Supercommutator (Hom-)superalgebras of right (Hom-)alternative superalgebras

TL;DR

This paper explores the algebraic structures derived from right (Hom-)alternative superalgebras, showing their supercommutator superalgebras form Hom-Bol superalgebras and establishing connections with Hom-Jordan and Hom-Lie supertriple systems.

Contribution

It introduces Hom-Bol superalgebras, proves their closure under self-morphisms, and demonstrates the supercommutator of right Hom-alternative superalgebras naturally forms a Hom-Bol superalgebra.

Findings

Supercommutator superalgebra of right alternative superalgebra is a Bol superalgebra

Hom-Bol superalgebras are closed under even self-morphisms

Supercommutator of right Hom-alternative superalgebra has a Hom-Bol structure

Abstract

It shown that the supercommutator superalgebra of a right alternative superalgebra is a Bol superalgebra. Hom-Bol superalgebras are defined and it is shown that they are closed under even self-morphisms. Any Bol superalgebra along with any even self-morphism is twisted into a Hom-Bol superalgebra. The supercommutator superalgebra of a right Hom-alternative superalgebra has a natural Hom-Bol structure. In order to prove this last result, the Hom-Jordan-admissibility of right Hom-alternative superalgebras is investigated and next Hom-Jordan supertriple systems are defined and their connection with Hom-Jordan superalgebras and Hom-Lie supertriple systems is considered.