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Proceedings of the American Mathematical Society
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
MathSciNet review: 1100662
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1100662-5
PII: S 0002-9939(1992)1100662-5
Article copyright: © Copyright 1992 American Mathematical Society