AI 中文总结
本文系统刻画多类李代数上的 Hom-Lie 结构，给出维数下界与余维数界，并计算具体代数实例，纠正文献错误，揭示其与转置 Poisson 结构及导子的联系。
AI 中文摘要
我们系统地描述了若干类 Lie 代数上的 $\Hom$-Lie 结构及其与转置 Poisson 结构的关系。我们的出发点是观察到定义恒等式的左边是一个交错三线性映射；这使得 $\mathcal{H}(\LL)$（即 $\LL$ 的 $\Hom$-Lie 结构集合）成为 $\End(\LL)$ 的一个子空间，并将其计算简化为对递增基向量三元组的处理。由此，在任意有限维数 $n$ 下，我们推导出下界 $\dim \mathcal{H}(\LL)\geq n \dim \mathcal{C}(\LL)$ 以及余维数界 $\binom{n}{3}\dim [\LL,[\LL,\LL]]$，并给出后者的改进形式。我们确定了幂零代数、振子代数和可解代数、$\sltwo \oplus A_{n-3}$、Witt 代数、Virasoro 代数和 Heisenberg--Virasoro 代数、所有三维代数、$M_n(\bF)$ 和 $T_n(\bF)$、类 Virasoro 代数及其 $q$-模拟，以及平面伽利略共形代数的 $\mathcal{H}(\LL)$。根据 Filippov 的一个结果，Lie 代数的每个 $\frac12$-导子，更一般地，每个 $\delta\neq0,1$ 的 $\delta$-导子，都是 $\Hom$-Lie 结构；我们观察到，更一般地，对于 $\delta\neq 0,1$，由次数至多为三的恒等式定义的任何代数簇（结合、Novikov、左对称、Leibniz 等）的 $\delta$-导子都是相应 $\Hom$-簇的 $\Hom$-结构。通过计算 Witt 型、Virasoro 型和 Schrödinger 型代数及其中心扩张的分次 $\Hom$-Lie 结构，并给出显式的提升障碍，我们表明 $\frac12$-导子包含于 $\Hom$-Lie 结构之中远非等式。在此过程中，我们纠正了文献中两个违反 Jacobi 恒等式的乘法表。
英文摘要
We give a systematic description of $\Hom$-Lie structures on several classes of Lie algebras and of their relations with transposed Poisson structures. Our starting point is the observation that the left-hand side of the defining identity is an alternating trilinear map; this makes $\mathcal{H}(\LL)$, the set of $\Hom$-Lie structures of $\LL$, a subspace of $\End(\LL)$ and reduces its computation to increasing triples of basis vectors. From this we derive, in every finite dimension $n$, the lower bound $\dim \mathcal{H}(\LL)\geq n \dim \mathcal{C}(\LL)$ and the codimension bound $\binom{n}{3}\dim [\LL,[\LL,\LL]]$, together with a refinement of the latter, and we determine $\mathcal{H}(\LL)$ for nilpotent, oscillator and solvable algebras, for $\sltwo \oplus A_{n-3}$, for the Witt, Virasoro and Heisenberg--Virasoro algebras, for all three-dimensional algebras, for $M_n(\bF)$ and $T_n(\bF)$, for the Virasoro-like algebra and its $q$-analogue, and for the planar Galilean conformal algebra. By a result of Filippov, every $\frac12$-derivation, and more generally every $δ$-derivation with $δ\neq0,1$, of a Lie algebra is a $\Hom$-Lie structure; we observe that, more generally, for $δ\neq 0,1$ the $δ$-derivations of an algebra of any variety defined by identities of degree at most three (associative, Novikov, left-symmetric, Leibniz, \ldots) are $\Hom$-structures of the corresponding $\Hom$-variety. We show that the inclusion of the $\frac12$-derivations into the $\Hom$-Lie structures is far from being an equality by computing the graded $\Hom$-Lie structures of Witt-, Virasoro- and Schrödinger-type algebras and of their central extensions, for which we give an explicit lifting obstruction. Along the way we correct two multiplication tables from the literature that violate the Jacobi identity.
Comments38 pages, Aristotle (Harmonic) was used
