The Pozhidaev and Cantarini--Kac Constructions of Simple n-Lie Algebras: Distinctions and Realizations
Abstract
Let n≥ 3, let H⊆ℂn be any additive subgroup spanning ℂn, and let 0≠ t∈ H. We study Pozhidaev's central simple n-Lie algebra P(H,t)=𝒜̃(H,t)/ℂe0 without a finite generation or discreteness assumption on H. Its inner derivation algebra is the simple generalized divergence-free Lie algebra 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 ℂ[H]. Finally, we realize Pozhidaev's second construction E(H) over ℂ as a W-algebra.
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BibTeX
@misc{cao2026pozhidaev,
title = {{The Pozhidaev and Cantarini--Kac Constructions of Simple $n$-Lie Algebras: Distinctions and Realizations}},
author = {Xinru Cao and Bakhrom A. Omirov and Yuhui Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07569},
eprint = {2609.07569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07569},
}
RIS
TY - UNPB
TI - The Pozhidaev and Cantarini--Kac Constructions of Simple n-Lie Algebras: Distinctions and Realizations
AU - Xinru Cao
AU - Bakhrom A. Omirov
AU - Yuhui Tan
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07569
UR - https://arxiv.org/abs/2609.07569
AB - Let n≥ 3, let H⊆ℂ^n be any additive subgroup spanning ℂ^n, and let 0≠ t∈ H. We study Pozhidaev's central simple n-Lie algebra P(H,t)=𝒜̃(H,t)/ℂe_0 without a finite generation or discreteness assumption on H. Its inner derivation algebra is the simple generalized divergence-free Lie algebra 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 ℂ[H]. Finally, we realize Pozhidaev's second construction E(H) over ℂ as a W-algebra.
ER -