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https://www.ias.ac.in/article/fulltext/pmsc/118/03/0351-0356

Infinite dimensional Lie algebras; split Lie algebras; roots.

We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such algebras 𝐿 is of the form $L=\mathcal{U}+\sum_j I_j$ with $\mathcal{U}$ a subspace of the abelian Lie algebra 𝐻 and any $I_j$ a well described ideal of 𝐿, satisfying $[I_j,I_k]=0$ if $j\neq k$. Under certain conditions, the simplicity of 𝐿 is characterized and it is shown that 𝐿 is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.

Antonio J Calderón Martín¹

1. Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Cádiz, Spain

Published

August 2008

Manuscript received

24 May 2007