A commutative algebra approach to multiplicative Hom-Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and $\textrm{HLie}_{m}(\mathfrak{g})$ be the affine variety of all multiplicative Hom-Lie algebras on $\mathfrak{g}$. We use a method of computational ideal theory to describe $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))$, showing that $\textrm{HLie}_{m}(\mathfrak{gl}_{2}(\mathbb{C}))$ consists of two 1-dimensional and one 3-dimensional irreducible components, and showing that $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))=\{\textrm{diag}\{\delta,\dots,\delta,a\}\mid \delta=1\textrm{ or }0,a\in\mathbb{C}\}$ 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 $\textrm{HLie}_{m}(\mathfrak{u}_{2}(\mathbb{C}))$ and $\textrm{HLie}_{m}(\mathfrak{u}_{3}(\mathbb{C}))$. We also study the derivation algebra $\textrm{Der}_{D}(\mathfrak{g})$ of a multiplicative Hom-Lie algebra $D$ on $\mathfrak{g}$ and under some hypotheses on $D$, we prove that the Hilbert series $\mathcal{H}(\textrm{Der}_{D}(\mathfrak{g}),t)$ is a rational function.