On the Dong Property for a binary quadratic operad

TL;DR

This paper explores the conditions under which a class of nonassociative algebras satisfy an analogue of the Dong Lemma, using operad theory to identify when the property holds for binary quadratic operads.

Contribution

It establishes necessary and sufficient conditions for the Dong Lemma to hold in various nonassociative algebras, including Novikov and Novikov--Poisson algebras, via operad criteria.

Findings

Novikov and Novikov--Poisson algebras satisfy the Dong Lemma

Operad-based criterion for Dong Lemma applicability

Black Manin product of Dong operads is also a Dong 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.