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The following <it>integrability</it> theorem for vertex operator algebras <it>V</it> satisfying some finiteness conditions (<it>C</it><inf>2</inf>-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra ℊ of the weight-one subspace <it>V</it><inf>1</inf> is isomorphic to the irreducible highest-weight \ℊ̂-module <it>L</it>(<it>k</it>, 0) for a positive integer <it>k</it>, and <it>V</it> is an integrable \ℊ̂-module. The case in which ℊ is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed.
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Dong et al. (2006) studied this question.
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