A smash product construction of nonlocal vertex algebras

A notion of vertex bialgebra and a notion of module nonlocal vertex algebra for a vertex bialgebra are studied and then a smash product construction of nonlocal vertex algebras is presented. For every nonlocal vertex algebra $V$ satisfying a suitable condition, a canonical bialgebra $B(V)$ is constructed such that primitive elements of $B(V)$ are essentially pseudo derivations and group-like elements are essentially pseudo endomorphisms. Furthermore, vertex algebras associated with Heisenberg Lie algebras as well as those associated with nondegenerate even lattices are reconstructed through smash products.