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Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point inS^1 and identifying the real line with the punctured circle, we consider thesubalgebra Sg of Lg of rapidly decreasing elements on R. We classify thetranslation-invariant 2-cocycles on Sg. We show that the ground staterepresentation of Sg is unique for each cocycle. These ground states correspondprecisely to the vacuum representations of Lg.

Ground state representations of loop algebras