Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On representations of affine Kac-Moody
groups and related loop groups

Author: Yu Chen
Journal: Trans. Amer. Math. Soc. 348 (1996), 3733-3743
MSC (1991): Primary 17B67, 20C15, 22E70
MathSciNet review: 1361638
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type $A_{n}^{(1)}$ for some $n\ge 1$. Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.

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

  • 1. A. Borel, J. Tits, Homomorphismes "abstraits" de groupes algébriques simples, Annals of Math. 97 (1973), 499--571. MR 47:5134
  • 2. R. Carter, Y. Chen, Automorphisms of affine Kac-Moody groups and related Chevalley groups, J. Algebra 155 (1993), 44--94. MR 94k:17032
  • 3. Y. Chen, On rational subgroups of reductive algebraic groups over integral domains, Math. Proc. Camb. Phil. Soc. 117 (1995), 203--212. MR 95m:20051
  • 4. C.Chevalley, Classification des groupes de Lie algébriques, Notes polycopiées, Inst. H. Poincaré, Paris, 1956/58.
  • 5. M. Demazure, A. Grothendiek, Schémas en groupes III, Lecture Notes in Math. 153, Springer-Verlag, Berlin-Heidelberg-New York, 1970. MR 43:223c
  • 6. H. Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 5--136. MR 83a:20057
  • 7. V. Kac, Infinite dimensional Lie algebras, Birkhäuser, Boston, 1983. MR 86h:17015
  • 8. V. Kac, D. Peterson, Regular functions on certain infinite-dimensional groups, Arithmetic and Geometry, Progress in Math. 36, Birkhäuser, Boston, 1983, p. (141--166). MR 86b:17010
  • 9. J. Morita, Tits' systems in Chevalley groups over Laurent polynomial rings, Tsukuba J. Math. 3 (1979), 41--51. MR 82a:20050
  • 10. D. Peterson, V. Kac, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA 80 (1983), 1778--1782. MR 84g:17017
  • 11. A. Pressley, G. Segal, Loop groups, Clarendon Press, Oxford, 1988. MR 88i:22049
  • 12. R. Steinberg, Lectures on Chevalley groups, Yale Univ. Lect. Notes, 1967. MR 57:6215
  • 13. J. Tits, Groupes associés aux algèbres de Kac-Moody, Séminaire Bourbaki, 700, Paris, 1987. MR 91c:22034
  • 14. J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), 542--573. MR 89b:17020

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 17B67, 20C15, 22E70

Retrieve articles in all journals with MSC (1991): 17B67, 20C15, 22E70

Additional Information

Yu Chen
Affiliation: Dipartimento di Matematica, Università di Torino, Via C. Alberto 10, 10123 Torino, Italy

Keywords: Kac-Moody group, loop group, Chevalley-Demazure group scheme, minimal representation
Received by editor(s): August 4, 1995
Additional Notes: Research supported in part by the Italian M.U.R.S.T. and C.N.R.-G.N.S.A.G.A
Dedicated: Dedicated to Professor G. Zacher on the occasion of his seventieth birthday
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society