Cocommutative vertex bialgebras

TL;DR

This paper characterizes cocommutative vertex bialgebras, showing they decompose into components related to vertex Lie algebras and establishing an equivalence of categories for connected cases.

Contribution

It proves that cocommutative connected vertex bialgebras are equivalent to vertex Lie algebras, providing a structural decomposition and explicit construction.

Findings

Decomposition of cocommutative vertex bialgebras into connected components.

Equivalence between cocommutative connected vertex bialgebras and vertex Lie algebras.

Explicit description of the coalgebra structure when the group of group-like elements is central.

Abstract

In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra $V$, it is proved that the set $G(V)$ of group-like elements is naturally an abelian semigroup, whereas the set $P(V)$ of primitive elements is a vertex Lie algebra. For $g\in G(V)$, denote by $V_g$ the connected component containing $g$. Among the main results, it is proved that if $V$ is a cocommutative vertex bialgebra, then $V=\oplus_{g\in G(V)}V_g$, where $V_{\bf 1}$ is a vertex subbialgebra which is isomorphic to the vertex bialgebra ${\mathcal{V}}_{P(V)}$ associated to the vertex Lie algebra $P(V)$, and $V_g$ is a $V_{\bf 1}$-module for $g\in G(V)$. In particular, this shows that every cocommutative connected vertex bialgebra $V$ is isomorphic to ${\mathcal{V}}_{P(V)}$ and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Furthermore, under the condition that $G(V)$ is a group and lies in the center of $V$, it is proved that $V={\mathcal{V}}_{P(V)}\otimes \C[G(V)]$ as a coalgebra where the vertex algebra structure is explicitly determined.