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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]].\nThe group $G$ is a *compact Lie group* if its underlying topological space is [[compact|0TVJZYB6SH]].\n\n## Remarks\n\nExample: the special unitary group $\\mathrm{SU}(2)$ is the set of complex matrices $\\begin{pmatrix} a \u0026 -\\bar b \\\\ b \u0026 \\bar a \\end{pmatrix}$ with $\\abs{a}^2 + \\abs{b}^2 = 1$.\nIt is a closed subgroup of $\\mathrm{GL}_2(\\C)$, so it is a Lie group.\nAs a space it is the unit sphere $S^3$ in $\\C^2 \\cong \\R^4$, which is closed and bounded, hence compact.\n\nNon-example: the additive group $\\R$ is a Lie group, but it is not compact, since the open cover by the intervals $(-n, n)$ has no finite subcover.\n\nA compact group need not be a Lie group.\nThe group $\\Z_p$ of $p$-adic integers is compact, but each neighbourhood of $0$ contains the nontrivial subgroup $p^k \\Z_p$ for $k$ large, so it is not a Lie group.\n\nEach compact Lie group is isomorphic to a closed subgroup of some unitary group $\\mathrm{U}(n)$.\n",
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