Abstract
We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSLn(q) and PSp2n(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSUn(2m).
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Andruskiewitsch, N.: An introduction to Nichols algebras. In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes, A. (eds.) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics, pp 135–195. Springer (2017)
Andruskiewitsch, N., Angiono, I.: On Finite dimensional Nichols algebras of diagonal type. Bull. Math. Sci. 7, 353–573 (2017)
Andruskiewitsch, N., Carnovale, G., García, G.A.: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSLn(q). J. Algebra 442, 36–65 (2015)
Andruskiewitsch, N., Carnovale, G., García, G.A.: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II. Unipotent classes in the symplectic groups. Commun. Contemp. Math. 18(4), 35 (2016). Article ID 1550053
Andruskiewitsch, N., Carnovale, G., García, G.A.: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSLn(q), to appear in Rev. Mat. Iberoam
Andruskiewitsch, N., Carnovale, G., García, G.A.: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg Groups, Preprint: arXiv:1812.11566v2
Andruskiewitsch , N., Fantino, F., Graña, M., Vendramin, L.: Finite-dimensional pointed Hopf algebras with alternating groups are trivial. Ann. Mat. Pura Appl. 190(4), 225–245 (2011)
Andruskiewitsch, N., Fantino, F., Graña, M., Vendramin, L.: Pointed Hopf algebras over the sporadic simple groups. J. Algebra 325, 305–320 (2011)
Bourbaki, N.: Groupes et algèbres de Lie Chap. IV, V, VI. Hermann, Paris (1968)
Chang, B.: The conjugate classes of Chevalley groups of type g2. J. Algebra 9, 190–211 (1968)
Cuntz, M., Heckenberger, I.: Finite Weyl groupoids. J. Reine Angew. Math. 702, 77–108 (2015)
Deriziotis, D.I., Michler, G.O.: Character tables and blocks of finite simple triality groups 3D4(q). Trans. Amer. Math. Soc. 303, 39–70 (1987)
Enomoto, H.: The conjugacy classes of Chevalley groups of type g2 over finite fields of characteristic 2 or 3. J. Fac. Sci. Univ. Tokyo, Sect. I 16, 497–512 (1970)
Geck, M.: Generalized Gelfand-Graev characters for Steinberg’s triality groups and their applications. Comm. Algebra 19(12), 3249–3269 (1991)
Himstedt, F.: Character tables of parabolic subgroups of Steinberg’s triality groups 3d4(2n). J. Algebra 316, 254–283 (2007)
Heckenberger, I., Schneider, H.-J.: Nichols algebras over groups with finite root system of rank two i. J. Algebra 324, 3090–3114 (2011)
Heckenberger, I., Vendramin, L.: The classification of Nichols algebras with finite root system of rank two. J. Europ. Math. Soc. 19(7), 1977–2017 (2017)
Humphreys, J.E.: Introduction to Lie algebras and representation theory. Springer GTM, 9 (1972)
Humphreys, JE: Conjugacy classes in semisimple algebraic groups. Amer. Math. Soc. Providence, RI (1995)
Malle, G., Testerman, D.: Linear algebraic groups and finite groups of lie type. Cambridge Studies in Advanced Mathematics, vol. 133 (2011)
Shinoda, K.-I.: The conjugacy classes of Chevalley groups of type (f4) over finite fields of characteristic 2. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 21, 133–159 (1974)
Shoji, T.: The conjugate classes of Chevalley groups of type f4 over finite fields of characteristic p≠ 2. J. Fac. Sci. Univ. Tokyo 21, 1–17 (1974)
Sommers, E.: B-stable ideals in the nilradical of a Borel subalgebra. Can. Math. Bull. 48(3), 460–472 (2005)
Springer, T.A.: Linear Algebraic Groups, 2nd Edition Progress in Math, vol. 9. Birkhäuser, Boston (1998)
Springer, T.A., Steinberg, R.: Conjugacy Classes, Seminar on Algebraic Groups and Related Finite Groups, pp. 167-266, Lect. Notes Math. 131, Springer (1970)
Steinberg, R.: Lectures on Chevalley Groups. Yale University Press, New Haven (1968)
Suzuki, M: Group Theory I, Grundlehren Math. Wiss, vol. 247. Springer, Berlin (1982)
Acknowledgments
At different stages of this project, Mauro Costantini, Paolo Papi and Jay Taylor, helped us with interesting conversations and precise references. We thank them a lot. We thank also the referee for his/her suggestions.
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Presented by: Peter Littelmann
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The work of N. A. was partially supported by CONICET, Secyt (UNC) and the MathAmSud project GR2HOPF. The work of G. C. was partially supported by Progetto di Ateneo dell’Università di Padova: CPDA125818/12. The work of G. A. G. was partially supported by CONICET, Secyt (UNLP) and ANPCyT-Foncyt 2014-0507. The results were obtained during visits of N. A. and G. A. G. to the University of Padova, and of G. C. to the University of Córdoba, partially supported by the bilateral agreement between these Universities, the Erasmus Mundus Action 2 Programme AMIDILA and the Visiting Scientist program of the University of Padova.
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Andruskiewitsch, N., Carnovale, G. & García, G.A. Finite-Dimensional Pointed Hopf Algebras Over Finite Simple Groups of Lie Type IV. Unipotent Classes in Chevalley and Steinberg Groups. Algebr Represent Theor 23, 621–655 (2020). https://doi.org/10.1007/s10468-019-09868-6
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DOI: https://doi.org/10.1007/s10468-019-09868-6