A Survey of Lie Groups and Lie Algebra with Applications and by Johan G. F. Belinfante

By Johan G. F. Belinfante

During this reprint version, the nature of the e-book, particularly its concentrate on classical illustration thought and its computational elements, has now not been replaced

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Conversely, every subalgebra of the Lie algebra of a Lie group may be identified with the Lie algebra of a connected Lie subgroup. Every closed subgroup of a Lie group is a Lie subgroup. However, a Lie subgroup of a Lie group need not be a closed subgroup. From the theory of topological groups it is known that every open subgroup is closed, and hence open subgroups of Lie groups are also Lie subgroups [64]. Similarly, a homomorphism between Lie groups in general need not be an analytic mapping.

Analytic homomorphisms of Lie groups induce homomorphisms of the corresponding Lie algebras. The homomorphism induced by a composition of analytic homomorphisms is the composition of their induced homomorphisms. This relation between homomorphisms of Lie groups and Lie algebras can be used to translate many results about Lie groups into related results about their corresponding Lie algebras. For example, the inclusion mapping for a Lie subgroup is an analytic homomorphism, and hence induces the corresponding embedding for their Lie algebras noted above.

This is because by multiplying together many elements very near to the identity element, we can obtain elements further away. Also, any neighborhood of the identity can be transported along any arc in arbitrarily small steps, much as one does analytic continuation. One may therefore ask whether the whole Lie group is determined by its Lie algebra, the answer being yes, provided that the Lie group is simply-connected [54], [192]. ) Each real Lie algebra is isomorphic to the Lie algebra of a simply-connected Lie group, which is uniquely determined to within an isomorphism.

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