五维幂零右交错代数的代数分类

中文摘要 AI 辅助

我们发展了复数域上幂零右交错代数的中心扩张方法（Skjelbred--Sund 方法），并应用该方法获得五维幂零右交错代数的代数分类。更精确地，我们对所有没有零化子分量（即不是较小代数与一维零乘积代数的直和）且不是 2 步幂零的复数五维幂零右交错代数进行了同构分类。每个这样的代数都是三维非平凡幂零右交错代数（通过二维空间）或四维非平凡幂零右交错代数（通过一维空间）的非分裂中心扩张。对于相关的三维和四维代数，我们计算了第二上同调空间、自同构群及其在第二上同调上的作用，并确定了所有给出非分裂扩张的轨道。所得列表包含 124 个代数和代数族；其中包含 29 个具有二维零化子的代数和 95 个具有一维零化子的代数。

英文摘要

We develop the method of central extensions (the Skjelbred--Sund method) for nilpotent right alternative algebras over the field of complex numbers and apply it to obtain the algebraic classification of five-dimensional nilpotent right alternative algebras. More precisely, we classify, up to isomorphism, all complex five-dimensional nilpotent right alternative algebras that have no annihilator component (that is, which are not a direct sum of a smaller algebra and a one-dimensional algebra with zero product) and which are not $2$-step nilpotent. Every such algebra is a non-split central extension of a nontrivial nilpotent right alternative algebra of dimension three (by a two-dimensional space) or of dimension four (by a one-dimensional space). For each of the relevant three- and four-dimensional algebras we compute the second cohomology space, the automorphism group and its action on the second cohomology, and we determine all orbits that give non-split extensions. The resulting list consists of $124$ algebras and families of algebras; it contains $29$ algebras with two-dimensional annihilator and $95$ algebras with one-dimensional annihilator.

发表机构

• New Uzbekistan University（新乌兹别克斯坦大学）
• Turan International University（图兰国际大学）

机构由 AI 辅助整理，请以论文原文为准。