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Equivariant vertex coalgebras, $C_2$-coalgebras and duality for diagonalisable group schemes

Antoine Caradot, Zongzhu Lin

TL;DR

The paper develops a framework for vertex (co)algebras and their representation theory within the category of rational $G_\Gamma$-modules, introducing $(G_\Gamma,\beta,\gamma_0)$-vertex (co)algebras and their (co)modules and analyzing how a group-equivariant loop algebra and continuous distributions govern their structure. It extends Zhu-style $C_2$-constructions to the $G_\Gamma$-equivariant setting, defining $C_2$-algebras and $C_2$-coalgebras and proving that $R(V)$ carries a $\beta$-commutative Poisson algebra while $\mathcal{R}(V)$ carries a $\beta$-cocommutative co-Poisson coalgebra. A central result is a duality between vertex algebras and vertex coalgebras in this non-rigid ambient category, together with corresponding dualities for their $C_2$-objects and modules/comodules, yielding a rich interplay between algebraic and coalgebraic perspectives. The work provides a categorical blueprint for studying equivariant vertex structures over derived stacks and suggests avenues for derived vertex algebras and geometric invariant theory in this context. Overall, it unifies operator-distribution viewpoints with categorical and Poisson-theoretic structures in a diagonalisable-group setting, laying foundations for further representation-theoretic and geometric applications.

Abstract

In this paper, we define vertex algebras and vertex coalgebras in the category of rational $G_Γ$-modules, where $G_Γ$ is the group scheme defined by the group algebra $\mathsf k Γ$ for an abelian group $Γ$. In this context, we introduce the notion of $C_2$-coalgebra for a vertex coalgebra. We prove that there exists a duality between vertex algebras and vertex coalgebras in the category of $G_Γ$-modules, and this duality establishes a connection between $C_2$-algebras and $C_2$-coalgebras. Moreover, we also investigate the relationship between their respective modules/comodules.

Equivariant vertex coalgebras, $C_2$-coalgebras and duality for diagonalisable group schemes

TL;DR

The paper develops a framework for vertex (co)algebras and their representation theory within the category of rational -modules, introducing -vertex (co)algebras and their (co)modules and analyzing how a group-equivariant loop algebra and continuous distributions govern their structure. It extends Zhu-style -constructions to the -equivariant setting, defining -algebras and -coalgebras and proving that carries a -commutative Poisson algebra while carries a -cocommutative co-Poisson coalgebra. A central result is a duality between vertex algebras and vertex coalgebras in this non-rigid ambient category, together with corresponding dualities for their -objects and modules/comodules, yielding a rich interplay between algebraic and coalgebraic perspectives. The work provides a categorical blueprint for studying equivariant vertex structures over derived stacks and suggests avenues for derived vertex algebras and geometric invariant theory in this context. Overall, it unifies operator-distribution viewpoints with categorical and Poisson-theoretic structures in a diagonalisable-group setting, laying foundations for further representation-theoretic and geometric applications.

Abstract

In this paper, we define vertex algebras and vertex coalgebras in the category of rational -modules, where is the group scheme defined by the group algebra for an abelian group . In this context, we introduce the notion of -coalgebra for a vertex coalgebra. We prove that there exists a duality between vertex algebras and vertex coalgebras in the category of -modules, and this duality establishes a connection between -algebras and -coalgebras. Moreover, we also investigate the relationship between their respective modules/comodules.
Paper Structure (14 sections, 35 theorems, 187 equations)

This paper contains 14 sections, 35 theorems, 187 equations.

Table of Contents

  1. The category of representations of diagonalisable groups
  2. The category of rational $G_\Gamma$-modules
  3. The loop algebra and continuous functions
  4. Vertex (co)algebras
  5. Vertex (co)algebra (co)modules
  6. The $C_2$-algebra and its associated modules
  7. The structure of the $C_2$-algebra
  8. The associated $C_2$-modules
  9. The $C_2$-coalgebra and its associated comodules
  10. The structure of the $C_2$-coalgebra
  11. The associated $C_2$-comodules
  12. Duality for $C_2$-(co)algebras and $C_2$-(co)modules
  13. The algebra-coalgebra correspondence
  14. The module-comodule correspondence

Key Result

Lemma 2.3

Let $V$ be a $\mathsf{k}$-vector space with $V^*$ the dual vector space, and consider a sequence $\{v_n\}_{n \in \mathbb{Z}}$ of elements in $V$. Let $z$ be a formal variable. Then

Theorems & Definitions (86)

  • Definition 1.1
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • ...and 76 more