Chapter

The <i>so</i>(<i>n</i>) algebra and Clifford numbers

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/9780199662920.003.0005
The so(n) algebra and Clifford numbers

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Spinor representations of so(n) algebras are obtained with the help of Clifford numbers. Starting from the definition of SO(n) as the group whose elements are orthogonal unimodular nxn matrices commutation relations that define the so(n) algebra are abstracted from the defining n-dimensional representation. Clifford numbers are defined and shown to be realizable in terms of 2mx2m matrices. Explicit realizations are described. Next, the chapter defines the special Clifford number γ 2m+1 and use it to obtain spinor representations of so(2m+1) and semispinor representations of so(2m). Explicit results are given for so(2), so(3) and so(4). The isomorphism between so(4) and the direct sum of two so(3)s is demonstrated. Biographical notes on Clifford and Schur are given.

Keywords: Clifford numbers; spinor; semispinor

Chapter.  4208 words. 

Subjects: Mathematical and Statistical Physics

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