On the construction of semisimple Lie algebras and Chevalley groups

Geck, Meinolf

doi:10.1090/proc/13600

On the construction of semisimple Lie algebras and Chevalley groups
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by Meinolf Geck

Proc. Amer. Math. Soc. 145 (2017), 3233-3247

DOI: https://doi.org/10.1090/proc/13600

Published electronically: February 22, 2017

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Abstract:

Let $\mathfrak {g}$ be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the “canonical basis” of the adjoint representation of $\mathfrak {g}$. Here, we present a variation of this idea which leads to a new, and quite elementary, construction of $\mathfrak {g}$ itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of $\mathfrak {g}$.

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