Representations On Hilbert Spaces
A representation of a Lie group G on a complex Hilbert space V is a group homomorphism Ψ:G → B(V) from G to B(V), the group of bounded linear operators of V which have a bounded inverse, such that the map G×V → V given by (g,v) → Ψ(g)v is continuous.
This definition can handle representations on infinite-dimensional Hilbert spaces. Such representations can be found in e.g. quantum mechanics, but also in Fourier analysis as shown in the following example.
Let G=R, and let the complex Hilbert space V be L2(R). We define the representation Ψ:R → B(L2(R)) by Ψ(r){f(x)} → f(r-1x).
See also Wigner's classification for representations of the Poincaré group.
Read more about this topic: Representation Of A Lie Group
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