海森堡超代数上的转置泊松超代数的分类

Classification of Transposed Poisson Superalgebras over Heisenberg Superalgebras

摘要: 本文主要研究了海森堡超代数上的转置泊松超代数的自同构和分类。首先介绍了海森堡超代数和转置泊松超代数的定义，然后对于任意维转置泊松超代数，计算了海森堡超代数上的转置泊松超代数的代数运算的特点。最后，研究海森堡超代数上的转置泊松超代数的分类，找到两个海森堡超代数上的转置泊松超代数同构的条件，利用转置泊松超代数的反对称双线性函数，得到任意维海森堡超代数上的转置泊松超代数的完整分类。

Abstract: This paper mainly investigates the automorphisms and classification of transposed Poisson superalgebras over Heisenberg superalgebras. Firstly, we introduce the definitions of Heisenberg superalgebras and transposed Poisson superalgebras. Then, for transposed Poisson superalgebras of arbitrary dimensions, we calculate the properties of algebraic operations of transposed Poisson superalgebras over Heisenberg superalgebras. Finally, we study the classification of such transposed Poisson superalgebras, obtain the isomorphism conditions between two transposed Poisson superalgebras over Heisenberg superalgebras, and further derive a complete classification of transposed Poisson superalgebras over Heisenberg superalgebras of arbitrary dimensions by virtue of antisymmetric bilinear functions on transposed Poisson superalgebras.