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I am trying to find out the adjoint and coadjoint actions of the semidirect product lie algebra action of [itex]X[/itex] ⋊ [itex]X[/itex] (sometimes the semidirect product sign is also represented as a capital S encapsulated in a cirlce). Where [itex]X[/itex] here represents the set of vector fields on a manifold M.

Now the group multiplication is known for G=H⋊K when H and K are both arbitrary groups or even for P=H⋊V where H is a group and V here represents a vector space.

So what I ask is, since a set of vector fields on a manifold M is not a group would it be justified to consider its semidirect product results (i.e. the group mutliplication in [itex]X[/itex] ⋊ [itex]X[/itex], adjoint and coadjoint actions etc) as specialisations of the general G=H⋊K ?

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# Semidirect Product

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