A commutative algebra approach to multiplicative Hom-Lie algebras

Abstract

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.

Keywords:

• Hom-Lie algebra
• general linear Lie algebra
• Heisenberg Lie algebra

2020 Mathematics Subject Classifications:

• 17B61
• 13P10
• 13P15

Acknowledgements

The authors would like to thank the two referees and the editor for their helpful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by NNSF of China (No. 11301061).