Reps of groups and reps of algebras

[metroplex021]

Hi folks - I have a couple of basic questions about fundamental
representations.

First of all, does every group have a set of fundamental
representations?

Secondly, I know that in the case of the (compact?) semi-simple groups, any other
representation of the group can be constructed by taking tensor
products of its L fundamental representations, where L is the group's
rank. Is this the case for all groups, or just the semi-simple ones?

Any help would be really appreciated. Thanks!

 

[metroplex021]

I've got what might be a couple of very basic questions on the fundamental representations of locally isomorphic semi-simple Lie groups and their relationship to representations of the corresponding Lie algebra. I know (from reading Cornwell, 'Group Theory in Physics') that every representation of a semi-simple Lie algebra exponentiates to give a representation of the unique, simply connected semi-simple Lie group associated with that algebra. (So for example, every representation of the algebra su(2) exponentiates to give a representation of the group SU(2).) I also know that, at least in some cases (such as with the group SO(3), which also has su(2) as its algebra), it is not the case that every representation of the algebra exponentiates to give a representation of a given non-simply-connected group associated with that algebra. The first non-trivial representation of SO(3), for example, is not obtainable by exponentiating the fundamental (2-dimensional) representation of su(2) (with highest weight 1/2), but is rather obtained by exponentiating the 3-dimensional representation of the algebra (with highest weight 1).

So I have two questions on the same theme as the above.

1. Is it always the case that the fundamental representation(s) of the various non-simply-connected semi-simple Lie groups associated with a given semi-simple Lie algebra are not obtainable by exponentiating the fundamental representation(s) of the algebra?

2. Do the fundamental representation(s) of each of the different non-simply-connected groups correspond to different representations of the algebra? (That is, will two groups that are locally but not globally isomorphic always have different fundamental representations?)

Any knowledge anyone can bring to bear on this would be really appreciated. (I'm pretty sure the second in particular is trivial, but I'm not sure as at the level I'm at in physics we pretty much always work with representations of the algebras and I'm not familiar with how they relate to the representations of the groups.) Thanks a lot.

 

[fresh_42]

Hi folks - I have a couple of basic questions about fundamental
representations.

First of all, does every group have a set of fundamental
representations?

No, simply because fundamental representations are only defined for semi-simple Lie groups.

Secondly, I know that in the case of the (compact?) semi-simple groups, any other
representation of the group can be constructed by taking tensor
products of its L fundamental representations, where L is the group's
rank. Is this the case for all groups, or just the semi-simple ones?

See above. Any representation is a direct sum of irreducible representations, which fundamental representations are.

Any help would be really appreciated. Thanks!

 

[fresh_42]

1. Is it always the case that the fundamental representation(s) of the various non-simply-connected semi-simple Lie groups associated with a given semi-simple Lie algebra are not obtainable by exponentiating the fundamental representation(s) of the algebra?

The general connection is given by
\begin{align}
\operatorname{Ad}(\varphi(g)) \circ D\varphi = D\varphi \circ \operatorname{Ad}(g)
\end{align}
for group homomorphisms. A group representation is a special group homomorphism $\varphi\, : \,G \longrightarrow GL(V)$

2. Do the fundamental representation(s) of each of the different non-simply-connected groups correspond to different representations of the algebra? (That is, will two groups that are locally but not globally isomorphic always have different fundamental representations?)

Any knowledge anyone can bring to bear on this would be really appreciated. (I'm pretty sure the second in particular is trivial, but I'm not sure as at the level I'm at in physics we pretty much always work with representations of the algebras and I'm not familiar with how they relate to the representations of the groups.) Thanks a lot.

Maybe this could help:
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-representations/
Otherwise I'd like to refer to the references at the end of the article.