Vertex coalgebras, comodules, cocommutativity and coassociativity

TL;DR

This paper introduces vertex coalgebras, explores their algebraic properties like cocommutativity and coassociativity, and discusses their structures and examples, especially in relation to vertex Lie algebras and enveloping algebras.

Contribution

It generalizes vertex operator coalgebras to vertex coalgebras and analyzes their properties and structures, including comodules and specific instances.

Findings

Vertex coalgebras generalize vertex operator coalgebras.

Cocommutativity and coassociativity require grading.

Examples include graded vertex coalgebras related to vertex Lie algebras.

Abstract

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc's vertex Lie algebra and (universal) enveloping vertex algebras.