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On the construction of semisimple Lie algebras and Chevalley groups


Author: Meinolf Geck
Journal: Proc. Amer. Math. Soc. 145 (2017), 3233-3247
MSC (2010): Primary 17B45; Secondary 20G40
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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Meinolf Geck
Affiliation: IAZ - Lehrstuhl für Algebra, Universität Stuttgart, Pfaffenwaldring 57, D–70569 Stuttgart, Germany
Email: meinolf.geck@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/proc/13600
Received by editor(s): May 2, 2016
Received by editor(s) in revised form: September 1, 2016
Published electronically: February 22, 2017
Dedicated: To George Lusztig on his $70$th birthday
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2017 American Mathematical Society