Characterization of Lie groups on the cotangent bundle of a Lie group (Q1085450)

The author gives a simple characterization, in differential geometric terms, of the groups K of dimension 2n which can be interpreted as semidirect products of an n-dimensional Lie group G by the group of translations of the dual space of its Lie algebra \(g^*\). In this case the natural symplectic 2-form F of K is always right invariant by G and the canonical cotangent group \((T^*G)_ c\) corresponding to the coadjoint representation c of \(G^ c\) into \(g^*\) can be characterized by a left invariance of this form under G. Finally it is shown that the canonical cotangent group \((T^*G)_ c\) has an invariant symplectic structure if and only if G is Abelian.