This is often written as even though for a general Lie group the expression on the right is not the product of three factors in any way. But for a matrix Lie group it is: in this case both as well as are canonically identified with matrices and the expression on the right is the product of these matrices.
Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself
which is simply the Lie bracket
Of a Hopf algebra on itself
Let be a commutative unital ring and be a Hopf -algebra with multiplication , unit map , comultiplication , counit and the antipode map . We can use Sweedler notation . The adjoint action of on is given by
and it makes not only an -module, but in fact a monoid in the monoidal category of -modules (usually called -module algebra).