Lie Groups and Lie Algebras

Avishek Adhikari () and Mahima Ranjan Adhikari
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Avishek Adhikari: Presidency University, Department of Mathematics
Mahima Ranjan Adhikari: Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)

Chapter Chapter 4 in Basic Topology 2, 2022, pp 291-353 from Springer

Abstract: Abstract This chapter conveys the basic principles governing the theory of Lie groups. A Lie group is a combination of three simultaneously different structures such as abstract group structure together with topological and manifold structures which are interrelated with each other by smooth functions. Lie groups consist of two most important special families: a family of differentiable manifolds and a family of topological groups. Their important examples include classical matrix groups $$ GL (n, \textbf{R}), ~ O (n, \textbf{R}), U (n, \textbf{C}), SL (n, \textbf{R}),$$ G L ( n , R ) , O ( n , R ) , U ( n , C ) , S L ( n , R ) , $$SL (n, \textbf{C}) $$ S L ( n , C ) and their Hermitian analogues. Almost all important groups of geometry and analysis are Lie groups. The theory of Lie groups studies topological groups, differentiable manifolds and Lie algebra. It establishes a relationship between a Lie group and its associated Lie algebra, especially, its Lie algebra of left-invariant vector fields, correspondence between subgroups of a Lie group and subalgebras of the associated Lie algebra and also correspondence between homomorphisms of Lie groups and homomorphisms of the associated Lie algebras. In this way, this theory provides a key link between Lie groups and Lie algebra. This link facilitates a study of Lie theory.

Date: 2022
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DOI: 10.1007/978-981-16-6577-6_4

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