Regularity of rational vertex operator algebras

Rational vertex operator algebras, which play a fundamental role in rational conformal field theory (see [BPZ] and [MS]), single out an important class of vertex operator algebras. Most vertex operator algebras which have been studied so far are rational vertex operator algebras. Familiar examples include the moonshine module V ♮ ([B], [FLM], [D2]), the vertex operator algebras VL associated with positive definite even lattices L ([B], [FLM], [D1]), the vertex operator algebras L(l, 0) associated with integrable representations of affine Lie algebras [FZ] and the vertex operator algebras L(cp,q, 0) associated with irreducible highest weight representations for the discrete series of the Virasoro algebra ([DMZ] and [W]). A rational vertex operator algebra as studied in this paper is a vertex operator algebra such that any admissible module is a direct sum of simple ordinary modules (see Section 2). It is natural to ask if such complete reducibility holds for an arbitrary weak module (defined in Section 2). A rational vertex operator algebra with this property is called a regular vertex operator algebra. One motivation for studying such vertex operator algebras arises in trying to understand the appearance of negative fusion rules (which are computed by the Verlinde formula) for vertex operator algebras L(l, 0) for certain rational l (cf. [KS] and [MW]). In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the rational vertex operator algebras V , L(l, 0) for positive integers l, L(cp,q, 0) and VL for positive definite even lattices L are regular. Our result for L(l, 0) implies that any restricted integrable module of level l for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. This result is expected to be useful in comparing the construction of tensor product of modules for L(l, 0) in [F] based on Kazhdan-Lusztig’s approach [KL] with the construction of tensor product of modules [HL] in this special case. We should remark that VL in general is a vertex algebra in the sense of [DL] if L is not positive definite. In this case we establish the complete reducibility of any weak module. Since the definition of vertex operator algebra is by now well-known, we do not define vertex operator algebra in this paper. We refer the reader to [FLM] and [FHL] for their elementary properties. The reader can find the details of the constructions of V ♮ and VL in [FLM], and L(l, 0) and L(cp,q, 0) in [DMZ], [DL], [FLM], [FZ], [L1] and [W].