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  "body": "\n## Definition\n\nA *real Lie group* is a [[group|G5S1M9]] $G$ that is also a finite-dimensional [[smooth manifold|F45VQ47GPV]], such that:\n\n1. the multiplication map $\\mu \\colon G \\times G \\to G$, $\\mu(g, h) = gh$, is [[smooth|0V33BNKZ9S]];\n2. the inversion map $\\iota \\colon G \\to G$, $\\iota(g) = g^{-1}$, is smooth.\n\nA *complex Lie group* is defined in the same way with a complex manifold and holomorphic maps.\nA *homomorphism of Lie groups* is a group homomorphism that is a smooth map.\n\n## Remarks\n\nExample: the circle $S^1 = \\set{z \\in \\C : \\abs{z} = 1}$ with complex multiplication.\nIt is a smooth submanifold of $\\C = \\R^2$, and in the angle coordinate the product and the inverse are $(\\theta, \\varphi) \\mapsto \\theta + \\varphi$ and $\\theta \\mapsto -\\theta$, which are smooth.\n\nExample: $GL_n(\\R)$ is an open subset of $\\R^{n^2}$, and the product of matrices is polynomial in the entries.\nBy Cramer's rule, the inverse is a rational function of the entries with nonvanishing denominator $\\det$.\n\nNon-example: the additive group $\\Q$ with the topology induced from $\\R$ is a topological group but not a Lie group.\nIt is not locally Euclidean, since no point of $\\Q$ has a neighbourhood homeomorphic to an open set of some $\\R^n$.\n\nThe two clauses can be merged: $G$ is a Lie group exactly when the map $(g, h) \\mapsto g h^{-1}$ is smooth.\nThe dimension of $G$ is its dimension as a manifold.\n",
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