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  "body": "\n## Example\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] and let $S^1$ be the circle.\nThe *loop group* of $G$ is the set\n$$LG = \\set{\\gamma \\colon S^1 \\to G : \\gamma \\text{ smooth}},$$\nwith pointwise multiplication $(\\gamma\\delta)(z) = \\gamma(z)\\delta(z)$.\nIt is a [[group|G5S1M9]].\nThe pointwise product and the pointwise inverse $z \\mapsto \\gamma(z)^{-1}$ of smooth loops are smooth, because multiplication and inversion of $G$ are smooth.\nThe identity element is the constant loop at $e$, and associativity holds pointwise.\n\nFor compact $G$, $LG$ is an infinite-dimensional Lie group and a [[Fréchet manifold|EBMCFQT0JN]].\nIts Lie algebra is the loop algebra $L\\mathfrak{g}$ of smooth maps $S^1 \\to \\mathfrak{g}$ with the pointwise bracket.\n\n## Remarks\n\nThe constant loops form a copy of $G$ inside $LG$.\nFor $G$ the [[circle group|04EAFVFTF7]] $U(1)$, a loop $\\gamma \\colon S^1 \\to U(1)$ has a winding number, and $LU(1)$ has one connected component for each integer.\nCentral extensions of loop groups of compact Lie groups are closely tied to affine Lie algebras and their representations.\n\n## Explanations\n\n### L1 human\n\nA loop group consists of all smooth closed paths in a Lie group, where two paths are multiplied by multiplying their values at each moment.\nIn the circle group, take a loop that runs once around and a loop that runs once around the other way.\nTheir product is a loop of winding number $0$.\nLoop groups are among the best understood infinite-dimensional groups, and they appear in conformal field theory and the theory of integrable systems.\n",
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