Semidirect Product Lie Algebra Action of X ⋊ X

[Hazmitaz]

Hello!

I am trying to find out the adjoint and coadjoint actions of the semidirect product lie algebra action of $X$ ⋊ $X$ (sometimes the semidirect product sign is also represented as a capital S encapsulated in a cirlce). Where $X$ 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 $X$ ⋊ $X$, adjoint and coadjoint actions etc) as specialisations of the general G=H⋊K ?

 

[pat_connell]

In other words, can we assume the adjoint and coadjoint actions of a semidirect product lie algebra action of X ⋊ X can be found by taking specialisations from the general group multiplication formulas for G=H⋊K or P=H⋊V? Thank you!