The Pozhidaev and Cantarini--Kac Constructions of Simple $n$-Lie Algebras: Distinctions and Realizations

Abstract

Let $n\ge 3$, 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 $P(H,t)=\tilde{\mathcal{A}}(H,t)/\mathbb{C}{e}_0$ 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.