On Abelian Coset Generalized Vertex Algebras

This paper studies the algebraic aspect of a general abelian coset theory with a work of Dong and Lepowsky as our main motivation. It is proved that the vacuum space Ω V (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 Dong and Lepowsky and that the vacuum space Ω W of a V-module W is a natural Ω V -module. The automorphism group Aut Ω V Ω V of the adjoint Ω V -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 Aut Ω V Ω V , certain quotient algebras [Formula: see text] of Ω V are constructed. Furthermore, certain functors among the category of V-modules, the category of Ω V -modules and the category of [Formula: see text]-modules are constructed and irreducible Ω V -modules and [Formula: see text]-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 [Formula: see text]-modules is semisimple.