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.
Abstract. This is the first in a series of papers in which we study vertex-algebraic structure of Feigin-Stoyanovsky's principal subspaces associated to standard modules for both untwisted and twisted affine Lie algebras. A key idea is to prove suitable presentations of principal subspaces, without using bases or even "small" spanning sets of these spaces. In this paper we prove presentations of the principal subspaces of the basic A (1) 1 -modules. These convenient presentations were previously used in work of Capparelli-Lepowsky-Milas for the purpose of obtaining the classical Rogers-Ramanujan recursion for the graded dimensions of the principal subspaces.
We discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of W-algebras and vertex algebras in general. More precisely, motivated by clues from conformal field theory, first, we are able to determine modularlike transformation properties of regularized partial and false theta functions. Then, after suitable identification of regularized partial/false theta functions with the characters of atypical modules for the singlet vertex algebra W(2, 2p − 1), we formulate a Verlinde-type formula for the fusion rules of irreducible W(2, 2p − 1)-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).
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