Compact Lie groups are smooth manifolds that are also groups, which are both compact and have a finite-dimensional Lie algebra. These groups have a rich structure and play a significant role in various areas such as representation theory, topology, and physics, due to their nice properties like every representation being completely reducible and the existence of a Haar measure. Compact Lie groups can be analyzed through harmonic analysis and their associated representations, connecting them to important concepts in representation theory.
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