Deforming vertex algebras by vertex bialgebras

TL;DR

This paper develops a framework for deforming vertex algebras using vertex bialgebras, introducing new structures and constructions, including quantum vertex algebras from lattice-associated vertex algebras.

Contribution

It introduces a method to deform vertex algebras via vertex bialgebras, expanding the theory of nonlocal vertex algebras and their modules.

Findings

Constructed deformations of vertex algebras with compatible module and comodule structures.

Developed ${ ext{phi}}$-coordinated quasi modules for smash product nonlocal vertex algebras.

Produced a family of quantum vertex algebras from lattice-based vertex algebras.

Abstract

This is a continuation of a previous study initiated by one of us on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ and give a construction of deformations of vertex algebras with a right $H$-comodule nonlocal vertex algebra structure and a compatible $H$-module nonlocal vertex algebra structure. We also give a construction of $\phi$-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.