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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