三维乘法简单BiHom-李代数上Nijenhuis算子的分类

中文摘要 AI 辅助

我们在复数域C上给出了三维乘法简单BiHom-李代数上Nijenhuis算子的完整分类。基于Saadaoui对此类代数的分类（分为L1、L2和L3三个族），我们推导出由等变条件和Nijenhuis恒等式产生的完整方程组。我们给出了所有解的显式矩阵表示，包括所有退化参数情形。我们在BiHom-李代数的上链复形上构造了BiHom-Frölicher-Nijenhuis括号，证明了该分次李代数的Maurer-Cartan元素恰好是Nijenhuis算子。这使我们能够定义Nijenhuis算子的上同调并研究形式形变。我们引入BiHom-NS-李代数作为Nijenhuis算子背后的代数结构。我们给出了显式的形变BiHom-李代数结构，并证明了不同族之间的非同构性。我们还研究了Nijenhuis算子的模空间，并对所有结果进行了计算验证。

英文摘要

We provide a complete classification of Nijenhuis operators on three-dimensional multiplicative simple BiHom-Lie algebras over the complex field C. Based on the classification of such algebras by Saadaoui into three families L1, L2, and L3, we derive the complete system of equations arising from the equivariance conditions and the Nijenhuis identity. We provide explicit matrix representations of all solutions, including all degenerate parameter cases. We construct the BiHom-Frölicher-Nijenhuis bracket on the cochain complex of a BiHom-Lie algebra, showing that Maurer-Cartan elements of this graded Lie algebra are precisely Nijenhuis operators. This enables us to define the cohomology of Nijenhuis operators and study formal deformations. We introduce BiHom-NS-Lie algebras as the algebraic structure underlying Nijenhuis operators. We provide explicit deformed BiHom-Lie algebra structures and prove the non-isomorphism of different families. We also study the moduli space of Nijenhuis operators and provide computational verification of all results.

发表机构

• Department of Mathematics, Faculty of Sciences, University of Sfax（萨法克斯大学理学院数学系）

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