Representation Theory of Local Current Algebras

We prove the following theorem: Every finite‐dimensional irreducible representation of the local current algebra [F ρ ( k 1 ), F σ ( k 2 )]=e ρσ τ F τ ( k 1 + k 2 ) , where the {e ρσ τ } are the structure constants of a semisimple Lie algebra L, has the form F ρ ( k )→ ∑ i=1 N e ikx i T ρ i , where T ρ i is a finite‐dimensional representation of L, and for i ≠ j, xi ; T ρ i commutes with T σ j for all ρ and σ.