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  "body": "\n## Statement\n\nLet $G$ and $H$ be real [[Lie groups|7JBRGCVJRB]] with Lie algebras $\\mathfrak{g}$ and $\\mathfrak{h}$, and [[exponential maps|H3QX1ZFC8K]] $\\exp$.\n\n1. For every Lie group homomorphism $\\varphi \\colon G \\to H$, the differential $d\\varphi_e \\colon \\mathfrak{g} \\to \\mathfrak{h}$ is a Lie algebra homomorphism, and $\\varphi(\\exp X) = \\exp(d\\varphi_e X)$ for all $X \\in \\mathfrak{g}$.\n2. If $G$ is connected, a Lie group homomorphism $\\varphi \\colon G \\to H$ is determined by $d\\varphi_e$.\n3. If $G$ is connected and simply connected, each Lie algebra homomorphism $\\psi \\colon \\mathfrak{g} \\to \\mathfrak{h}$ is $d\\varphi_e$ for a unique homomorphism $\\varphi \\colon G \\to H$.\n4. The map $K \\mapsto \\operatorname{Lie}(K)$ is a bijection from the connected [[Lie subgroups|NM8XCPS1Q3]] $K$ of $G$ onto the Lie subalgebras of $\\mathfrak{g}$.\n5. Every finite-dimensional real Lie algebra is the Lie algebra of a connected and simply connected Lie group, unique up to isomorphism.\n\nHere simply connected means that the [[fundamental group|S1XK3HXHFX]] is trivial.\n\n## Remarks\n\nPart 1 follows from the chain rule, since $\\varphi$ maps one-parameter subgroups to one-parameter subgroups.\nPart 2 holds because a connected Lie group is generated by any neighbourhood of $e$, and $\\exp$ covers a neighbourhood of $e$.\nPart 3 is proved by integrating the graph of $\\psi$, a subalgebra of $\\mathfrak{g} \\oplus \\mathfrak{h}$, to a connected Lie subgroup of $G \\times H$ by part 4.\nIts projection to $G$ is a covering map, hence an isomorphism since $G$ is simply connected.\nPart 4 rests on the Frobenius theorem applied to the left-invariant distribution spanned by a subalgebra.\nPart 5 is [[Lie's third theorem|P6QNEK44RX]].\nSimple connectivity in part 3 is necessary.\nThe identity map of $\\R$ is the differential of no homomorphism from the circle group $\\R / \\Z$ to $\\R$, since the only such continuous homomorphism is trivial.\n\n## Explanations\n\n### L1 human\n\nEvery Lie group has a Lie algebra, a linear object that records how the group looks near its identity.\nThis theorem says that, for connected groups, almost all information about homomorphisms and subgroups can be read off the Lie algebra.\nFor simply connected groups the dictionary is perfect: Lie groups and Lie algebras then carry the same information.\n\n### L2 human\n\nIn practice one computes with Lie algebras and lifts the answers to groups.\nFor example, a representation of $\\mathfrak{su}(2)$ gives a representation of the simply connected group $SU(2)$ by part 3, while for $SO(3)$ only some of them lift.\nParts 3 and 5 together say that $G \\mapsto \\mathfrak{g}$ is an equivalence of categories between connected simply connected real Lie groups and finite-dimensional real Lie algebras.\nA connected Lie group with Lie algebra $\\mathfrak{g}$ is the quotient of the simply connected one by a discrete central subgroup.\n",
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