# Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference.

Most of the time physicists are sloppy and don't distinguish groups and algebras properly. However, although we talk about groups all the time I wasn't able to come up with an instant where we don't actually care only about the corresponding Lie algebra.

As an example, physicists usually talk about the Poincare group. However, the thing we are really interested in is the corresponding complexified Lie algebra, which actually belongs to the universal covering group

$$SL(2,\mathbb{C}) \ltimes \mathbb{R}(3,1).$$

Now, the other kind of symmetry that is important in fundamental physics is gauge symmetry. However, again the global structure doesn't seem to be important. To quote from Witten's Physics and Geometry:

“Experiment tells us more directly about the Lie algebra of G than about G itself. When I say that G contains the subgroup SU(3) X SU(2) x U(1), I really mean only that the Lie algebra of G contains that of SU(3) X SU(2) X U(1); there is no claim about the global form of G. For the same reason, in later comments I will not be very precise in distinguishing different groups that have the same Lie algebra.”

• It is untrue that we are "really" interested in the complexified Lie algebra. It's just the case that when we look at the projective representations of a Lie group, this is equivalent to looking at the unitary representations of the Lie algebra for groups without non-trivial central extensions by $\mathrm{U}(1)$. When you're looking at other things, the group structure may well be relevant - just think of how bosonics reps are "true" reps of the Poincaré group, while the fermionic reps are only reps of the universal cover. This seems to be asking for an open-ended list of such examples. – ACuriousMind Dec 28 '16 at 15:06
• This question (v3) seems like a list question. – Qmechanic Dec 28 '16 at 15:28
• @ACuriousMind 1) We construct the irreps of the complexified Lie algebra and this yields all important representations. If we do the usual representation theory for the Lorentz group, we miss the half-integer spin reps. Thus, I don't see how and why the group structure should be important here. It is not as if we miss some group irreps, if we consider irreps of the Lie algebra. Bosonic reps are true reps of the complexified Lie algebra, too. 2.) Fundamental physics is a narrow field and therefore I don't see how one could end up with an "open-ended" list. – jak Dec 28 '16 at 16:34
• By "fundamental physics", do you mean "high energy theory"? – Danu Dec 28 '16 at 19:42
• – ACuriousMind Jul 22 at 15:52