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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