二维anti-Zinbiel超代数的分类

Classification of Two-Dimensional anti-Zinbiel Hyperalgebras

摘要: 本文研究anti-Zinbiel超代数的构造和二维anti-Zinbiel超代数的分类。首先给出anti-Zinbiel超代数的定义，并建立了与交换结合超代数之间的关系。进一步地，利用二维交换结合代数的分类，我们得到二维anti-Zinbiel超代数在同构意义下的完全分类。最后，引入交换结合超代数关于表示的强anti-O-算子与强anti-Rota-Baxter算子的定义，利用它们构造anti-Zinbiel超代数结构，并给出了低维例子。

Abstract: This paper focuses on the construction of anti-Zinbiel superalgebras and the classification of two-dimensional anti-Zinbiel superalgebras. First, we give the definition of anti-Zinbiel superalgebras and establish their relationship with commutative associative superalgebras. Furthermore, by utilizing the classification of two-dimensional commutative associative superalgebras, we obtain the complete classification of two-dimensional anti-Zinbiel superalgebras up to isomorphism. Finally, we introduce the definitions of strong anti-operators and strong anti-Rota-Baxter operators on the representations of commutative associative superalgebras, construct the structures of anti-Zinbiel superalgebras by virtue of these operators, and present corresponding low-dimensional examples.