Cocommutative vertex bialgebras

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∈G(V), denote by Vg the connected component containing g. Among the main results, it is proved that if V is a cocommutative vertex bialgebra, then V=⊕g∈G(V)Vg, where V1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra VP(V) associated to the vertex Lie algebra P(V), and Vg is a V1-module for g∈G(V). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP(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=VP(V)⊗C[G(V)] as a coalgebra where the vertex algebra structure is explicitly determined.