Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

Abstract

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.

Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A

“…[1,2,25]). In this paper we study one similar special case, for which we verify the conjecture from [4].…”

“…One can say that these vertex operator algebras are rational in the category O. In [4], authors gave a conjecture that vertex operator algebras L(k, 0), for all admissible levels k, are rational in the category O. In the case of vertex operator algebras associated to affine Lie algebras of type B…”

“…Algebra A(L(− 1 2 (l + 1), 0)) is infinite-dimensional in this case. Using methods from [2,4,24], we get that irreducible A(L(− 1 2 (l + 1), 0))-modules from the category O are in one-to-one correspondence with zeros of the certain set of polynomials P 0 . By determining a basis for the vector space P 0 , we obtain the classification of irreducible A(L(− 1 2 (l + 1), 0))-modules from the category O.…”

“…Vertex operator algebras associated to affine Lie algebras of type A (1) 1 with arbitrary admissible level have been studied in [4,7]. In these cases vertex operator algebra L(k, 0) has finitely many irreducible weak modules from the category O and every weak L(k, 0)-module from the category O is completely reducible.…”

Embeddings of Vertex Operator Algebras Associated to Orthogonal Affine Lie Algebras

“…relation (4.1)). Vertex operator algebras associated to affine Lie algebras with admissible levels have also recently been studied in [A1,A2,AM,DLM].…”