Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Projections in Kac-Moody Lie algebras


Authors: Kailash C. Misra and Mohan S. Putcha
Journal: Proc. Amer. Math. Soc. 116 (1992), 351-359
MSC: Primary 17B67; Secondary 22E67
DOI: https://doi.org/10.1090/S0002-9939-1992-1100662-5
MathSciNet review: 1100662
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathfrak{g}$ be a Kac-Moody Lie algebra and $ \mathcal{B}$ its set of positive Borel subalgebras. If $ \mathfrak{b} \in \mathcal{B}$ and $ \mathfrak{p}$ is a parabolic subalgebra, let $ {\operatorname{proj} _\mathfrak{p}}(\mathfrak{b}) = \mathfrak{p} \cap \mathfrak{b} + {\mathfrak{r}_n}(\mathfrak{p})$ where $ {\mathfrak{r}_n}(\mathfrak{p})$ denotes the nilradical of $ \mathfrak{p}$. In this paper we consider the idempotent maps $ {E_{\mathfrak{p},{\mathfrak{p}^ - }}} = {\operatorname{proj} _\mathfrak{p}} \circ {\operatorname{proj} _{{\mathfrak{p}^ - }}}:\mathcal{B} \to \mathcal{B}$, where $ \mathfrak{p}$ and $ {\mathfrak{p}^ - }$ are opposite parabolic subalgebras with $ \mathfrak{p}$ being of positive type. We consider the semigroup $ M = M(\mathfrak{g})$ generated (with respect to composition) by the maps $ {E_{\mathfrak{p},{\mathfrak{p}^ - }}}$. In particular we show that the maximal subgroups of $ M$ are closely related to proper Levi subgroups of the Kac-Moody group associated with $ \mathfrak{g}$.


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

  • [1] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55-150. MR 0207712 (34:7527)
  • [2] V. G. Kac, Infinite dimensional Lie algebras, 2nd ed., Cambridge Univ. Press, Cambridge, 1985. MR 823672 (87c:17023)
  • [3] V. G. Kac and D. H. Peterson, Defining relations of certain infinite dimensional groups, Elie Cartan et les Mathématiques d'Aujourd'hui, Astérisque 1985, 165-208. MR 837201 (87j:22027)
  • [4] D. H. Peterson and V. G. Kac, Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), 1778-1782. MR 699439 (84g:17017)
  • [5] M. S. Putcha, The monoid generated by projections in an algebraic group, J. Algebra 128 (1989), 45-54. MR 1031911 (91a:20049)
  • [6] J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., vol. 386, Springer-Verlag, New York, 1974. MR 0470099 (57:9866)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 17B67, 22E67

Retrieve articles in all journals with MSC: 17B67, 22E67


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1100662-5
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society