The Pozhidaev and Cantarini--Kac Constructions of Simple $n$-Lie Algebras: Distinctions and Realizations
Year: 2026
Type: preprint
Abstract: Let $n\geq 3$, let $H\subseteq\mathbb{C}^n$ be any additive subgroup spanning $\mathbb{C}^n$, and let $0\ne t\in H$. We study Pozhidaev's central simple $n$-Lie algebra $P(H,t)=\widetilde{\mathcal A}(H,t)/\mathbb{C}e_0$ without a finite generation or discreteness assumption on $H$. Its inner derivation algebra is the simple generalized divergence-free Lie algebra $\mathcal S(0,0,n;t,H)$. We prove that its space of inner-equivariant symmetric products vanishes and that every $1/n$-derivation is a scalar multiple of the identity. Using these invariants, we show that $P(H,t)$ is not isomorphic to any simple nonabelian $n$-Lie algebra defined on the underlying spaces of the $S$, $W$, or $SW$ constructions recorded by Cantarini and Kac. We also show that $P(H,t)$ is the quotient by the constants of the derived algebra of an explicit $S$-algebra on $\mathbb{C}[H]$. Finally, we realize Pozhidaev's second construction $E(H)$ over $\mathbb{C}$ as a $W$-algebra.
Source: arXiv (Cornell University)
Authors: Cao, Xinru, Omirov, Bakhrom A., Tan, Yuhui
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211992378