发表机构
Oles Honchar Dnipro National University（奥列斯·洪恰尔第聂伯罗国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了任意域上维数至多三的泊松代数结构，通过结合平方和导李代数等内部结构，给出了维数一、二的完全分类和维数三的结构分类，并揭示了特征2等特殊现象。
AI 中文摘要
我们研究了任意域上维数至多为三的泊松代数的结构。我们的方法基于其相伴的交换结合代数和李代数的内部结构，特别关注结合平方$P^2$、导李代数$[P,P]$、李中心、结合零化子、幂等元、理想及可分解性。我们在维数一和二上获得了完全分类，并在维数三上给出了结构分类。在维数二上，我们证明了结合乘法和李乘法不能同时非零。在维数三上，分类根据导李代数的维数和位置来组织，在李乘法平凡的情形下，则根据$P^2$的维数来组织。任意域的设置导致了在复数域上不会出现的现象。特别地，二次和三次域扩张自然地出现在分类中，一些族依赖于对称双线性形式的等价类以及基域上三维李代数的结构，并且特征$2$给出了一个额外的具有两种乘法均非零的泊松代数族。在复数域上，所得分类在基变换和记号调整的意义下，归结为维数至多三的已知分类。
英文摘要
We investigate the structure of Poisson algebras of dimensions at most three over an arbitrary field. Our approach is based on the internal structure of the associated commutative associative and Lie algebras, with particular emphasis on the associative square $P^2$, the derived Lie algebra $[P,P]$, the Lie center, the associative annihilator, idempotents, ideals and decomposability. We obtain a complete classification in dimensions one and two and give a structural classification in dimension three. In dimension two, we prove that the associative and Lie multiplications cannot be simultaneously non-zero. In dimension three, the classification is organized according to the dimension and position of the derived Lie algebra and, in the case of trivial Lie multiplication, according to the dimension of $P^2$. The arbitrary-field setting leads to phenomena which do not occur over the complex field. In particular, quadratic and cubic field extensions appear naturally in the classification, some families depend on equivalence classes of symmetric bilinear forms and on the structure of three-dimensional Lie algebras over the ground field, and characteristic $2$ gives an additional family of Poisson algebras with both multiplications non-zero. Over the complex field, the resulting classification specializes, up to changes of basis and notation, to the known classifications in dimensions at most three.
