1 7 D ec 1 99 9 Decomposition of the vertex operator algebra V √

A weight two vector v of a vertex operator algebra is called a conformal vector with central charge c if the component operators Lv(n) for n ∈ Z of Y (v, z) = ∑ n∈Z Lv(n)z −n−2 satisfy the Virasoro algebra relation with central charge c. In this case, the vertex operator subalgebra Vir(v) generated by v is isomorphic to a Virasoro vertex operator algebra with central charge c ([FZ], [M]). Let V2R be the vertex operator algebra associated with √ 2 times an ordinary root lattice R of type Al, Dl, or El. In [DLMN], several sets of mutually orthogonal conformal vectors in V2R were constructed and studied. It was shown that the Virasoro element of V2R can be written as a sum of l + 1 mutually orthogonal conformal vectors ω , 1 ≤ i ≤ l + 1. Since these conformal vectors are mutually orthogonal, the subalgebra T generated by them is a tensor product T = ⊗ i=1Vir(ω) and Vir(ω) is isomorphic to L(ci, 0) with ci the central charge of ω . Here, we denote by L(c, h) the irreducible highest weight module for the Virasoro algebra with highest weight h ∈ C and central charge c.Moreover, V2R is completely reducible as a T -module and each irreducible direct summand is in the form ⊗ i=1L(ci, hi). Hence one can study the structure of the vertex operator algebra V2R from a point of view that V √ 2R is a T -module. Along this line, the decomposition of V2A2 and V √ 2A3 into a direct sum of irreducible T -modules has been determined in [KMY] and [DLY]. See also [DMZ], in which the moonshine module V ♮ is studied as a module for L( 2 , 0)⊗48. In this paper we determine the decomposition of V2Dl into a sum of irreducible T modules for general l. There are many ways to choose a set of the mutually orthogonal