= Lie group–Lie algebra correspondence
{c}
{title2=$G\longmapsto T_eG,\quad\phi\longmapsto d_e\phi$}
Differentiation sends <Lie groups> to <Lie algebras> and smooth <group homomorphisms> to bracket-preserving <linear maps>. On connected <simply connected> <Lie groups>, this is an equivalence with finite-dimensional real <Lie algebras>: the <Lie third theorem> gives existence, and <integration of a Lie algebra homomorphism> gives unique maps. General connected <groups> require discrete central quotient data, and disconnected <groups> additionally require component and conjugation data.
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