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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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