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PreprintarXiv licencearXiv

Extending Structures for n-Lie Algebras

  • Tao Zhang
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Abstract

Motivated by the Casas-Loday-Pirashvili maps concerning representations of n-Leibniz algebras, we give an intrinsic characterization of the cohomology of n-Lie algebras with coefficient in a representation space V. We investigated the cohomology theory of n-Lie algebras in three different ways: cohomology theory of Leibniz algebras, infinitesimal deformation and abelian extension. In the main part of this paper, we solve the extending problems for n-Lie algebras. A necessary and sufficient unified-product criterion is obtained. The case of crossed products, sparse non-abelian extensions, matched pairs are studied as special cases. We also investigate the factorizations, deformation maps, and complements problem for n-Lie algebras. An appendix removes the general non-reduced unified product with all intermediate mixed components.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026extending,
  title = {{Extending Structures for $n$-Lie Algebras}},
  author = {Tao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.05051},
  eprint = {2609.05051},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05051},
}
RIS
TY  - UNPB
TI  - Extending Structures for n-Lie Algebras
AU  - Tao Zhang
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.05051
UR  - https://arxiv.org/abs/2609.05051
AB  - Motivated by the Casas-Loday-Pirashvili maps concerning representations of n-Leibniz algebras, we give an intrinsic characterization of the cohomology of n-Lie algebras with coefficient in a representation space V. We investigated the cohomology theory of n-Lie algebras in three different ways: cohomology theory of Leibniz algebras, infinitesimal deformation and abelian extension. In the main part of this paper, we solve the extending problems for n-Lie algebras. A necessary and sufficient unified-product criterion is obtained. The case of crossed products, sparse non-abelian extensions, matched pairs are studied as special cases. We also investigate the factorizations, deformation maps, and complements problem for n-Lie algebras. An appendix removes the general non-reduced unified product with all intermediate mixed components.
ER  -