Actually, the Peter-Weyl Theorem is a generalization of the Pontryagin duality, and by extending the definition of the homomorphisms from the circle group to generalized unitary matricies asserts the existence of an algebraic dual group even if the group is not necessarily abelian. I referenced this in a paper last year for defining a non-abelian Fourier transform to solve the hidden subgroup problem. To quote wikipedia: "In mathematics <http://en.wikipedia.org/wiki/Mathematics>, the *Peter–Weyl theorem* is a basic result in the theory of harmonic analysis<http://en.wikipedia.org/wiki/Harmonic_analysis>, applying to topological groups<http://en.wikipedia.org/wiki/Topological_group> that are compact <http://en.wikipedia.org/wiki/Compact_group>, but are not necessarily abelian."
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