李仿代数的仿射幂零性与Engel定理
Affine Nilpotency and Engel's Theorem for Lie Affgebras
AI总结：
该研究引入李仿代数的ideaf、仿射幂零性等概念，刻画其结构性质，建立李仿代数与收缩李代数的幂零性关联，并证明一类李仿代数的Engel型定理。
AI中文摘要：
我们研究了作为李代数仿射类似物的李仿代数的若干结构性质。我们引入了ideaf（理想的仿射对应物）的概念，并建立了左、右ideaf的刻画。我们进一步研究了中心、商结构与乘积ideaf，在合适条件下证明了李仿代数的中心是一个ideaf。随后我们提出了仿射幂零性的概念，建立了李仿代数的幂零性与其收缩李代数的幂零性之间的联系。作为应用，我们证明了形如$\boldsymbol{\frak{a}}(\frak{g};\boldsymbol{\u03ba}=2\boldsymbol{\u03bb},\boldsymbol{\u03bb},s)$的李仿代数的Engel型定理。
英文摘要：
We study some structural properties of Lie affgebras as affine analogues of Lie algebras. We introduce the notion of an ideaf, the affine counterpart of an ideal, and establish characterizations of left and right ideafs. We further study centers, quotient structures, and product ideafs, proving, under suitable conditions, that the center of a Lie affgebra is an ideaf. We then develop the notion of affine nilpotency and establish connections between the nilpotency of Lie affgebras and that of their retracted Lie algebras. As an application, we prove an Engel-type theorem for Lie affgebras of the form $\mathfrak{a}(\mathfrak{g};κ=2λ,λ,s)$.
