Vertex algebras generated by Lie algebras

In this paper we introduce a notion of vertex Lie algebra U, in a way a "ha lf" of vertex algebra structure sufficient to construct the corresponding l ocal Lie algebra L(U) and a vertex algebra V(U). We show that we may consid er U as a subset U subset of 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 vert ex Lie algebras extends uniquely to a homomorphism V(U) --> V of vertex alg ebras. Ln the second part of paper we study under what conditions on struct ure constants one can construct a vertex Lie algebra U by starting with a g iven commutator formula for fields. (C) 1999 Elsevier Science B.V. All righ ts reserved. 1991 Math. Subj. Class. 17B69.