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  "body": "\n## Summary\n\nThe complexification of a real [[Lie group|7JBRGCVJRB]] $G$ is a complex Lie group $G_\\C$ with a homomorphism $G \\to G_\\C$.\nEvery continuous homomorphism of $G$ to a complex Lie group factors through it, in a unique holomorphic way.\nIt is the group analogue of passing from a real Lie algebra $\\mathfrak{g}$ to $\\mathfrak{g} \\otimes_\\R \\C$.\nFor example $\\mathrm{SL}_n(\\C)$ is the complexification of both $\\mathrm{SL}_n(\\R)$ and $\\mathrm{SU}(n)$.\nIt links real groups, compact groups and complex groups, as in Weyl's unitary trick.\n\n## Explanations\n\n### L1 human\n\nA real Lie group is described by real parameters, such as angles or real matrix entries.\nIts complexification lets these parameters become complex numbers.\nFor the circle of angles $e^{i\\theta}$, letting $\\theta$ be complex gives all nonzero complex numbers $\\C^\\times$.\nSo $\\C^\\times$ is the complexification of the circle group.\nAny continuous homomorphism from the circle to a complex Lie group extends in exactly one way to a holomorphic homomorphism on $\\C^\\times$.\n",
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