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On the structure of Kac–Moody algebras

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  20 May 2020

Timothée Marquis
Timothée Marquis*
Affiliation:
Université catholique de Louvain, IRMP, Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique
*
e-mail: timothee.marquis@uclouvain.be
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Abstract

Let A be a symmetrisable generalised Cartan matrix, and let $\mathfrak {g}(A)$ be the corresponding Kac–Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak {g}(A)$: given two homogeneous elements $x,y\in \mathfrak {g}(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak {g}(A)$.

Keywords

Kac-Moody algebrassolvable subalgebrasnilpotent subalgebras

MSC classification

Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Secondary: 17B30: Solvable, nilpotent (super)algebras

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1124 - 1152
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

F.R.S.-FNRS post-doctoral researcher.

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