On Structure and Central Extensions of $(n+1)$-Lie Algebras Induced by $n$-Lie Algebras

TL;DR

This work analyzes how $(n+1)$-Lie algebras induced by $n$-Lie algebras, via generalized trace maps, inherit structure, cohomology, and central-extension properties. It establishes that induced algebras are always solvable and that nilpotency of the base $n$-Lie algebra transfers to the induced $(n+1)$-Lie algebra, with detailed relationships between nilpotency classes. Central extensions lift from $n$-Lie algebras to their induced algebras through explicit cocycle transformations, and the cohomology of the induced algebras is connected to that of the original via 1- and 2-cocycle correspondences. The paper also provides a dimension-bounded classification of $n$-Lie algebras induced by $(n-1)$-Lie algebras (up to dim $n+2$), integrating Filippov and Bai classifications and supplying concrete low-dimensional examples.

Abstract

The purpose of this paper is to investigate $(n+1)$-Lie algebras induced by $n$-Lie algebras and trace maps. We highlight a comparison of their structure properties (solvability, nilpotency) and the cohomology groups as well as central extensions. Moreover, we provide for dimensions $n$, $n+1$ and $n+2$, the classification of $n$-Lie algebras which are induced by $(n-1)$-Lie algebras.

Theorems & Definitions (65)

• Definition 1.1
• Definition 1.2: Fundamental object
• Proposition 1.3
• Remark 1.4
• Definition 1.5
• Lemma 1.6
• Definition 1.7
• Definition 1.8
• Definition 1.9
• Definition 1.10