A commutative algebra approach to multiplicative Hom-Lie algebras

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