A commutative algebra approach to multiplicative Hom-Lie algebras

We use a method of commutative algebra to describe the affine variety $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))$ all multiplicative Hom-Lie algebras on general linear Lie $\mathfrak{gl}_{n}(\mathbb{C})$, showing that $\textrm{HLie}_{m}(\mathfrak{gl}_{2}(\mathbb{C}))$ consists two 1-dimensional and one 3-dimensional irreducible components. also prove $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))=\{\textrm{diag}\{\delta,\dots,\delta,a\}\mid \delta=1\textrm{ or }0,a\in\mathbb{C}\}$ for $n\geqslant 3$.