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  "body": "\n## Summary\n\nA compact Lie group is a [[Lie group|7JBRGCVJRB]] whose underlying manifold is compact.\nExamples are the circle, the tori, the orthogonal groups $\\mathrm{O}(n)$ and the unitary groups $\\mathrm{U}(n)$.\nThese groups carry a finite invariant measure, so averaging over the group works as for finite groups.\nAs a result their representation theory is as clean as that of finite groups.\nTheir structure is described by [[maximal tori|3JVGZVVTB0]], root systems and Weyl groups.\n\n## Explanations\n\n### L1 human\n\nA compact Lie group is a group of symmetries that is also a smooth, closed and bounded shape.\nThe rotations of the plane form a circle, and the circle is the simplest example.\nThe rotations of space form the group $\\mathrm{SO}(3)$, a compact space of dimension $3$.\nBecause the group is bounded, one can take the average of a function over all its elements.\nThis averaging trick makes many questions about these groups as easy as for finite groups.\n",
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