Extension of vertex operator algebra VH₄(ℓ,0)

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We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra H₄ with some natural conditions. It turns out the representation theory of H₄ is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra VH₄(ℓ,0) by the even lattice L. We give the structure of the extension VH₄(ℓ,0)⊗ [L] and its irreducible modules via irreducible representations of VH₄(ℓ,0) viewed as a vertex algebra.