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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$.\nFor $g \\in G$, the *left translation* by $g$ is the diffeomorphism $L_g \\colon G \\to G$, $L_g(h) = gh$.\nA [[vector field|0299VCW1TH]] $X$ on $G$ is *left-invariant* if\n$$(dL_g)_h (X_h) = X_{gh} \\quad \\text{for all } g, h \\in G,$$\nwhere $(dL_g)_h$ is the [[differential|6P9PXQDSEN]] of $L_g$ at $h$.\n\n## Remarks\n\nExample: on $G = GL_n(\\R)$, open in the space $M_n(\\R)$ of matrices, fix $A \\in M_n(\\R)$ and put $X_g = gA$.\nLeft translation by $g$ is the linear map $h \\mapsto gh$, so its differential is again $v \\mapsto gv$, and $(dL_g)_h(X_h) = g h A = X_{gh}$.\n\nNon-example: on the same group, the field $Y_g = Ag$ with $A \\ne 0$ is not left-invariant in general.\nFor $n = 2$, $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $g = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 2 \\end{pmatrix}$, one gets $(dL_g)_e(Y_e) = gA \\ne Ag = Y_g$.\nIt is right-invariant instead.\n\nThe map $X \\mapsto X_e$ is a linear isomorphism from the space of left-invariant vector fields onto the [[tangent space|E7A5DHFT01]] $T_eG$; the inverse sends $v$ to the field $g \\mapsto (dL_g)_e(v)$.\nThe [[Lie bracket|68FGQEBD6J]] of two left-invariant vector fields is left-invariant.\nSo $T_eG$ inherits a bracket, and with it becomes the Lie algebra of $G$, written $\\mathfrak{g}$ or $\\operatorname{Lie}(G)$.\nEvery left-invariant vector field is complete: its integral curves are defined on all of $\\R$.\n",
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