Lie Group v Lie algebra representation

[gentsagree]

Hi y'all,

This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer.

I have a Lie group homomorphism $\rho : G \rightarrow GL(n, \mathbb{C}) \hspace{0.5cm}(1)$
and a Lie Algebra homomorphism $\hat{\rho} : g \rightarrow gl(n, \mathbb{C}) \hspace{1cm}(2)$

which are the group and algebra representations on the space of nxn matrices viewed as a vector space.

Now, the difference in the images of these two maps is that in (1) it has to be a group, so formally it is defined as the group of Automorphisms on a vector space, Aut(V), whereas in (2) it has to be an algebra, and I've read somewhere this is defined formally as End(W), and seen it written as "the Lie algebra of Endomorphisms of a vector space W".

Two questions:

- How does one properly define the Lie Algebra representation? I'm not quite sure I understand what is going on with the Endomorphisms being an algebra?

- If, in my original maps above, g is the Lie algebra of G, and I rewrite the maps as

$\rho : G \rightarrow Aut(V) \hspace{0.5cm}(3)$
$\hat{\rho} : g \rightarrow End(W) \hspace{1cm}(4)$

What situation am I representing if I choose V=W ? Does demanding V=W output the "analogous" representation, say the fundamental of the group and the fundamental of the algebra?

 

[fresh_42]

Hi y'all,

This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer.

I have a Lie group homomorphism $\rho : G \rightarrow GL(n, \mathbb{C}) \hspace{0.5cm}(1)$
and a Lie Algebra homomorphism $\hat{\rho} : g \rightarrow gl(n, \mathbb{C}) \hspace{1cm}(2)$

which are the group and algebra representations on the space of nxn matrices viewed as a vector space.

Right. Usually it's required for Lie Groups that the representation is analytic, too.

Now, the difference in the images of these two maps is that in (1) it has to be a group, so formally it is defined as the group of Automorphisms on a vector space, Aut(V) ...

One usually writes $GL(V)$ instead of $Aut(V)$ to emphasize the multiplication rather than linearity. $Aut()$ is more a group notation: $Aut(G)$, e.g.

..., whereas in (2) it has to be an algebra, and I've read somewhere this is defined formally as End(W), and seen it written as "the Lie algebra of Endomorphisms of a vector space W".

Which is just $gl(W)$ or $gl(n,ℂ)$ if $W$ is an n-dimensional complex vector space. The Lie multiplication here is defined by $[X,Y] = XY - YX$.

- How does one properly define the Lie Algebra representation? I'm not quite sure I understand what is going on with the Endomorphisms being an algebra?

A Lie algebra representation $(V,φ)$ of the Lie algebra $L$ is a Lie algebra homomorphism from $L$ into $gl(V)$. i.e. $φ([X,Y]) = [φ(X),φ(Y)] = φ(X)φ(Y) - φ(Y)φ(X)$.

- If, in my original maps above, g is the Lie algebra of G, and I rewrite the maps as

$\rho : G \rightarrow Aut(V) \hspace{0.5cm}(3)$
$\hat{\rho} : g \rightarrow End(W) \hspace{1cm}(4)$

Unusual (s.a.) but ok. $End(W)$ is more sensible here than $Aut(V)$.

What situation am I representing if I choose V=W ?

Representations on the same vector space.

Does demanding V=W output the "analogous" representation, say the fundamental of the group and the fundamental of the algebra?

What do you mean by fundamental? There are connections between inner group automorphisms of $G$ and the adjoint representation (left multiplication) of $g$ through the exponential map.

Edit: AFAIK are the weights in the classification of Lie algebra representations, esp. the $sl_2$, used in QFT to determine eigenstates in the SM. (I apologize if that's wrong, I'm no physicist.)

 

[dextercioby]

In physics, especially quantum physics, one speaks of representations on (complex, separable) Hilbert spaces and the connection between these morphisms is provided by Stone's theorem. Then we have subtleties of domain, continuity of representations, etc.