Affinization of algebraic structures: Leibniz algebras

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.