## Semi-simple lie groups and their fundamental representations

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I've got a really basic question on the representation theory of semi-simple Lie groups. I know that a rank-R semi-simple Lie group possesses R fundamental representations. But is the relation between semi-simple Lie groups and their fundamental representations injective? That is, can two distinct semi-simple Lie groups possess the same set of fundamental representations?

I have always just assumed that the answer was no, but I don't know how to prove it ( in my defense I'm an undergrad physicist, not a mathematician, by trade!) Any straightening out of this matter would be most appreciated.

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I don't understand this question. A (finite-dimensional) representation of a Lie group $G$ is a homomorphism $G \to \mathrm{GL}(V)$ for some vector space $V$. In other words, it's not just the vector space $V$, but includes the map. Hence if you have two non-isomorphic Lie groups $G$ and $G'$, what does it mean to say that they have the same set of (fundamental) representations? – José Figueroa-O'Farrill Jul 4 2010 at 12:35
OK, I see what you mean: in physics we often (I suppose lazily) refer to the vector space itself as the representation. Thus it is often said that the up, down and strange quarks constitute a fundamental 'representation' of the group SU3, even though we're just talking about the particles (ie the 3 states in that vector space). So I suppose what I really want to know is whether, if you have two non-isomorphic semi-simple groups (like SU2 and SO3), is it possible for the weight diagrams corresponding to the fundamental representations be the same in each case? – fourthinternational Jul 4 2010 at 13:07

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Are you only interested in compact (connected) Lie groups? For example, how about the compact and split real forms of a common complex (connected) semisimple Lie group (such as ${\rm{SU}}_n$ and ${\rm{SL}}_n(\mathbb{R})$)? – Boyarsky Jul 4 2010 at 13:30
The terminology has to be a little more precise here: (1) You seem to be talking only about compact semisimple Lie groups, whose irreducible representations are all finite dimensional and correspond naturally to those of a corresponding complex group or its Lie algebra. (2) To get all of the fundamental representations you have to work with a simply connected group, which can cover various proper homomorphic images having the same rank but sharing only some of the irreducible representations. Note too that a weight diagram lives in a space whose dimension equals the rank of the given group. – Jim Humphreys Jul 4 2010 at 13:31
Thanks for the clarifications. Yes, I'm only interested in connected semi-simple Lie groups. As for compactness, I'm pretty sure a restriction to compact groups would be fine in this context (since the sort of particle physics reps I'm directly interested in are finite-dimensional and unitary). – fourthinternational Jul 4 2010 at 13:44
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## 1 Answer

It seems to me that you need to think more about what it is you really want to know.

First, taking your question at face value. It sounds like a version of 20 Questions that starts with "I'm thinking of a semisimple Lie algebra". It is not clear what questions I am allowed to ask. My overall impression is that we could have an involved discussion about the rules and arrive at the point where I have a set of questions you find acceptable and which allow me to determine the Lie algebra. For example, I find it plausible that the rank and the list of dimensions of fundamental representations determine the Lie algebra. I don't feel inclined to attempt a proof. Any such proof would rely heavily on the classification of simple Lie algebras and tricks involving the lists of dimensions of fundamental representations.

I would also question whether this is the right question from the point of view of the physics. First we don't know how many fermions there are. All we can say is that we have done these scattering experiments at these energies and this is the list of particles we have seen. Second if we think we have found all particles then the gauge bosons form the adjoint representation so we know the dimension of the adjoint representation. Is this information you would disclose in the game of 20 Questions? Thirdly I don't know of any reason why the fermions should be a fundamental representation.

As I understand it supersymmetry does impose strong conditions on the representations. My understanding is that there are good physical arguments for restricting the spin to be at most 2 (or maybe less?). I would be interested in seeing these various physical conditions listed and it would then be challenging problem to classify the solutions. This must be known in the physics community.

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Well, what I really want to know is if you knew you had a complete set of fundamental representations of a (compact, connected) semi-simple Lie group, but didn't know what the group was, you could figure out what the group was from the information that these were its fundamental representations. (I heard that groups are classified entirely by their 'weight lattices' and 'root lattices', and since these are determined by the fundamental weights, a complete set of fundamental weights should be sufficient to work backwards to identifying the group. But I'd like a reference for this. – fourthinternational Jul 4 2010 at 22:07
You also need to say simply-connected (you would have saved some trouble if you had put "Lie algebra" instead of "Lie group"). – Bruce Westbury Jul 5 2010 at 0:13
The real issue is: what precisely do you mean by "having a complete set of fundamental representations"? – Bruce Westbury Jul 5 2010 at 0:15
Your comments on weight and root lattices sound confused. Have you attended a course or read a book on the classification of complex simple Lie algebras? – Bruce Westbury Jul 5 2010 at 0:21

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It sounds as though you need a tutorial. There are numerous books on this subject (look in QA252.3 in your library). This site works best for focused questions. – Bruce Westbury Jul 5 2010 at 7:57
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