On the Dong Property for a binary quadratic operad

Abstract

The classical Dong Lemma for distributions over a Lie algebra lies in the foundation of vertex algebras theory. In this paper, we find necessary and sufficient condition for a variety of nonassociative algebras with binary operations to satisfy the analogue of the Dong Lemma. In particular, it turns out that Novikov and Novikov--Poisson algebras satisfy the Dong Lemma. The criterion is stated in the language of operads, so we determine for which binary quadratic operads the Dong Lemma holds true. As an application, we show the black Manin product of Dong operads is also a Dong operad.