On abelian 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 $Ω_{V}$ (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 $\C^{\times}$. For certain subgroups $A$ of $\Aut_{Ω_{V}}Ω_{V}$, certain quotient algebras $Ω_{V}^{A}$ of $Ω_{V}$ are constructed. Furthermore, certain functors among the category of $V$-modules, the category of $Ω_{V}$-modules and the category of $Ω_{V}^{A}$-modules are constructed and irreducible $Ω_{V}$-modules and $Ω_{V}^{A}$-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 $Ω_{V}^{A}$-modules is semisimple.