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 V g 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 ) V g , where V 1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra V P (V ) associated to the vertex Lie algebra P (V ), and V g is a V 1 -module for g ∈ G(V ). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to 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 = V P (V ) ⊗ C[G(V )] as a coalgebra where the vertex algebra structure is explicitly determined.