Chapter

Casimir operators for orthogonal groups

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/9780199662920.003.0011
Casimir operators for orthogonal groups

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Casimir operators for orthogonal groups are defined. Rank for semisimple groups is defined and shown to equal m for SO(2m) and SO(2m+1). It is shown that there are m independent Casimirs and a set of them is presented in the form of polynomials in the generators of degree 2k, 1 ≤ km. For SO(2m) the Casimir of degree 2m must be replaced in the integrity basis by a Casimir of degree m defined using the invariant ε tensor. This special Casimir plays a crucial role in distinguishing conjugate representations that are inequivalent. Biographical notes on Pfaff are given.

Keywords: Casimir operators; rank; integrity basis; invariant ε tensor

Chapter.  1814 words. 

Subjects: Mathematical and Statistical Physics

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