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Deforming vertex algebras by vertex bialgebras

Naihuan Jing, Fei Kong, Haisheng Li, Shaobin Tan

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 $φ$-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.

Deforming vertex algebras by vertex bialgebras

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 -comodule nonlocal vertex algebra for a nonlocal vertex bialgebra and give a construction of deformations of vertex algebras with a right -comodule nonlocal vertex algebra structure and a compatible -module nonlocal vertex algebra structure. We also give a construction of -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.

Paper Structure

This paper contains 9 sections, 43 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. Smash product nonlocal vertex algebras
  3. Nonlocal vertex algebras and quantum vertex algebras
  4. Smash product nonlocal vertex algebras
  5. More on nonlocal vertex algebras $\mathfrak D_{Y_{V}^{H}}^\rho (V)$
  6. $\phi$-coordinated quasi modules for smash product nonlocal vertex algebras
  7. Deformations of lattice vertex algebras $V_L$
  8. Vertex algebra $V_{L}$ and associative algebra $A(L)$
  9. Vertex bialgebra $B_{L}$

Key Result

Lemma 2.2

Let $V$ be a nonlocal vertex algebra. Then the following weak associativity holds: For any $u,v,w\in V$, there exists $l\in {\mathbb{N}}$ such that

Theorems & Definitions (93)

  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 83 more