We study a W -algebra of central charge 2(k − 1)/(k + 2), k = 2, 3, . . . , contained in the commutant of a Heisenberg algebra in a simple affine vertex operator algebra L(k, 0) of type A (1) 1 with level k. We calculate the operator product expansions of the W -algebra. We also calculate some singular vectors in the case k 6 and determine the irreducible modules and Zhu's algebra. Furthermore, the rationality and the C 2 -cofiniteness are verified for such k.The argument heavily depends on singular vectors v 0 , v 1 and v 2 of weight 8, 9 and 10, respectively in W and on singular vectors u r = (W 3 1 ) r u 0 of weight k + 1 + r, r = 0, 1, 2, 3 in W. It seems that we can take W 3 1 v 0 and (W 3 1 ) 2 v 0 in place of v 1 and v 2 , respectively. However, we do not verify it. The importance of u 0 is clear from the degenerate case, namely, the case k = 2, 3, 4, for u 0 is a scalar multiple of W 3 , W 4 or W 5 in such a case. It would be difficult to express u r , r = 0, 1, 2, 3 in terms of ω, W 3 , W 4 and W 5 for an arbitrary k. We should take a different approach for a general case.Our notation is fairly standard [12,20]. Let V be a vertex operator algebra and (M, Y M ) be its module. Then Y M (v, x) = n∈Z v n x −n−1 is the vertex operator associated with v ∈ V . The linear operator v n on M is called a component operator. For a subalgebra U of V and a subset S of M, let U · S = span{u n w | u ∈ U , w ∈ S, n ∈ Z}, which is the U -submodule of M generated by S.Part of the results in this paper was announced in [6]. We remark that N 0 (resp. K 0 ) is denoted by W (resp. W) in [6]. In this paper, we distinguish N 0 and W (resp. K 0 and W) clearly to avoid confusion. W and its singular vectorsLet {h, e, f } be a standard Chevalley basis of sl 2 . Thus [h, e] = 2e, [h, f ] = −2 f , [e, f ] = h for the bracket and h, h = 2, e, f = 1, h, e = h, f = e, e = f , f = 0 for the normalized Killing form.We fix an integer k 2. Let V (k, 0) = V sl 2 (k, 0) be a Weyl module for the affine Lie algebra sl 2 = sl 2 ⊗ C[t, t −1 ] ⊕ CC with level k, that is, a generalized Verma module for sl 2 with level k and highest weight 0. Let 1 be its canonical highest weight vector, which is called the vacuum vector. Then sl 2 ⊗ C[t] acts as 0 and C acts as k on 1. We denote by h(n), e(n) and f (n) the operators on V (k, 0) induced by the action of h ⊗t n , e ⊗t n and f ⊗t n , respectively. Thus h(n)1 = e(n)1 = f (n)1 = 0 for n 0 andfor a, b ∈ {h, e, f }. The elements h(−i 1 ) · · · h(−i p )e(− j 1 ) · · · e(− j q ) f (−m 1 ) · · · f (−m r )1, (2.2) i 1 · · · i p 1, j 1 · · · j q 1, m 1 · · · m r 1 form a basis of V (k, 0).Let a(x) = n∈Z a(n)x −n−1 for a ∈ {h, e, f } and

Abstract:The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.

The W 3 algebra of central charge 6/5 is realized as a subalgebra of the vertex operator algebra V √ 2A2 associated with a lattice of type √ 2A 2 by using both coset construction and orbifold theory. It is proved that W 3 is rational. Its irreducible modules are classified and constructed explicitly. The characters of those irreducible modules are also computed.

Abstract. We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E 8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E 8 lattice obtained by removing one node from the extended E 8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V √ 2E 8. There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of U coincides with the algebra described in his Table 3. There is a canonical automorphism of U of order |E 8 /L|. Such an automorphism can be extended to the Leech lattice VOA V Λ , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of U will be discussed in detail. It is expected that if U is actually contained in the Moonshine VOA V , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A (1) 1 of level k coincides with a certain W -algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.