Abstract
A class of unitary representations of the group of maps from a Riemannian manifold to a compact semisimple Lie group which are a modification of the so-called energy representations is considered. Their irreducibility is proved in any dimension ≥2. This extends the known results of irreducibility for certain cases in dimension 2. Next, the corresponding algebra representations are studied and their irreducibility in dimension 2 is also proved. These are extended in a natural way to representations of certain central extensions of the algebras. Very explicit formulas are given, particularly on the two-dimensional torus, for the algebra extensions and the representations.
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