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On the construction of semisimple Lie algebras and Chevalley groups


Author: Meinolf Geck
Journal: Proc. Amer. Math. Soc. 145 (2017), 3233-3247
MSC (2010): Primary 17B45; Secondary 20G40
DOI: https://doi.org/10.1090/proc/13600
Published electronically: February 22, 2017
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Abstract: Let $ \mathfrak{g}$ be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the ``canonical basis'' of the adjoint representation of  $ \mathfrak{g}$. Here, we present a variation of this idea which leads to a new, and quite elementary, construction of $ \mathfrak{g}$ itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of $ \mathfrak{g}$.


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  • [1] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238; MR 0453824
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). MR 0453824
  • [3] Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
  • [4] C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. (2) 7 (1955), 14-66 (French). MR 0073602
  • [5] Claude Chevalley, Certains schémas de groupes semi-simples, Séminaire Bourbaki, Vol. 6, Exp. No. 219, Soc. Math. France, Paris, 1995, pp. 219-234 (French). MR 1611814
  • [6] Karin Erdmann and Mark J. Wildon, Introduction to Lie algebras, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2006. MR 2218355
  • [7] M. Geck, Minuscule weights and Chevalley groups, Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemp. Math., Amer. Math. Soc., Providence, RI, to appear.
  • [8] James E. Humphreys, Introduction to Lie algebras and representation theory, second printing, revised, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. MR 499562
  • [9] Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532
  • [10] George Lusztig, On quantum groups, J. Algebra 131 (1990), no. 2, 466-475. MR 1058558, https://doi.org/10.1016/0021-8693(90)90187-S
  • [11] George Lusztig, Quantum groups at roots of $ 1$, Geom. Dedicata 35 (1990), no. 1-3, 89-113. MR 1066560, https://doi.org/10.1007/BF00147341
  • [12] George Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257-296. MR 1013053, https://doi.org/10.2307/1990988
  • [13] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
  • [14] G. Lusztig, Study of a $ \mathbf {Z}$-form of the coordinate ring of a reductive group, J. Amer. Math. Soc. 22 (2009), no. 3, 739-769. MR 2505299, https://doi.org/10.1090/S0894-0347-08-00603-6
  • [15] G. Lusztig, On conjugacy classes in the Lie group $ E_8$, notes from a talk given at the AMS-RMS joint meeting in Alba Iulia, 2013; arXiv:1309.1382.
  • [16] George Lusztig, The canonical basis of the quantum adjoint representation, J. Comb. Algebra 1 (2017), no. 1, 45-57. MR 3589909, https://doi.org/10.4171/JCA/1-1-2
  • [17] Liangang Peng and Jie Xiao, Root categories and simple Lie algebras, J. Algebra 198 (1997), no. 1, 19-56. MR 1482975, https://doi.org/10.1006/jabr.1997.7152
  • [18] Konstanze Rietsch, The infinitesimal cone of a totally positive semigroup, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2565-2570. MR 1401752, https://doi.org/10.1090/S0002-9939-97-03931-2
  • [19] Claus Michael Ringel, Hall polynomials for the representation-finite hereditary algebras, Adv. Math. 84 (1990), no. 2, 137-178. MR 1080975, https://doi.org/10.1016/0001-8708(90)90043-M
  • [20] Jean-Pierre Serre, Complex semisimple Lie algebras, translated from the French by G. A. Jones, reprint of the 1987 edition. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. MR 1808366
  • [21] Robert Steinberg, Lectures on Chevalley groups, notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. MR 0466335
  • [22] J. Tits, Sur les constantes de structure et le théorème d'existence des algèbres de Lie semi-simples, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 21-58 (French). MR 0214638

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Additional Information

Meinolf Geck
Affiliation: IAZ - Lehrstuhl für Algebra, Universität Stuttgart, Pfaffenwaldring 57, D–70569 Stuttgart, Germany
Email: meinolf.geck@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/proc/13600
Received by editor(s): May 2, 2016
Received by editor(s) in revised form: September 1, 2016
Published electronically: February 22, 2017
Dedicated: To George Lusztig on his $70$th birthday
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2017 American Mathematical Society

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