A commutative algebra approach to multiplicative Hom-Lie algebras

Abstract: 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.