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About the theory of n-Lie algebras. (Russian) Zbl 0647.17001
An n-Lie algebra over a field F is a linear space A over F with an n-ary polylinear operation [,...,] which is antisymmetric for all arguments and satisfies an identity similar to the Jacobi one in Lie algebras: \[ [[x_ 1,...,x_ n],y_ 2,...,y_ n]=\sum^{n}_{i=1}[x_ 1,...[x_ i,y_ 2,...,y_ n],...,x_ n]. \] This notion was introduced by V. T. Filippov [Sib. Mat. Zh. 26, No.6, 126-140 (1985; Zbl 0585.17002)].
There are many ways to define solvability and nilpotence for ideals of n- Lie algebras with \(n>2\) which are discussed by the author. A subspace I of A is called an ideal if \([I,A,...,A]\subset I\). An ideal I of A is called k-solvable \((2\leq k\leq n)\) if \(I^{(r,k)}=0\) for some \(r>0\), where \(I^{(0,k)}=I\), \[ I^{(s+1,k)}=[I^{(s,k)},...,I^{(s,k)},\quad A,...,A] \] for \(s\geq 0,\) (k-times \(I^{(s,k)},\) \(n-k\)-times A). An ideal I of A is called k- nilpotent \((2\leq k\leq n)\) if \(I^{r,k}=0\) for some \(r\geq 1\) where \(I^{1,k}=I\), \(I^{s+1,k}=[I^{s,k},I,...,I,A,...,A]\) (\(k-1\)-times I, \(n-k\)-times A). Further the n-Lie modules and representations of n-Lie algebras are considered. Several classical results for Lie algebras, in particular the well known Engel theorem and Cartan criterion for solvability are generalized to the case of n-Lie algebras.
Reviewer: G.I.Zhitomirskij

MSC:
17A42 Other \(n\)-ary compositions \((n \ge 3)\)
17B99 Lie algebras and Lie superalgebras
17D99 Other nonassociative rings and algebras
17B30 Solvable, nilpotent (super)algebras
20N15 \(n\)-ary systems \((n\ge 3)\)
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