Compact group

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In the following we will assume all groups are Hausdorff spaces.

Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include

The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.

Given any compact Lie group G one can take its identity component G₀, which is connected. The quotient group G/G₀ is the group of components π₀(G) which must be finite since G is compact. We therefore have a finite extension

Meanwhile, for connected compact Lie groups, we have the following result:

Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.)

Finally, every compact, connected, simply-connected Lie group K is a product of finitely many compact, connected, simply-connected simple Lie groups K_i each of which is isomorphic to exactly one of the following: