A characterization of vertex algebras associated to even lattices
One of the first and best known classes of vertex operator algebras are those associated to even lattices: Let \({\mathfrak h}\) be a finite dimensional abelian Lie algebra and \(L \subset {\mathfrak h}\) a positive definite even lattice. Then the affinization \(\widehat {\mathfrak h} = \mathbb{C} [t,t^{-1}] \otimes {\mathfrak h} \oplus \mathbb{C} c\) is a sum of \({\mathfrak h} \cong {\mathfrak h} \otimes t^0\) and the Heisenberg algebra \(\widetilde {\mathfrak h}'\), and its representation space \(V_{({\mathfrak h}, L)} = \mathbb{C} [L] \otimes S (\widehat {\mathfrak h}^-)\) has the structure of a simple vertex operator algebra. Moreover, \(V = V_{({\mathfrak h}, L)}\) has the following properties: (1) any vector of \({\mathfrak h} \subset V\) is primary of weight one and \(\widetilde {\mathfrak h}'\) is associated to \({\mathfrak h}\) in terms of vertex operator algebra structure, (2) \(V\) is a sum of generalized \({\mathfrak h}\)-weight spaces \(V^\alpha\) which are completely reducible \(\widetilde {\mathfrak h}'\)-modules, and (3) \(V^0\) is an irreducible \(\widetilde {\mathfrak h}'\)-module. In the paper under review it is proved that conditions (1)--(3) characterize this class of algebras: if a simple vertex operator algebra \(V\) contains a weight one primary subspace \({\mathfrak h}\) such that for the associated Heisenberg algebra \(\widetilde {\mathfrak h}'\) conditions (2) and (3) hold, then \(V\) is isomorphic to some \(V_{({\mathfrak h}, L)}\).
scientific article; zbMATH DE number 763944