View source for Lie group, exponential

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''Lie group of type $(E)$''

A real finite-dimensional [[Lie group|Lie group]] $G$ for which the [[Exponential mapping|exponential mapping]] $\exp\colon \mathfrak{g} \to G$, where $\mathfrak{g}$ is the [[Lie algebra|Lie algebra]] of $G$, is a [[Diffeomorphism|diffeomorphism]].

Any exponential Lie group is solvable and simply connected and its Lie algebra is an exponential Lie algebra (cf. [[Lie algebra, exponential|Lie algebra, exponential]]). The class of exponential Lie groups is closed with respect to passing to connected subgroups, forming quotient groups by a connected normal subgroup and forming finite direct products, but it is not closed with respect to extensions. A supersolvable Lie group (in particular, a nilpotent Lie group) is exponential if it is simply-connected (cf. [[Lie group, supersolvable|Lie group, supersolvable]])

The intersection of connected subgroups of an exponential Lie group is connected. The centralizer of an arbitrary subset is also connected. A simply-connected Lie group is exponential if and only if it has no quotient groups containing the universal covering group of motions of the Euclidean plane as a subgroup.

====References====
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|valign="top"|{{Ref|Be}}|| P. Bernal,   et al.,   "Répresentation des groupes de Lie résolubles" , Dunod  (1972)
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 |valign="top"|{{Ref|Di}}|| J. Dixmier,   "L'application exponentielles dans les groupes de Lie résolubles"  ''Bull. Soc. Math. France'' , '''85'''  (1957)  pp. 113–121
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|valign="top"|{{Ref|Sa}}|| M. Saitô,   "Sur certains groupes de Lie résolubles I, II"  ''Sci. Papers Coll. Gen. Educ. Univ. Tokyo'' , '''7'''  (1957)  pp. 1–11; 157–168
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