Vertex algebras generated by Lie algebras

In this paper we introduce a notion of vertex Lie algebra U , in a way a “half” of vertex algebra structure sufficient to construct the corresponding local Lie algebra L(U) and a vertex algebra V(U). We show that we may consider U as a subset U ⊂ V(U) which generates V(U) and that the vertex Lie algebra structure on U is induced by the vertex algebra structure on V(U). Moreover, for any vertex algebra V a given homomorphism U → V of vertex Lie algebras extends uniquely to a homomorphism V(U) → V of vertex algebras. In the second part of paper we study under what conditions on structure constants one can construct a vertex Lie algebra U by starting with a given commutator formula for fields.