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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.

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Please, clarify what you mean by "fundamental representations" of a (semisimple) Lie group. This term is usually reserved for irreducible representations of a semisimple Lie algebra whose highest weights are primitive elements of the semigroup of dominant integral weights. –  Victor Protsak Aug 23 '10 at 19:33
    
Dear fourthinternational, Just to elaborate on Victor Protsak's comment, "fundamental representation" has a technical meaning in Lie theory (which is the one that Jim uses in his answer, for example), which seems to be slightly at odds with the way you are using it. I think that your usage may be closer to what people call the "standard representation", i.e. a group of $2\times 2$ matrices, like $SU(2)$, has a standard 2-dimensional rep'n, and a group of $3\times 3$ matries, like $SO(3)$ has a standard 3-dimensional representation. As Victor suggests, you may want to clarify what you mean. –  Emerton Aug 23 '10 at 20:56
    
Besides Victor's request for clarification of language, I'd add a request to clarify the meaning of the second question. Taken at face value I'd say the answer to that is clearly no. The first paragraph of the question makes more sense to me though expressed from the physics viewpoint. –  Jim Humphreys Aug 23 '10 at 21:00

2 Answers 2

up vote 2 down vote accepted

Let us consider for simplicity the complex semi-simple case. As you mention, every irreducible representation $\rho$ of a semi-simple finite dimensional Lie algebra $g$ integrates to a representation of the corresponding simply connected Lie group $G$. But even if we suppose $\ker\rho=0$, the representation of $G$ may well have a kernel, a finite central subgroup $H$ of $G$. So we obtain representations of $G/H'$ where $H'$ is a finite central subgroup contained in $H$, but not of other groups with Lie algebra $g$.

So one could ask: given a Lie group $G'$ with Lie algebra $g$, when does a representation $\rho$ of $g$ with highest weight $\Lambda$ integrates to a representation of $G'$? The answer: if and only if $\Lambda$ belongs to the character lattice of a maximal torus $T$ of $G$. (Recall that the character lattice of $T$ is the discrete additive subgroup of the dual of the Lie algebra of $T$ spanned by the differentials of the homomorphisms $T\to\mathbf{C}^{\ast}$.)

One can also ask a similar question: given a representation $\rho$ of $g$ with $\ker\rho =0$ and highest weight $\Lambda$, how to compute the kernel of the resulting representation $R$ of the simply connected group $G$? The answer is as follows. The dual of the Lie algebra $t$ of $T$ contains the weight lattice $P$ and the root lattice $Q$. Recall that the weight lattice is the lattice spanned by the fundamental weights and the root lattice is spanned by the roots i.e. the weights of the adjoint representation. We have $Q\subset P$. Consider the dual lattices $Q^\vee\subset P^\vee$; these are formed by all elements $a$ of the Lie algebra of $T$ such that $l(a)\in\mathbf{Z}$ for all $l\in P$, resp. all $l\in Q$.

Inside $P^\vee$ there is a sublattice formed by all $a\in t$ such that $\Lambda(a)\in\mathbf{Z}$; it contains $Q^\vee$ and the quotient by $Q^\vee$ is naturally identified with the kernel of $R$ (by exponentiating). There is a similar result when $\rho$ is reducible -- in that case we just consider the sublattice formed by all elements of the tangent space such that all weights that take integral values on them.

Example: when $G=SL_2(\mathbf{C})$, we can identify $Q$ with the sublattice of $\mathbf{R}$ spanned by 1 and $R$ will be spanned by $\frac{1}{2}$. Then the representations with integral weight will have kernel $\mathbf{Z}/2$ and the representations with half-integral weight will have trivial kernel.

Hopefully one of these two questions also covers the questions of the posting.

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Thank you so much for this. I will have to read up on it all but this provides me with an indispensable orientation. As for the definition of 'fundamental representation of a group' that I'm using: well I guess that should have been the first question I asked. The problem is that in a lot of particle physics books (which is where I'm coming from) representations of groups are often referred to when I think that what I really being talked about is representations of algebras. (i.e. the two are often not kept distinct and that's caused me a lot of confusion.) –  fourthinternational Aug 25 '10 at 11:24
    
Perhaps it would be easier to see what it is I'm trying to get at by just stating the issue I'm trying to straighten out. I'm reading Ne'eman's classic paper on the Eightfold Way in which he identifies su(3) as the algebra for the quark flavours. He notes that this algebra generates several groups, of which SU(3) and the adjoint group SU(3)/Z3 are two. (Here of course SU(3) is the unique simply connected group.) He then says that these two groups imply different 'fundamental representations': the 3 for SU(3) and the 8 for SU(3)/Z3. –  fourthinternational Aug 25 '10 at 11:30
    
Now, what can be meant by this isn't clear, as of course it is only the 3-dimensional representations that are the fundamental reps of the algebra (and here I understand by a fundamental rep one whose highest weight is a fundamental weight, with the latter defined in terms of its scalar products with the simple roots of the algebra.) Hence I thought that maybe they corresponded to the fundamental representations of the respective groups under the exponential mapping. –  fourthinternational Aug 25 '10 at 11:43
    
But maybe that guess isn’t right after all. Perhaps, then, someone could advise me of how to state the relationship between the 3 rep of su(3) and the group SU(3), and the 8 rep of su(3) and SU(3)/Z3; likewise between the 2 rep of su(2) and SU(2), and the 3 rep of su(2) and SO(3). (Is it right to say, for example, that these reps of the algebra exponentiate to give the ‘first non-trivial’ representation of the corresponding group? But that’s just a guess as well.) –  fourthinternational Aug 25 '10 at 11:48
    
Ne’eman’s talk seems to imply that there is a special relationship between these reps of the algebras and these groups, but the way he phrases it (in terms of ‘fundamental representations’) lacks a clear meaning to me. For now let me just refer to the relationship that SU(3)/Z3 stands to the 8 of su(3) and that SO(3) stands to the 3 of su(2) as the ‘special relationship’. So my first question is how to characterize this special relationship in more precise terms than Ne’eman has. –  fourthinternational Aug 25 '10 at 11:48

To supplement algori's answer: The first interesting case (for the Lie algebra of type $A_1$ over $\mathbb{C}$) is already noted in the question: the odd dimensional irreducible representations exponentiate to both the simply connected and the adjoint group, but not the even dimensional ones including the single fundamental representation. In type $B_2$ there are two fundamental representations, of dimensions 4 and 5. The latter one gives the standard realization of the adjoint group as a special orthogonal group in 5 variables, while the former one is the "spin" representation of the simply connected covering group but is not a representation of the adjoint group. As algori notes, which fundamental representations exponentiate to which group depends on the position of its highest weight in the weight lattice of a maximal torus of the group relative to the root lattice.

However, in all cases basic Lie theory guarantees that representations belonging to different highest weights (including the fundamental ones) are non-isomorphic.

In the case of real semisimple Lie algebras and groups, similar but more complicated things happen as seen in a number of math and physics textbooks.

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