Malcev classification for the variety of left-symmetric algebras
A. Ryskeldin, B. Sartayev
TL;DR
The paper classifies subvarieties of left-symmetric algebras via Malcev's framework and computes their Koszul dual operads, establishing connections to well-known operads such as the alternative, assosymmetric and Zinbiel families. For three subfamilies $\mathcal{LS}_{\mathfrak{A}_1}$, $\mathcal{LS}_{\mathfrak{B}_1}$, and $\mathcal{LS}_{\mathfrak{A}_2}$, it determines the dual operads, constructs explicit bases for the free algebras, and derives polynomial identities for the commutator and anti-commutator up to degree $4$. The duals are shown to align with the alternative, assosymmetric, and a variant of the alternative (with a left-center identity) operad, with additional degree-$4$ identities appearing in certain cases; the work also places these results within a broader diagram of operads and discusses links to Zinbiel and Perm algebras. Overall, the results illuminate the structural relationships among Malcev subvarieties of left-symmetric algebras and provide concrete bases and dimension data for their Koszul duals, enabling explicit computations and further applications in deformation theory and invariant theory.
Abstract
In this paper, we study three classes of subvarieties inside the variety of left-symmetric algebras. We show that these subvarieties are naturally related to some well-known varieties, such as alternative, assosymmetric and Zinbiel algebras. For certain subvarieties of the varieties of alternative and assosymmetric algebras, we explicitly construct bases of the corresponding free algebras. We then define the commutator and anti-commutator operations on these algebras and derive a number of identities satisfied by these operations in all degrees up to $4$.