Lie groups / P. M. CohnType de document : MonographieCollection : Cambridge tracts in mathematics and mathematical physics, 46Langue : anglais.Pays : Grande Bretagne.Éditeur : Cambridge : Cambridge University Press, 1965Description : 1 vol. (VI-164 p.) : appendix ; 22 cmISBN : 9780521092982.ISSN : 0068-6824.Bibliographie : Notes bibliogr. Index.Sujet MSC : 22Exx, Topological groups, Lie groups, Lie groups
22-01, Topological groups, Lie groups, Instructional exposition (textbooks, tutorial papers, etc.)En-ligne : Zentralblatt | MathSciNet
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Notes bibliogr. Index
This is a concise and elementary textbook on the beginnings of the theory of Lie groups, covering the so-called Lie's fundamental theorems. It also includes the theory of universal covering groups. The readers are required to have only elementary knowledge about groups, vector spaces, topology, and analysis. Numbers of examples are inserted to help the readers.
Chapter 1 introduces the elementary notions in the theory of (real) analytic manifolds. Chapter 2 introduces the definitions of topological groups, Lie groups, local Lie groups, analytic subgroups of Lie groups, followed by discussions on one-parameter subgroups.
Chapter 3 treats algebras of infinitesimal right translations, that is, infinitesimal transformations which are left invariant, of Lie groups and of analytic subgroups, in particular one-parameter subgroups. These are also generalized to the case of Lie groups of transformations. Chapter 4 treats differential forms, exterior differentiation, their behavior under analytic mappings, Maurer-Cartan forms (or left-invariant forms), and their relations with infinitesimal right translations.
Chapter 5 is concerned with Lie's fundamental theorems. They are formulated roughly in the following fashion. The first and its converse: The multiplication functions of local Lie groups are characterized by the fact that they satisfy a certain type of partial differential equation. The second and its converse: The necessary and sufficient conditions for a set of infinitesimal transformations to be the algebra of infinitesimal translations of a local Lie group. The third and its converse: The abstract algebraic characterization of algebras of infinitesimal translations of local Lie groups and the determination up to local isomorphism of a local Lie group by its algebra.
The main subjects discussed in Chapter 6 are the following: The canonical charts; the fact that continuous homomorphisms of Lie groups are analytic; the correspondence between analytic subgroups of a Lie group and subalgebras of its Lie algebra; the fact that the closed subgroups of Lie groups are analytic; homomorphism, kernel and quotient of Lie groups; homomorphism of Lie algebras and their relations with homomorphism of Lie groups; the adjoint representation of a Lie group.
The final chapter is about the universal covering group. The book closes with an appendix, in which a theorem on the integration of a system of total differential equations used in Chapter 4 is proved. (MathSciNet)