Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of a Lie group ${\displaystyle G}$ to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras.

The ordinary exponential function of mathematical analysis is a special case of the exponential map when ${\displaystyle G}$ is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.

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