Analogs of the Cartan criteria for \(n\)-Lie algebras
type: Item
instance of: scholarly article
title: Analogs of the Cartan criteria for \(n\)-Lie algebras
zbMATH Open document ID: 0860.17001
DOI: 10.1007/BF02341871
author: Sh. M. Kasymov
published in: Algebra and Logic
publication date: 1996-08-25T00:00:00Z
full work available at URL: https://eudml.org/doc/187725
review text: According to \textit{V. T. Filippov} [Sib. Mat. Zh. 26, No. 6, 126-140 (1985; Zbl 0585.17002)], an \(n\)-Lie algebra \((n\geq 2)\) over a field \(F\) of characteristic 0 is a linear space \(A\) over \(F\) with \(n\)-ary polylinear operation \([.,\dots,.]\) which is skewsymmetric by all arguments and satisfies to the generalized Jacobi identity \[ \bigl[ [x_1,\dots, x_n],y_2,\dots, y_n\bigr]= \sum^n_{i=1} \bigl[x_1,\dots, [x_i,y_2,\dots, y_n],\dots, x_n\bigr]. \] A subspace \(I\) of \(A\) is called an ideal of \(A\) if \([I,A,\dots, A]\subseteq I\). An ideal \(I\) of an \(n\)-Lie algebra \(A\) is called solvable in the sense of V. T. Filippov if \(I^{(r)}=0\) for some \(r\geq 0\) where \(I^{(0)}= I\), \(I^{(s+1)}= [I^{(s)}, I^{(s)},\dots, I^{(s)}]\). An ideal \(I\) of an \(n\)-Lie algebra \(A\) is called solvable in the sense of E. N. Kuz'min if \(I^{(r)}=0\) for some \(r\geq 0\) where \(I^{(0)}=I\), \(I^{(s+1)}= [I^{(s)}, I^{(s)}, A,\dots,A]\). It is shown that Cartan's criteria for finite-dimensional Lie algebras to be semisimple and solvable are fully adaptable to \(n\)-Lie algebras, provided that ideals of an \(n\)-Lie algebra are understood to be solvable in the sense of Kuz'min. Specifically, a characterization of the Kuz'min radical in terms of a trace form associated with some representation is presented. It is analogous to the characterization one has in the case of Lie algebras. Another analog of the Cartan theorem is proved for \(n\)-Lie algebras which are solvable in the sense of Filippov.
Mathematics Subject Classification ID: 17A42
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 907268
zbMATH Keywords: solvable
zbMATH Keywords: Cartan's criteria
zbMATH Keywords: semisimple
zbMATH Keywords: \(n\)-Lie algebras
zbMATH Keywords: Kuz'min radical
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
Wikidata QID: Q115392470
cites work: Q3710656
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cites work: Q4006520
OpenAlex ID: W2035820107
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description: scientific article; zbMATH DE number 907268