Lie group, exponential

Lie group of type $(E)$

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

Any exponential Lie group is solvable and simply connected and its Lie algebra is an exponential Lie algebra (cf. 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)

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.

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