On the Dong Property for a binary quadratic operad

Kolesnikov P.S. Sartayev B.K.
1 April 2026 Academic Press Inc.

Journal of Algebra
2026 #691 428 - 452 pp.

The classical Dong Lemma for distributions over a Lie algebra lies in the foundation of the theory of vertex and conformal algebras. 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 for alternative, Novikov, and Novikov–Poisson algebras the Dong Lemma holds true. The criterion is stated in the language of operads, so we determine for which binary quadratic operads the Dong Lemma holds in the corresponding class of algebras. As an application, we show the black Manin product of such Dong operads is also a Dong operad.

Dong lemma , Identity , Manin product , Operad

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Sobolev Institute of Mathematics, Akad. Koptyug prosp., 4, Novosibirsk, 630090, Russian Federation
Narxoz University, Zhandossov str., 55, Almaty, 050035, Kazakhstan
SDU University, Abylai Khan str., 1/1, Kaskelen, 040900, Kazakhstan

Sobolev Institute of Mathematics
Narxoz University
SDU University

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