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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] and let $k$ be a field.\nThe *cohomology* of $G$ with coefficients in $k$ is the graded $k$-algebra $H^*(G; k) = \\bigoplus_{n \\ge 0} H^n(G; k)$.\nHere $H^n(G; k)$ is the [[singular cohomology|6N7BC70CX0]] of the underlying manifold of $G$, and the product is the [[cup product|JX47MS1R1H]].\nWhen $G$ is connected and each $H^n(G; k)$ is finite-dimensional, the multiplication $\\mu \\colon G \\times G \\to G$ also gives a coproduct $\\mu^*$, which makes $H^*(G; k)$ a [[Hopf algebra|WJXK0V9NE9]].\n\n## Remarks\n\nExample: $G = S^1$, the circle group.\nThen $H^0(S^1; \\R) = \\R$, $H^1(S^1; \\R) = \\R x$, and all other groups vanish.\nSo $H^*(S^1; \\R)$ is the exterior algebra on one generator $x$ of degree $1$, with $\\mu^*(x) = x \\otimes 1 + 1 \\otimes x$.\n\nNon-example: the cohomology of $G$ as an abstract group, $H^*(BG^\\delta; k)$, or the continuous cohomology of $G$, is a different object.\nFor $G = S^1$ the continuous cohomology with coefficients in $\\R$ vanishes in positive degrees, while $H^1(S^1; \\R) \\ne 0$.\n\nFor $k = \\R$, the de Rham theorem identifies $H^*(G; \\R)$ with the [[de Rham cohomology|GX74C07T0G]] of $G$.\nIf $G$ is compact and connected, averaging over $G$ shows that each class has a left- and right-invariant representative.\nThis gives an isomorphism $H^*(G; \\R) \\cong H^*(\\mathfrak{g}; \\R)$ with the Lie algebra cohomology of the Lie algebra $\\mathfrak{g}$.\n",
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