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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$ and Lie algebra $\\mathfrak{g} = T_eG$, the [[tangent space|E7A5DHFT01]] at $e$.\nFor $g \\in G$, let $L_g \\colon G \\to G$, $L_g(h) = gh$, be left translation.\nThe *(left) Maurer–Cartan form* is the $\\mathfrak{g}$-valued [[one-form|02Z5B2HA3F]] $\\omega$ on $G$ with\n$$\\omega_g = (dL_{g^{-1}})_g \\colon T_gG \\to \\mathfrak{g},$$\nfor each $g \\in G$, where $(dL_{g^{-1}})_g$ is the [[differential|6P9PXQDSEN]] of $L_{g^{-1}}$ at $g$.\n\n## Remarks\n\nExample: for $G = GL_n(\\R)$, left translation by $g^{-1}$ is the linear map $V \\mapsto g^{-1}V$ of $M_n(\\R)$.\nSo $\\omega_g(V) = g^{-1} V$, written $\\omega = g^{-1}\\, dg$.\nThen $d\\omega = -g^{-1}\\,dg\\, g^{-1} \\wedge dg = -\\omega \\wedge \\omega$, which is the Maurer–Cartan equation for matrix groups.\n\nNon-example: the form $\\omega'_g = (dR_{g^{-1}})_g$, with $R_g(h) = hg$, also takes values in $\\mathfrak{g}$, but it is not the left Maurer–Cartan form.\nFor $GL_n(\\R)$ it is $dg\\, g^{-1}$, and at $g = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 2 \\end{pmatrix}$, $V = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ one gets $V g^{-1} = \\tfrac12 V \\ne V = g^{-1}V$.\nIt is the right Maurer–Cartan form.\n\nThe form $\\omega$ is left-invariant: $L_h^*\\omega = \\omega$ for all $h \\in G$.\nFor each [[left-invariant vector field|SNA3DVPK0C]] $X$, the function $\\omega(X)$ is constant, equal to $X_e$.\nIn general it satisfies $d\\omega + \\tfrac12[\\omega \\wedge \\omega] = 0$.\n",
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