ON ABELIAN COSET GENERALIZED VERTEX ALGEBRAS

Abstract

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 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 Dong and Lepowsky and that the vacuum space ΩW of a V-module W is a natural ΩV-module. The automorphism group Aut ΩVΩV 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 Aut ΩVΩV, certain quotient algebras [Formula: see text] of ΩV are constructed. Furthermore, certain functors among the category of V-modules, the category of ΩV-modules and the category of [Formula: see text]-modules are constructed and irreducible ΩV-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.