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

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