On abelian coset generalized vertex algebras

This paper studies the algebraic aspect of a general abelian coset theory with [DL2] as our main motivation. It is proved that the vacuum space ΩV (or the space of highest weight vectors) of a Heisenberg algebra in a general vertex operator algebra V has a natural generalized vertex algebra structure in the sense of [DL2] and that the vacuum space ΩW of a V -module W is a natural ΩV -module. The automorphism group AutΩV ΩV of the adjoint ΩV -module is studied and it is proved to be a central extension of a certain torsion free abelian group by C×. For certain subgroups A of AutΩV ΩV , certain quotient algebras ΩV of ΩV are constructed. Furthermore, certain functors among the category of V -modules, the category of ΩV -modules and the category of Ω A V -modules are constructed and irreducible ΩV -modules and Ω A V -modules are classified in terms of irreducible V -modules. If the category of V -modules is semisimple, then it is proved that the category of ΩV -modules is semisimple.