标题： 代数结构的仿射化：李布尼茨代数 显示英文标题

标题： Affinization of algebraic structures: Leibniz algebras

摘要： 线性代数结构的仿射化一般过程通过Leibniz代数的情况进行说明。 具体来说，给出了仿射Leibniz括号的定义，即在仿射空间上的一种双仿射运算，在每个切向量空间上成为（双线性）Leibniz括号，这是通过一个称为Leibnizian的三重仿射运算来实现的。 一个带有这种运算的仿射空间被称为Leibniz仿代数。 表明任何Leibniz代数都可以扩展为一个Leibniz仿代数族。 根据Leibnizian的选择不同，引入了不同类型的Leibniz仿代数。 这些包括：导数型，捕捉线性Leibniz括号的导子性质；齐次型，基于Leibnizian最简单且限制最少的选择；李型，包括在[R.R. Andruszkiewicz, T. Brzeziński & K. Radziszewski, Lie affgebras vis-à-vis Lie algebras, Res. Math. 80 (2025), art. 61.]中引入的所有李仿代数。 每种类型都通过具有指定Leibniz代数纤维的例子进行说明。 显示英文摘要

摘要： 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.