m at h . Q A ] 1 8 M ar 1 99 9 Decomposition of the vertex operator algebra V √ 2 A 3

A conformal vector with central charge c in a vertex operator algebra is an element of weight two whose component operators satisfy the Virasoro algebra relation with central charge c. Then the vertex operator subalgebra generated by the vector is isomorphic to a highest weight module for the Virasoro algebra with central charge c and highest weight 0 (cf. [M]). Let V2Al be the vertex operator algebra associated with a lattice √ 2Al, where √ 2Al denotes √ 2 times an ordinary root lattice of type Al. Motivated by the problem of looking for maximal associative algebras of the Griess algebra [G], a class of conformal vectors in V2Al were studied and constructed in [DLMN]. It was shown in [DLMN] that the Virasoro element ω of V2Al is decomposed into a sum of l + 1 mutually orthogonal conformal vectors ω; 1 ≤ i ≤ l+1 with central charge ci = 1−6/(i+2)(i+3) for 1 ≤ i ≤ l and cl+1 = 2l/(l + 3). The vertex operator subalgebra generated by conformal vector ω i is exactly the irreducible highest weight module L(ci, 0) for the Virasoro algebra. The vertex operator subalgebra T = Tl generated by these conformal vectors is isomorphic to a tensor product ⊗ i=1L(ci, 0) of the Virasoro vertex operator algebras L(ci, 0) and V2Al is a direct sum of irreducible T -submodules. In this paper we determine the decomposition of V2A3 into the direct sum of irreducible T -modules completely. The direct summands have been determined [KMY] in the case l = 2. For general l only the direct summands with minimal weights at most two are known [Y].