Pre-vertex algebras, pre-Poisson vertex algebras and their deformation quantizations

Abstract

In this paper, we introduce pre-vertex algebras and pre-Poisson vertex algebras as preLie-algebras in the chiral and classical pseudo-tensor categories, respectively. We show that a pre-vertex algebra gives rise to a vertex algebra, that every Rota-Baxter operator on a vertex algebra induces a pre-vertex algebra, and that every pre-vertex algebra can be embedded into a vertex algebra equipped with a Rota-Baxter operator. Moreover, we prove that pre-vertex algebras are equivalent to dendriform vertex algebras. In the Poisson setting, we demonstrate that pre-Poisson vertex algebras are obtained from Rota-Baxter operators on Poisson vertex algebras, from filtrations of pre-vertex algebras, and as classical limits of pre-vertex formal deformations of differential Zinbiel algebras. This extends the classical relationships among Rota-Baxter operators, pre-Lie algebras, and Poisson algebras to the framework of vertex algebras.