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Many of the physical symmetry groups are type I and unimodular. The unitary representations of type I second countable groups in separable Hilbert spaces can be given in a direct integral form which is convenient from a physical point of view. Is there any physically relevant symmetry group that is neither type I nor unimodular except for the ax+b group of affine transformations. Especially, are the Galileo and Poincare groups or their covering groups type I? Could you recommend any particular referencies?

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Is this really supposed to be tagged as "representation" and "theory", or should it have been a single tag "representation-theory"? – Andrej Bauer Nov 11 '10 at 9:00
It was proved by Dixmier that every connected Lie group is Type I, which would seem to answer your question – Yemon Choi Nov 11 '10 at 9:33
@Yemon: Didn't Dixmier show that there are connected (solvable) Lie groups of Type I whose universal cover is not of Type I? In fact, there's a 1961 paper of his in Comptes Rendus to this effect. – José Figueroa-O'Farrill Nov 11 '10 at 20:52
Jose: ah, I must have misremembered. Certainly any semisimple real Lie group is Type I, that's been known since Harish-Chandra/Godement/Stinespring. – Yemon Choi Nov 13 '10 at 18:35
Thinking a bit harder about the Dixmier paper I had in mind: if I recall correctly it shows that inside every connected Lie group G there is a normal closed Lie subgroup H which is unimodular and Type I, and such that G/H is abelian. – Yemon Choi Nov 13 '10 at 18:35

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