representation, ∞-representation?
symmetric monoidal (∞,1)-category of spectra
An adjoint action is an action by conjugation .
The adjoint action of a group on itself is the action given by
The adjoint action of a Lie group on its Lie algebra is for each the derivative of this action in the second argument at the neutral element of
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.
Since this is a linear action, it is called the adjoint representation of a Lie group. The associated bundles with respect to this representation are called adjoint bundles.
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
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).
Last revised on August 28, 2014 at 04:06:52. See the history of this page for a list of all contributions to it.