User twistor59 has addressed the part regarding the "generator" terminology, but let me give a bit more detail on the second part of the question. I'm going to restrict the discussion to matrix Lie groups for simplicity.
Some background.
Given a Lie group G with Lie algebra g, there exist two mappings Ad and ad, both are called "adjoint." In particular for all g∈G and for all X,Y∈g, we define Adg:g→g and adX by
Adg(X)=gXg−1,adX(Y)=[X,Y]
The mapping
Ad which takes an element
g∈G and maps it to
Adg is a representation of
G acting on
g, while the mapping
ad which takes an element
X∈g and maps it to
adX is a representation of
g acting on itself.
In other words, Ad is a Lie group representation while ad is a Lie algebra representation, but they both act on the Lie algebra which is a vector space.
Aside.
In response to user Christoph's comment below. Note that if we define the conjugation operation conj by
conjg(h)=ghg−1
Then for matrix Lie groups (which I initially stated I was restricting the discussion to for simplicity) we have
ddt|t=0conjg(etX)=AdgX
Addressing the question.
Having said all of this, in my experience (in high energy theory), physicists usually are referring to ad, the Lie algebra representation. In fact, you'll often see it written in physics texts that
generators Ta of the Lie algebra furnish the adjoint representation provided (Ta)bcb=fabcab.
where the f's are the structure constants of the Lie algebra with respect to the basis Ta;
[Ta,Tb]=fabcabTc
But notice that
adTa(Tb)=[Ta,Tb]=fabcabTc
which shows that the matrix representations of the generators in the Lie algebra representation
ad precisely have entries given by the structure constants.
Addendum (May 22, 2013).
Let a Lie-algebra valued field ϕ on a manifold M be given. If the field transforms under the representation Ad (which is a representation of the group acting on the algebra) then we have
ϕ(x)→Adg(ϕ(x))=gϕ(x)g−1
But recall that (
see here)
Ad is related to
ad (a representation on the
algebra acting on itself) as follows: Write an element of the Lie group as
g=eX for some
X in the algebra (here we assume that
G is connected) then
Adg(ϕ(x))=eadXϕ(x)=ϕ(x)+adX(ϕ(x))+O(X2)
so that the corresponding "infinitesimal" transformation law is
δϕ(x)=adX(ϕ(x))
So when talking about a field transforming under the adjoint representation,
Ad and
ad in some sense have the same content;
ad is the "infinitesimal" version of
Ad
This post imported from StackExchange Physics at 2014-03-22 17:13 (UCT), posted by SE-user joshphysics