Hom-Bol algebras

TL;DR

This paper introduces Hom-Bol algebras as a twisted extension of Bol algebras, explores their properties, and provides examples based on the classification of two-dimensional Bol algebras.

Contribution

It extends the concept of Bol algebras to Hom-Bol algebras, including their derived structures and closure properties, with explicit examples.

Findings

Hom-Bol algebras generalize Bol algebras via twisting.

The category of Hom-Bol algebras is closed under derived operations.

Examples are constructed from two-dimensional Bol algebras.

Abstract

Hom-Bol algebras are defined as a twisted generalization of (left) Bol algebras. Hom-Bol algebras generalize multiplicative Hom-Lie triple systems in the same way as Bol algebras generalize Lie triple systems. The notion of an $n$th derived (binary) Hom-algebra is extended to the one of an $n$th derived binary-ternary Hom-algebra and it is shown that the category of Hom-Bol algebras is closed under taking $n$th derived Hom-algebras. It is also closed by self-morphisms of binary-ternary Hom-algebras. Every Bol algebra is twisted into a Hom-Bol algebra. Relying on the well-known classification of real two-dimensional Bol algebras, examples of Hom-Bol algebras are given.