Affinization of algebraic structures: Leibniz algebras

TL;DR

This paper introduces a general method for extending Leibniz algebras into affine structures called Leibniz affgebras, exploring different types and their properties through examples.

Contribution

It defines affine Leibniz brackets and introduces Leibniz affgebras, extending Leibniz algebras into new affine algebraic structures with various types.

Findings

Leibniz affgebras generalize Leibniz algebras through affine structures.

Different types of Leibniz affgebras are characterized by the choice of Leibnizian.

Examples illustrate the structure and properties of each Leibniz affgebra type.

Abstract

A general procedure of affinization of linear algebra structures is illustrated by the case of Leibniz algebras. Specifically, the definition of an affine Leibniz bracket, that is, a bi-affine operation on an affine space that at each tangent vector space becomes a (bi-linear) Leibniz bracket in terms of a tri-affine operation called a Leibnizian, is given. An affine space together with such an operation is called a Leibniz affgebra. It is shown that any Leibniz algebra can be extended to a family of Leibniz affgebras. Depending on the choice of a Leibnizian different types of Leibniz affgebras are introduced. These include: derivative-type, which captures the derivation property of linear Leibniz bracket; homogeneous-type, which is based on the simplest and least restrictive choice of the Leibnizian; Lie-type which includes all Lie affgebras introduced in [R.R. Andruszkiewicz, T. Brzezi\'nski & K. Radziszewski, Lie affgebras vis-\`a-vis Lie algebras, Res. Math. 80 (2025), art. 61.]. Each type is illustrated by examples with prescribed Leibniz algebra fibres.