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 $\Omega_{V}$ (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 $\Omega_{W}$ of a $V$-module $W$ is a natural $\Omega_{V}$-module. The automorphism group $\Aut_{\Omega_{V}}\Omega_{V}$ of the adjoint $\Omega_{V}$-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 $\Aut_{\Omega_{V}}\Omega_{V}$, certain quotient algebras $\Omega_{V}^{A}$ of $\Omega_{V}$ are constructed. Furthermore, certain functors among the category of $V$-modules, the category of $\Omega_{V}$-modules and the category of $\Omega_{V}^{A}$-modules are constructed and irreducible $\Omega_{V}$-modules and $\Omega_{V}^{A}$-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 $\Omega_{V}^{A}$-modules is semisimple.