The Adjoint Representation
Since Lie groups are groups, they have representations — homomorphisms to the general linear group of some vector space or another. But since is a Lie group, we can use this additional structure as well. And so we say that a representation of a Lie group should not only be a group homomorphism, but a smooth map of manifolds as well.
As a first example, we define a representation that every Lie group has: the adjoint representation. To define it, we start by defining conjugation by . As we might expect, this is the map — that is, . This is a diffeomorphism from back to itself, and in particular it has the identity as a fixed point: . Thus the derivative sends the tangent space at back to itself: . But we know that this tangent space is canonically isomorphic to the Lie algebra . That is, . So now we can define by . We call this the “adjoint representation” of .
To get even more specific, we can consider the adjoint representation of on its Lie algebra . I say that is just itself. That is, if we view as an open subset of then we can identify . The fact that and both commute means that , meaning that and are “the same” transformation, under this identification of these two vector spaces.
Put more simply: to calculate the adjoint action of on the element of corresponding to , it suffices to calculate the conjugate ; then
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