Cocommutative vertex bialgebras

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 $gin G(V)$, 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 $V=oplus_{gin G(V)}V_g$, 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 $gin G(V)$. 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.