Chapter

Cartan classification of semisimple algebras

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0017
Cartan classification of semisimple algebras

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The Cartan classification of semisimple algebras relies on the observation that the number of ways that one can draw in the r-dimensional Euclidean root space a so-called root diagram is very limited. For rank one there is only one root diagram possible, the corresponding algebra being A 1B 1C 1 in Cartan’s notation. For rank 2 there are four different root diagrams possible, corresponding to the algebras A 2, B 2C 2, D 2A 1A 1 and G 2. By generalizing these one obtains the infinite series An, Bn, Cn and Dn of rank n algebras, which are the classical algebras. G 2 cannot be generalized but other exceptional algebras exist, namely F 4 and E 6, E 7 and E 8.

Keywords: Cartan classification; root diagrams; classical algebras; exceptional algebras

Chapter.  3923 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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