Malcev dialgebras

TL;DR

This paper introduces Malcev dialgebras, explores their defining identities, and establishes their connection to Leibniz triple systems, advancing the understanding of nonassociative algebraic structures.

Contribution

It applies Kolesnikov's algorithm to define Malcev dialgebras, verifies their identities computationally, and links them to Leibniz triple systems, revealing new structural insights.

Findings

Identities of Malcev dialgebras are equivalent to degree <= 4 identities in alternative dialgebras.

Any special identity for Malcev dialgebras has degree at least 7.

Malcev dialgebras can be endowed with a trilinear operation forming Leibniz triple systems.

Abstract

We apply Kolesnikov's algorithm to obtain a variety of nonassociative algebras defined by right anticommutativity and a `noncommutative' version of the Malcev identity. We use computational linear algebra to verify that these identities are equivalent to the identities of degree <= 4 satisfied by the dicommutator in every alternative dialgebra. We extend these computations to show that any special identity for Malcev dialgebras must have degree at least 7. Finally, we introduce a trilinear operation which makes any Malcev dialgebra into a Leibniz triple system.