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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 22: Topological groups, Lie groups

Introduction

Lie groups are an important special branch of group theory. They have algebraic structure, of course, and yet are also subsets of space, and so have a geometry; moreover, portions of them look just like Euclidean space, making it possible to do analysis on them (e.g. solve differential equations). Thus Lie groups and other topological groups lie at the convergence of the different areas of pure mathematics. (They are quite useful in application of mathematics to the sciences as well!)

History

Applications and related fields

For transformation groups, See 54H15, 57SXX, 58-XX. For abstract harmonic analysis, See 43-XX

Subfields

• 22A: Topological and differentiable algebraic systems, For topological rings and fields, see 12JXX, 13JXX, 16W80; for dual spaces of operator algebras and topological groups, See 47D35
• 22B: Locally compact abelian groups (LCA groups)
• 22C05: Compact groups
• 22D: Locally compact groups and their algebras
• 22E: Lie groups, For the topology of Lie groups and homogeneous spaces, see 57-XX, 57SXX, 57TXX; for analysis thereon, See 43-XX, 43A80, 43A85, 43A90
• 22F: Noncompact transformation groups [new in 2000]

Section 22 replaces a section 21 used until 1958.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Survey articles: Montgomery, Deane: "What is a topological group?", Amer. Math. Monthly 52, (1945). 302--307. MR7,114e

Morris, Sidney A. "Duality and structure of locally compact abelian groups ... for the layman", Math. Chronicle 8 (1979), 39--56. MR81a:22003

There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period. Topological groups are covered well, Lie groups per se hardly at all.

Bourbaki, Nicolas, "Lie groups and Lie algebras. Chapters 1--3" Springer-Verlag, Berlin-New York, 1989. 450 pp. ISBN3-540-50218-1 MR89k:17001

L. S. Pontrjagin "Topological groups", (English translation) Gordon and Breach, New York, 1966; MR 34 #1439

Software and tables

Index to relevant programs from Computer Physics Communications since 1969 (mostly representation theory).

Schur, a stand alone C program for interactively calculating properties of Lie groups and symmetric functions.

Other web sites with this focus

• Here are the AMS and Goettingen resource pages for area 22.

Selected topics at this site

• What are Lie Groups and how are they used in mathematics and physics?
• How do Lie groups and algebras fit in among the branches of mathematics?
• Example of connection between a Lie group and its Lie algebra -- SO(3)
• Connections among algebraicity, centers, and the fundamental group for Lie groups.
• The Baker-Campbell-Hausdorff formula relating products in a Lie group and in its Lie algebra (and the Poincaré-Birkhoff-Witt theorem).
• Comparison of semisimple, reductive, etc. for Lie algebras and groups
• Local isomorphisms (covering maps) of Lie groups and the Spin groups
• Generators for the symplectic group.
• Constructions of representations of SU(5), SO(5)