Dynkin diagrams

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI:
Dynkin diagrams

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Dynkin diagrams are described and used to show that the only allowed algebras are the ones described in the preceding chapter. This is accomplished by showing that Dynkin diagrams can contain no loops, no more than three lines can emanate from a circle and, finally, that a set of circles connected by a single line can be shrunk to a single circle. This approach is alternate to exploiting the properties of the Cartan matrix with its Cartan integers. The ADE algebras are collectively referred to as simply laced and it is noted that the A-D-E classification arises in a number of other situations. In conclusion, symmetries of the Dynkin diagrams are discussed and folding is defined.

Keywords: Dynkin diagrams; Cartan matrix; Cartan integers; a-d-e classification; simply laced; folding

Chapter.  2961 words. 

Subjects: Mathematical and Statistical Physics

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