Extending Structures for $n$-Lie Algebras

Tao Zhang

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arXiv · 2609.05051

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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Tao Zhang. 2026-09-20. Extending Structures for $n$-Lie Algebras. https://arxiv.org/abs/2609.05051

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