Chapter

Classical 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/9780199662920.003.0012
Classical groups

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Classical groups consist of the orthogonal, the unitary and the symplectic groups. These groups can be defined in terms of linear transformations that leave invariant certain quadratic forms. It follows that these are groups whose elements are matrices. The chapter shows that the following unified approach is possible: the classical groups are unitary groups over a field F; they are the orthogonal group for F = R, unitary group for F = C and unitary symplectic group for F = H. Using this unified approach the dimension and connectivity properties are obtained for all the classical groups in one fell swoop. I also note that several of the generalized orthogonal groups are of particular interest in Physics: the Lorentz group, the de Sitter group and the Liouville group. Biographical notes on Lorentz, de Sitter, Liouville, Maxwell and Thomas are given.

Keywords: classical groups; orthogonal groups; unitary groups; symplectic groups; quadratic form

Chapter.  3105 words. 

Subjects: Mathematical and Statistical Physics

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