Compact group - Wikipedia

Abstraction: Topological groups with compact topology and their complete representation theory

Key points:

• Every compact group carries a unique (normalized) Haar measure, making it unimodular; this enables averaging arguments analogous to finite group sums
• Peter-Weyl theorem: L2(K) decomposes as orthogonal direct sum of finite-dimensional matrix-entry spaces for all irreducible representations
• Classification of compact connected simply-connected Lie groups: finite products of USp(n), SU(n), Spin(n), and five exceptional groups G2, F4, E6, E7, E8
• Irreducible finite-dimensional representations classified by dominant analytically integral highest weights (theorem of the highest weight)
• Weyl character formula gives the character of a representation explicitly from its highest weight; used as a key step in classifying representations
• Non-Lie compact groups (e.g., profinite groups, Galois groups) are inverse limits of compact Lie groups via the Peter-Weyl theorem

Connections: Representation Theory · Group Theory · Lie Algebras · Harmonic Analysis

Source: http://en.wikipedia.org/wiki/Compact_group