Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and ${\mathrm{HLie}}_m(\mathfrak{g})$ be the affine variety of all multiplicative Hom-Lie algebras on $\mathfrak{g}$. We use a method of computational ideal theory to describe ${\mathrm{HLie}}_m({\mathfrak{g}\mathfrak{l}}_n(\mathbb{C}))$, showing that ${\mathrm{HLie}}_m({\mathfrak{g}\mathfrak{l}}_2(\mathbb{C}))$ consists of two 1-dimensional and one 3-dimensional irreducible components and ${\mathrm{HLie}}_m({\mathfrak{g}\mathfrak{l}}_n(\mathbb{C}))=\{\mathrm{diag}\{\delta ,\dots ,\delta ,a\}\mid \delta =1\mathrm{or}0,a\in \mathbb{C}\}$ for $n⩾3$. We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra ${\mathfrak{h}}_{2n+1}(\mathbb{C})$ and characterize the affine varieties ${\mathrm{HLie}}_m({\mathfrak{u}}_2(\mathbb{C}))$ and ${\mathrm{HLie}}_m({\mathfrak{u}}_3(\mathbb{C}))$. We also study the derivation algebra ${\mathrm{Der}}_D(\mathfrak{g})$ of a multiplicative Hom-Lie algebra D on $\mathfrak{g}$ and, under some hypotheses on D, we prove that the Hilbert series $\mathcal{H}({\mathrm{Der}}_D(\mathfrak{g}),t)$ is a rational function.

让$\mathfrak{G}$是有限维复李代数和${\mathrm{H谎言}}_{米}(\mathfrak{G})$是所有乘法 Hom-Lie 代数的仿射变体$\mathfrak{G}$. 我们使用计算理想理论的方法来描述${\mathrm{H谎言}}_{米}({\mathfrak{G}\mathfrak{升}}_n(\mathbb{C}))$, 表明${\mathrm{H谎言}}_{米}({\mathfrak{G}\mathfrak{升}}_{\mathrm{2个}}(\mathbb{C}))$由两个 1 维和一个 3 维不可约成分组成，并且${\mathrm{H谎言}}_{米}({\mathfrak{G}\mathfrak{升}}_n(\mathbb{C}))=\{\mathrm{诊断}\{\delta ,\dots \dots ,\delta ,A\}\forall \delta =\mathrm{1个}\mathrm{或者}0,A\in \mathbb{C}\}$为了$n⩾\mathrm{3个}$. 我们在海森堡李代数上构建了一个新的乘法 Hom-Lie 代数族${\mathfrak{H}}_{\mathrm{2个}n+\mathrm{1个}}(\mathbb{C})$并表征仿射品种${\mathrm{H谎言}}_{米}({\mathfrak{你}}_{\mathrm{2个}}(\mathbb{C}))$和${\mathrm{H谎言}}_{米}({\mathfrak{你}}_{\mathrm{3个}}(\mathbb{C}))$. 我们还学习推导代数${\mathrm{德}}_{丁}(\mathfrak{G})$乘法 Hom-Lie 代数D在$\mathfrak{G}$并且，根据关于D 的一些假设，我们证明了希尔伯特级数$\mathcal{H}({\mathrm{德}}_{丁}(\mathfrak{G}),吨)$是有理函数。