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Vertex algebras associated with even lattices. (English) Zbl 0807.17022
The theory of vertex algebras has been developed in recent years. It was motivated by the representation theory of affine Lie algebras, the representation theory of the Monster simple group, and by conformal field theory in physics. For example, one can attach a vertex algebra $$V_{L_ 0}$$ to any even lattice $$L_ 0$$. This paper studies the classification of irreducible modules for the vertex algebra $$V_{L_ 0}$$. The main theorem is that such irreducible modules are isomorphic to the untwisted Fock space (the tensor product of a symmetric algebra and a group algebra) associated with a coset of $$L_ 0$$ in $$L$$, the $$\mathbb{Z}$$- dual lattice of $$L_ 0$$.
This result was partly known, in the case of a positive definite even lattice $$L_ 0$$, however the author gives a clear, rigorous algebraic proof of it in the case of a lattice $$L_ 0$$ that is not necessarily positive definite.

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures
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