22: Topological groups, Lie groups
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!)
For transformation groups, See 54H15, 57SXX, 58-XX. For abstract harmonic analysis, See 43-XX
Section 22 replaces a section 21 used until 1958.
Browse all (old) classifications for this area at the AMS.
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
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.