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

TL;DR

This paper develops a duality theory for vertex algebras and coalgebras within the category of rational modules over diagonalisable group schemes, introducing $C_2$-coalgebras and exploring their module relationships.

Contribution

It introduces the notion of $C_2$-coalgebras for vertex coalgebras and establishes a duality between vertex algebras and coalgebras in the context of $G_ ext{Gamma}$-modules.

Findings

Established duality between vertex algebras and coalgebras.

Defined $C_2$-coalgebras in the vertex coalgebra setting.

Explored module and comodule relationships for these structures.

Abstract

In this paper, we define vertex algebras and vertex coalgebras in the category of rational $G_\Gamma$-modules, where $G_\Gamma$ is the group scheme defined by the group algebra $\mathsf k \Gamma$ for an abelian group $\Gamma$. 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 $G_\Gamma$-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.