Chapter

Generalities

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/9780199662920.003.0001
Generalities

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Generalities refers to the definition of basic concepts. The goal here is to define group, field, vector space and algebra. The chapter starts with the definitions of associativity, identity, inverse and closure and use these to define group. This is followed by the definition of subgroup and invariant subgroup. The cyclic group is introduced to provide an example of a finite group. Next, homomorphism, isomorphism, realization and representation are defined. A canonical description of an N-dimensional vector space is given in terms of Nx1 column matrices. In conclusion, the commutator of two matrices is defined and the Jacobi identity is presented. Biographical notes on Galois, Abel and Jacobi are given.

Keywords: group; field; vector space; algebra; isomorphism; representation; Jacobi identity

Chapter.  2514 words. 

Subjects: Mathematical and Statistical Physics

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