Editing Adjoint representation (section)

== Variants and analogues ==
The adjoint representation can also be defined for [[algebraic group]]s over any field.<!-- even for an elliptic curve? -->{{clarify|give a definition|date=November 2018}}

The '''[[Coadjoint representation|co-adjoint representation]]''' is the [[contragredient representation]] of the adjoint representation.  [[Alexandre Kirillov]] observed that the [[orbit (group theory)|orbit]] of any vector in a co-adjoint representation is a [[symplectic manifold]].  According to the philosophy in [[representation theory]] known as the '''orbit method''' (see also the [[Kirillov character formula]]), the irreducible representations of a Lie group ''G'' should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case of [[nilpotent Lie group]]s.