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Abstract. We investigate vertex operator algebras L(k, 0) associated with modular-invariant representations for an affine Lie algebra A(1) 1 , where k is an'admissible' rational number. We show that VOA L(k, 0) is rational in the category O and find all irreducible representations in the category of weight modules.

Motivated by [On the triplet vertex algebra W(p), Adv. Math. 217 (2008) 2664-2699, for every finite subgroup Γ ⊂ PSL(2, C) we investigate the fixed point subalgebra W(p) Γ of the triplet vertex W(p), of central charge 1 − 6(p−1) 2 p , p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, C). First, we prove the C 2 -cofiniteness of the Am-fixed subalgebra W(p) Am . Then we construct a family of W(p) Am -modules, which are expected to form a complete set of irreducible representations. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m = 2. We also present a rigorous proof of the fact that the full automorphism group of W(p) is PSL(2, C).

Let M(1) be the vertex algebra for a single free boson. We classify irreducible modules of certain vertex subalgebras of M(1) generated by two generators. These subalgebras correspond to the W(2, 2p − 1)-algebras with central charge 1 − 6(p − 1) 2

For every p ജ 2, we obtained an explicit construction of a family of W͑2,2p −1͒ modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual W͑2,2p −1͒ modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic W͑2,2p −1͒ modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as "logarithmic conformal field theory" of central charge c p,1 =1 − ͓6͑p −1͒ 2 / p͔, p ജ 2. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra W͑2,͑2p −1͒ 3 ͒ and other logarithmic models.