I am studying relativistic quantum mechanics and I have encountered the concept of projective representation for a group.

I have read in http://groupprops.subwiki.org/wiki/Projective_representation that the fact that a projective representation of G cannot be changed into a normal representation (without "up to phase" composition) is related to the second group cohomology of G. (Here I do not mean the second cohomology group, the "topological" one, but this http://groupprops.subwiki.org/wiki/Second_cohomology_group_for_trivial_group_action).

Now, it should be true that both SO(3) and O(3,1) (the homogeneous Lorentz group) admit intrisically projective representations (e.g the usual isomorphism $\ SO(3) = SU(2)/\{Id,-Id\}$, should be the projective representation for SO(3)).

I therefore would like to know how to compute the cohomology groups of SO(3) and O(3,1) to check that they are not trivial. If you have any suggestions for books explaining projective representations they are welcome as well.

  • 2
    $\begingroup$ Search for papers on «(co)homology of Lie groups made discrete». You will find several, starting from one by Milnor, which is very famous. Marcel Bökstedt, Morten Brun and Johan Dupont have one about SO(n) and S(1,n) but for homology, not cohomology (the homology case is important for the determination of the so called «scissor groups»; googling and mathscinet should get you to the existing information about cohomology) You'll notice soon that this (co)homology groups are quite non-trivial to determine! $\endgroup$ – Mariano Suárez-Álvarez May 10 '14 at 8:22
  • $\begingroup$ What is more relevant for representation theory of (connected) Lie groups $G$ is not the 2nd cohomology of $G$, but its second "continuous cohomology". This one is very much computable by means of Lie algebra cohomology. For instance, $H^2_{cont}(G)=0$ for every compact Lie group $G$. See mathoverflow.net/questions/78999/… for discussion and references. $\endgroup$ – Moishe Kohan May 11 '14 at 18:02

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