Wells exact sequences for n-Lie algebras
Abstract
We study derivations and automorphisms of arbitrary non-abelian extensions of n-Lie algebras through their full crossed-product data. Their relative center is a canonical coefficient module, and a third cohomology class obstructs completion of data satisfying the mixed Filippov identities. When this class vanishes, a relative second cohomology space classifies the remaining extension data. We prove a non-abelian good criterion, construct the Lie algebra and group of compatible pairs, and obtain derivation and automorphism Wells sequences with explicit lifting equations. The derivation obstruction is a Lie algebra $1$-cocycle; the additive automorphism obstruction is a crossed homomorphism. We distinguish the ordinary center from the relative center, and outer derivations of the kernel from the mixed action required. Sparse crossed products are treated, including the representation class of their second mixed operation and the additional equations governing changes of sparse sections.
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BibTeX
@misc{zhang2026wells,
title = {{Wells exact sequences for $n$-Lie algebras}},
author = {Tao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10561},
eprint = {2610.10561},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10561},
}
RIS
TY - UNPB
TI - Wells exact sequences for n-Lie algebras
AU - Tao Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.10561
UR - https://arxiv.org/abs/2610.10561
AB - We study derivations and automorphisms of arbitrary non-abelian extensions of n-Lie algebras through their full crossed-product data. Their relative center is a canonical coefficient module, and a third cohomology class obstructs completion of data satisfying the mixed Filippov identities. When this class vanishes, a relative second cohomology space classifies the remaining extension data. We prove a non-abelian good criterion, construct the Lie algebra and group of compatible pairs, and obtain derivation and automorphism Wells sequences with explicit lifting equations. The derivation obstruction is a Lie algebra $1$-cocycle; the additive automorphism obstruction is a crossed homomorphism. We distinguish the ordinary center from the relative center, and outer derivations of the kernel from the mixed action required. Sparse crossed products are treated, including the representation class of their second mixed operation and the additional equations governing changes of sparse sections.
ER -