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Representations of certain non-rational vertex operator algebras of affine type

In this paper we study a series of vertex operator algebras of integer level associated to the affine Lie algebra $A_l ^1$. These vertex operator algebras are constructed by using the explicit construction of certain singular vectors in the universal affine vertex operator algebra $N_l(n− 2, 0)$ at the integer level. In the case n=1 or l=2, we explicitly determine Zhu's algebras and classify all irreducible modules in the category. In the case l=2, we show that the vertex operator algebra $N_2(n− 2, 0)$ contains two linearly independent singular vectors of the same conformal weight.

vertex operator algebra; generalized Verma module; singular vectors; Affine Kac– Moody algebra; Zhu's algebra; category O

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