Editing Projective representation (section)

===Infinite-dimensional projective unitary representations: Bargmann's theorem===
On the other hand, [[Valentine Bargmann|Bargmann's]] theorem states that if the second [[Lie algebra cohomology]] group <math>H^2(\mathfrak g; \mathbb R)</math> of <math>\mathfrak g</math> is trivial, then every projective unitary representation of <math>G</math> can be de-projectivized after passing to the universal cover.<ref>{{harvnb|Bargmann|1954}}</ref><ref>{{harvnb|Simms|1971}}</ref> More precisely, suppose we begin with a projective unitary representation <math>\rho</math> of a Lie group <math>G</math>. Then the theorem states that <math>\rho</math> can be lifted to an ordinary unitary representation <math>\hat\rho</math> of the universal cover <math>\hat G</math> of <math>G</math>. This means that <math>\hat\rho</math> maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level, <math>\hat\rho</math> descends to <math>G</math>—and that the associated projective representation of <math>G</math> is equal to <math>\rho</math>.

The theorem does not apply to the group <math>\mathbb R^{2n}</math>—as the previous example shows—because the second cohomology group of the associated commutative Lie algebra is nontrivial. Examples where the result does apply include semisimple groups (e.g., [[Representation theory of SL2(R)|SL(2,R)]]) and the [[Poincaré group]]. This last result is important for [[Wigner's classification]] of the projective unitary representations of the Poincaré group.

The proof of Bargmann's theorem goes by considering a [[central extension (mathematics)|central extension]] <math>H</math> of <math>G</math>, constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group <math>G\times U(\mathcal H)</math>, where <math>\mathcal H</math> is the Hilbert space on which <math>\rho</math> acts and <math>U(\mathcal H)</math> is the group of unitary operators on <math>\mathcal H</math>. The group <math>H</math> is defined as
:<math>H = \{(g, U) \mid \pi(U) = \rho(g)\}.</math>
As in the earlier section, the map <math>\phi: H \rightarrow G</math> given by <math>\phi(g, U) = g</math> is a surjective homomorphism whose kernel is <math>\{(e, cI) \mid |c| = 1\},</math> so that <math>H</math> is a central extension of <math>G</math>. Again as in the earlier section, we can then define a linear representation <math>\sigma</math> of <math>H</math> by setting <math>\sigma(g, U) = U</math>. Then <math>\sigma</math> is a lift of <math>\rho</math> in the sense that <math>\rho\circ\phi = \pi\circ\sigma</math>, where <math>\pi</math> is the quotient map from <math>U(\mathcal H)</math> to <math>PU(\mathcal H)</math>.

A key technical point is to show that <math>H</math> is a ''Lie'' group. (This claim is not so obvious, because if <math>\mathcal H</math> is infinite dimensional, the group <math>G\times U(\mathcal H)</math> is an infinite-dimensional [[topological group]].) Once this result is established, we see that <math>H</math> is a one-dimensional Lie group central extension of <math>G</math>, so that the Lie algebra <math>\mathfrak h</math> of <math>H</math> is also a one-dimensional central extension of <math>\mathfrak g</math> (note here that the adjective "one-dimensional" does not refer to <math>H</math> and <math>\mathfrak{h}</math>, but rather to the kernel of the projection map from those objects onto <math>G</math> and <math>\mathfrak{g}</math> respectively). But the cohomology group <math>H^2(\mathfrak g; \mathbb R)</math> [[Lie algebra cohomology#Cohomology in small dimensions|may be identified]] with the space of one-dimensional (again, in the aforementioned sense) central extensions of <math>\mathfrak g</math>; if <math>H^2(\mathfrak g; \mathbb R)</math> is trivial then every one-dimensional central extension of <math>\mathfrak g</math> is trivial. In that case, <math>\mathfrak h</math> is just the direct sum of <math>\mathfrak g</math> with a copy of the real line. It follows that the universal cover <math>\tilde H</math> of <math>H</math> must be just a direct product of the universal cover of <math>G</math> with a copy of the real line. We can then lift <math>\sigma</math> from <math>H</math> to <math>\tilde H</math> (by composing with the covering map) and finally restrict this lift to the universal cover <math>\tilde G</math> of <math>G</math>.