An abelian Lie group has commutative multiplication. Its Lie algebra is an abelian Lie algebra, and its Lie exponential map is a group homomorphism from the additive Lie algebra. Conversely, an additive Lie exponential map forces the identity component of a Lie group to be abelian, but does not force all disconnected components to commute.
An **Abelian Lie group** is a type of Lie group in which the group operation is commutative. This means that for any two elements \( g \) and \( h \) in the group \( G \), the following property holds: \[ g \cdot h = h \cdot g \] where \( \cdot \) represents the group operation.
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