4-type subvarieties of the variety of associative algebras
A. Kunanbayev, B. Sartayev
TL;DR
The paper investigates four Mal'cev-type subvarieties of the variety of associative algebras, defined by degree-3 identities, and analyzes their structure through operad theory using white and black Manin products to connect them with noncommutative Novikov, dendriform, assosymmetric, and left-alternative operads. By employing polarization and Rota–Baxter embeddings, it shows how commutator and anti-commutator operations yield classical Lie- and Jordan-type identities within these subvarieties, and it derives explicit identities (up to degree 4) and a notable degree-5 relation in the first-type case. The work demonstrates universal embeddings into differential and RB-augmented algebras, clarifies Koszulness properties (notably non-Koszul for several duals), and provides a cohesive operadic framework tying together disparate algebraic structures. It also outlines several open problems, including complete identity classifications for the commutator and anti-commutator in the first-type case and embedding questions for metabelian Lie and Jordan algebras via these subvarieties, with potential implications for identifying Jordan elements in free associative algebras.
Abstract
In this paper, we consider four types of subvarieties of the variety of associative algebras. We study these subvarieties from the point of view of operads and show their connections with well-known classes of algebras, such as dendriform algebras and noncommutative Novikov algebras. Finally, we define the commutator and anti-commutator operations on these algebras and derive several identities satisfied by these operations.