Coadjoint representation

Definition for a Lie group over a topological field
The coadjoint representation of a Lie group $$G$$ over a topological field $$K$$ is defined as the defining ingredient::contragredient representation to the defining ingredient::adjoint representation of $$G$$. Note that if $$\mathfrak{g}$$ denotes the Lie algebra of $$G$$, the adjoint representation of $$G$$ is on $$\mathfrak{g}$$ and the coadjoint representation is on the dual vector space $$\mathfrak{g}^*$$.

Definition for a Lazard Lie group
The coadjoint representation of a Lazard Lie group, viewed as simply being over $$\mathbb{Z}$$, is its natural representation on the Pontryagin dual of its Lie algebra that arises from the adjoint action.