J. Cuadra and E. Meir, Non-existence of Hopf orders for a twist of the alternating and symmetric groups. J. London Math. Soc. (2) 100 (2019), no. 1, 137-158. · Zbl 1490.16075 |
| [55] | M. Movshev, Twisting in group algebras of finite groups. Funct. Anal. Appl. 27 (1994), 240-244. References · Zbl 0813.20005 |
| [56] | P. Balmer, The spectrum of prime ideals in a tensor triangulated category, J. Reine Angew. Math. 588 (2005), 149-168. · Zbl 1080.18007 |
| [57] | P. Balmer, Spectra, spectra, spectra-tensor triangular spectra versus Zariski spectra of endomorphism rings, AGT, 10, no. 3, (2010), 1521-1563. · Zbl 1204.18005 |
| [58] | T. Barthel, D. J. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, Lattices over finite group schemes and stratification, arxiv:2307.16271 (2023) |
| [59] | D. J. Benson, J. F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Math., 153, (1997) 59-80. · Zbl 0886.20007 |
| [60] | A. B. Buan, H. Krause, and O. Solberg, Support varieties: an ideal approach, Homology, Homotopy Appl., 9(1):45-74, 2007. · Zbl 1118.18005 |
| [61] | R-O. Buchweitz, Maximal Cohen Macaulay modules and Tate cohomology, Math Surveys Monogr., 262, AMS (2021) · Zbl 1505.13002 |
| [62] | E. M. Friedlander and J. Pevtsova, Π-supports for modules for finite groups schemes, Duke Math. J. 139 (2007), 317-368. · Zbl 1128.20031 |
| [63] | E. Lau, The Balmer spectrum of certain Deligne-Mumford stacks, Compositio Math, (2023). · Zbl 1525.14024 |
| [64] | D. Nakano, K. Vashaw, and M. Yakimov, Noncommutative tensor triangular geometry, arxiv:1909.04304 (2021). |
| [65] | C. Negron, J. Pevtsova, Hypersurface support and prime ideal spectra for stable categories, Ann K-theory, 8, no.1, (2023) 25-79. · Zbl 1523.16040 |
| [66] | J. Rickard, Derived categories and stable equivalence , JPAA 61 (1989) no. 3, 303-317 · Zbl 0685.16016 |
| [67] | W. van der Kallen, A Friedlander-Suslin theorem over a noetherian base ring, Transforma-tion groups, (2023) References |
| [68] | S. Fomin, K. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Progr. Math.172, Birkhäuser, (1999), 146-182. · Zbl 0940.05070 |
| [69] | G. A. García, A. García Iglesias, Finite dimensional pointed Hopf algebras over S 4 , Israel Journal of Math. 183, (2011), 417-444. · Zbl 1231.16023 |
| [70] | Kapranov, M.; Schechtman, V. Shuffle algebras and perverse sheaves. Pure Appl. Math. Q. 16 (2020), no. 3, 573-657. · Zbl 1464.14022 |
| [71] | Milinski, H.-J. Schneider, Pointed indecomposable Hopf algebras over Coxeter groups, New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 215-236. · Zbl 1093.16504 |
| [72] | J. E. Roos, Some non-Koszul Algebras, in: Advances in Geometry, J. L. Brylinksi, V. Nistor, B. Tsygan, P. Xu, eds, Progress in Mathematics 172 (1999), 385-389. References · Zbl 0962.16022 |
| [73] | J. Baez and M. Neuchl, Higher-dimensional algebra. I. Braided monoidal 2-categories. Adv. Math. 121 (1996), no. 2, 196-244. · Zbl 0855.18008 |
| [74] | B. Day and R. Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), 99-157. · Zbl 0910.18004 |
| [75] | D. Gepner and R. Haugseng. Enriched ∞-categories via non-symmetric ∞-operads. Adv. Math. 279 (2015), 575-716. · Zbl 1342.18009 |
| [76] | M. Kapranov and V. Voevodsky, 2-categories and Zamolodchikov tetrahedra equations. Al-gebraic groups and their generalizations: quantum and infinite-dimensional methods (Uni-versity Park, PA, 1991), 177-259, Proc. Sympos. Pure Math., 56, Part 2, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0809.18006 |
| [77] | M. Kapranov and V. Voevodsky, Braided monoidal 2-categories and Manin-Schechtman higher braid groups. J. Pure Appl. Algebra 92 (1994), no. 3, 241-267. · Zbl 0791.18010 |
| [78] | L. Liu, A. Mazel-Gee, D. Reutter, C. Stroppel, and P. Wedrich. A braided monoidal (∞, 2)-category of Soergel bimodules, arXiv:2401.02956 |
| [79] | R. Rouquier. Khovanov-Rozansky homology and 2-braid groups. In: Categorification in geometry, topology, and physics. Vol. 684. Contemp. Math. AMS, 2017, pp. 147-157. · Zbl 1365.57021 |
| [80] | W. Soergel. The combinatorics of Harish-Chandra bimodules. J. Reine Angew. Math. vol. 429 (1992) · Zbl 0745.22014 |
| [81] | C. Stroppel. Categorification: tangle invariants and TQFTs. ICM-International Congress of Mathematicians. Vol. II. Plenary lectures, 1312-1353. References · Zbl 1536.57014 |
| [82] | H. H. Andersen, J. C. Jantzen and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p. Astérisque 220 (1994). · Zbl 0802.17009 |
| [83] | N. Andruskiewitsch, I. Angiono and F. Rossi Bertone, Lie algebras arising from Nichols algebras of diagonal type. Int. Math. Res. Not. 4 (2023), 3424-3459. · Zbl 1531.16028 |
| [84] | C. Vay, Linkage principle for small quantum groups. arXiv:2310.00103 (2023). References |
| [85] | N. Andruskiewitsch, G. Carnovale, and G. A. García. Finite-dimensional pointed Hopf al-gebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups. Manuscripta Math., 166(3-4):605-647, 2021. · Zbl 1493.16028 |
| [86] | N. Andruskiewitsch, G. Carnovale, and G. A. García. Finite-dimensional pointed Hopf alge-bras over finite simple groups of Lie type VII. Semisimple classes in PSLn(q) and PSp 2n (q). J. Algebra, 639:354-397, 2024. · Zbl 1539.16021 |
| [87] | N. Andruskiewitsch, F. Fantino, M. Graña, and L. Vendramin. Finite-dimensional pointed Hopf algebras with alternating groups are trivial. Ann. Mat. Pura Appl. (4), 190(2):225-245, 2011. · Zbl 1234.16019 |
| [88] | N. Andruskiewitsch, F. Fantino, M. Graña, and L. Vendramin. Pointed Hopf algebras over the sporadic simple groups. J. Algebra, 325:305-320, 2011. · Zbl 1217.16026 |
| [89] | N. Andruskiewitsch and M. Graña. From racks to pointed Hopf algebras. Adv. Math., 178(2):177-243, 2003. · Zbl 1032.16028 |
| [90] | G. Carnovale and M. Costantini. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VI. Suzuki and Ree groups. J. Pure Appl. Algebra, 225(4):Paper No. 106568, 19, 2021. · Zbl 1487.16033 |
| [91] | I. Heckenberger. Classification of arithmetic root systems. Adv. Math., 220(1):59-124, 2009. · Zbl 1176.17011 |
| [92] | I. Heckenberger, E. Meir, L. Vendramin. Simple Yetter-Drinfeld modules over groups with prime dimension and a finite-dimensional Nichols algebra. arXiv:2306.02989, 2023. |
| [93] | I. Heckenberger and L. Vendramin. The classification of Nichols algebras over groups with finite root system of rank two. J. Eur. Math. Soc. (JEMS), 19(7):1977-2017, 2017. References · Zbl 1391.16036 |
| [94] | Jordan S. Ellenberg, TriThang Tran, and Craig Westerland, Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields, 2023. |
| [95] | Gunter Malle, On the distribution of Galois groups. II, Experiment. Math. 13 (2004), no. 2, 129-135. MR 2068887 References · Zbl 1099.11065 |
| [96] | R. Green, Generalized Temperley-Lieb algebras and decorated tangles, J. Knot Theory Ram-ifications 7 (1998) · Zbl 0926.20005 |
| [97] | H. Queffelec and P. Wedrich, Extremal weight projectors, Math. Res. Lett. 25 (2018) · Zbl 1475.57024 |
| [98] | C. Stroppel and Z. Wojciechowski, Diagrammatics for the smallest quantum coideal and Jones-Wenzl projectors, (2024), forthcoming References |
| [99] | I Angiono. Distinguished pre-Nichols algebras. Transform. Groups 21, 1-33 (2016). · Zbl 1355.16028 |
| [100] | N. Andruskiewitsch, I. Angiono and M. Yakimov. Poisson orders on large quantum groups. Adv. Math. 428 (2023), 109134, 66pp. · Zbl 1530.16029 |
| [101] | A. A. Belavin and V. G. Drinfeld. Triangle equations and simple Lie algebras. In Soviet Sci. Rev. Section C Math. Phys. Rev. 4, Harwood, Chur, Switzerland, (1984), 93-165. · Zbl 0553.58040 |
| [102] | G. Brown. Properties of a 29-dimensional simple Lie algebra of characteristic three. Math. Ann. 261, 487-492 (1982). · Zbl 0485.17005 |
| [103] | K. A. Brown and I. Gordon. Poisson orders, symplectic reflection algebras and representa-tion theory. J. Reine Angew. Math. 559, 193-216 (2003). · Zbl 1025.17007 |
| [104] | C. De Concini and V. Kac. Representations of quantum groups at roots of 1. Operator algebras, unitary representations, enveloping algebras and invariant theory (Paris, 1989), 471-506, Progr. Math. 92, Birkhäuser, Boston (1990). · Zbl 0738.17008 |
| [105] | C. De Concini, V. Kac, C. Procesi. Quantum coadjoint action. J. Am. Math. Soc. 5, 151-189 (1992). · Zbl 0747.17018 |
| [106] | C. De Concini and C. Procesi. Quantum groups. In: D-modules, representation theory and quantum groups, Lecture Notes in Math. 1565 31-140, Springer (1993). · Zbl 0795.17005 |
| [107] | I. Heckenberger. The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164, 175-188 (2006). · Zbl 1174.17011 |
| [108] | I. Heckenberger. Classification of arithmetic root systems. Adv. Math. 220, 59-124 (2009). · Zbl 1176.17011 |
| [109] | V. Kac and B. Weisfeiler. Exponentials in Lie algebras of characteristic p. Math. USSR Izv. 5, 777-803 (1971). Reporter: Thomas Letourmy · Zbl 0252.17003 |
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