AI 中文总结
本文研究预辛左对称代数的结构性质，引入Milnor预辛代数子类，揭示其与辛李代数的关联及双扩张构造，完成维数≤4的这类代数的完整分类。
AI 中文摘要
预辛左对称代数$\boldsymbol{\textit{A}}$是一类左对称代数，其带有非退化斜对称双线性形式$\boldsymbol{\textit{ω}}$，使得所有左乘算子关于$\boldsymbol{\textit{ω}}$对称。在此设定下，其基础的伴随李代数$(\boldsymbol{\textit{A}}^-,\boldsymbol{\textit{ω}})$构成平坦$T$-辛李代数。本文系统研究预辛左对称代数的结构性质，特别引入一类特殊子类，命名为“Milnor预辛代数”，并证明任何换位理想非退化的预辛左对称代数必然属于该子类。接下来，研究与辛李代数相关的Levi-Civita积，证明该积总能生成右对称代数，且当且仅当该积是结合的时，它构成左对称代数。此外，通过左对称代数的表示给出辛李代数的刻画，并建立这类结构的构造方法——$T^*$-扩张。进一步，借助交换结合代数为预辛左对称代数建立双扩张程序，证明每个换位理想退化的这类代数均可通过该扩张过程重构；更一般地，证明任何预辛左对称代数要么是Milnor预辛代数，要么可通过从Milnor预辛代数出发的有限次连续双扩张得到。作为这些结构结果的具体应用，本文对维数小于等于4的预辛左对称代数给出完整分类。
英文摘要
A pre-symplectic left-symmetric algebra $\mathcal{A}$ is a left-symmetric algebra endowed with a nondegenerate skew-symmetric bilinear form $ω$ such that all left multiplication operators are symmetric with respect to $ω$. In this setting, the underlying subadjacent Lie algebra $(\mathcal{A}^{-},ω)$ forms a flat $T$-symplectic Lie algebra. This paper provides a systematic investigation into the structural properties of pre-symplectic left-symmetric algebras. In particular, we introduce a distinguished subclass termed \emph{Milnor pre-symplectic algebras}, and prove that any pre-symplectic left-symmetric algebra whose commutator ideal is nondegenerate necessarily belongs to this subclass. Next, we investigate the Levi-Civita product associated with symplectic Lie algebras. We show that this product always yields a right-symmetric algebra, and we prove that it forms a left-symmetric algebra if and only if it is associative. Furthermore, we provide a characterization of symplectic Lie algebras in terms of representations of left-symmetric algebras, and conclude by establishing a construction method for these structures known as the $T^*$-extension. Furthermore, we develop a double extension procedure for pre-symplectic left-symmetric algebras by means of commutative associative algebras. We show that every such algebra with a degenerate commutator ideal can be reconstructed via this extension process. More generally, we show that any pre-symplectic left-symmetric algebra is either a Milnor pre-symplectic algebra or can be obtained through a finite sequence of successive double extensions starting from a Milnor pre-symplectic algebra. As a concrete application of these structural results, we provide a complete classification of pre-symplectic left-symmetric algebras of dimension less than or equal to $4$.
