Metric $n$-Lie Algebras
Ruipu Bai, Wanqing Wu, Zhenheng Li

TL;DR
This paper investigates the structure of metric $n$-Lie algebras over complex numbers, focusing on their Levi decomposition, minimal ideals, and invariant forms, revealing new structural relationships and conditions for semisimplicity.
Contribution
It provides new structural insights into metric $n$-Lie algebras, including relationships between radicals, minimal ideals, and invariant forms, and characterizes conditions for the absence of strong semisimple ideals.
Findings
$ ext{dim}( ext{space of invariant forms} \\geq m( ext{G}) + 1$
$ ext{centralizer of R} = ext{sum of minimal ideals}$
Condition for no strong semisimple ideals: $ ext{R}^ot \\subseteq ext{R}$
Abstract
We study the structure of a metric -Lie algebra over the complex field . Let be the Levi decomposition, where is the radical of and is a strong semisimple subalgebra of . Denote by the number of all minimal ideals of an indecomposable metric -Lie algebra and the orthogonal complement of . We obtain the following results. As -modules, is isomorphic to the dual module of The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on equals that of the vector space of certain linear transformations on ; this dimension is greater than or equal to . The centralizer of…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Advanced Algebra and Geometry
