# Lie groups and Lie algebras

# Lie groups and Lie algebras

Lie groups and Lie algebras are defined. *SU*(2), the group whose elements are 2x2 unitary unimodular matrices is described providing an example of a 3-dimensional Lie group. Infinitesimal generators are defined and used to provide a basis for a vector space that leads to the Lie algebra. Structure constants are introduced and shown to provide the so-called adjoint representation. The Cartan metric tensor is defined and Cartan’s criterion for semisimplicity is described. After showing that *SU*(2) is compact *SL*(2,*R*) — the group whose elements are 2x2 real unimodular matrices — is introduced to give an example of a non-compact group. Biographical notes on Euler, Lie and Cartan are given.

*Keywords:*
Lie groups, Lie algebras, infinitesimal generators, structure constants, adjoint representation, Cartan metric

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .