Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Study of a $\mathbf Z$-form of the coordinate ring of a reductive group

Author: G. Lusztig
Journal: J. Amer. Math. Soc. 22 (2009), 739-769
MSC (2000): Primary 20G99
Published electronically: March 31, 2008
MathSciNet review: 2505299
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show how the theory of canonical bases in modified universal enveloping algebras can be used to develop the theory of Chevalley groups over any commutative ring with $1$.

References [Enhancements On Off] (What's this?)

  • Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
  • Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
  • C. Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955), 14–66 (French). MR 73602, DOI
  • Claude Chevalley, Certains schĂ©mas de groupes semi-simples, SĂ©minaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 219, 219–234 (French). MR 1611814
  • [DG]DG M. Demazure and A. Grothendieck, SchĂ©mas en groupes, SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie 1962/64 (SGA3), Lecture Notes in Mathematics 151–153, Springer Verlag, 1970.
  • Anthony Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR 1315966
  • Bertram Kostant, Groups over $Z$, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 90–98. MR 0207713
  • George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, BirkhĂ€user Boston, Inc., Boston, MA, 1993. MR 1227098
  • George Lusztig, Quantum groups at $v=\infty $, Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993) Progr. Math., vol. 131, BirkhĂ€user Boston, Boston, MA, 1995, pp. 199–221. MR 1373004
  • Ya. S. SoÄ­belâ€Čman, Algebra of functions on a compact quantum group and its representations, Algebra i Analiz 2 (1990), no. 1, 190–212 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 1, 161–178. MR 1049910
  • Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 20G99

Retrieve articles in all journals with MSC (2000): 20G99

Additional Information

G. Lusztig
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
MR Author ID: 117100

Received by editor(s): September 19, 2007
Published electronically: March 31, 2008
Additional Notes: The author was supported in part by the National Science Foundation
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.