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A sufficient condition for nilpotence of the commutant of the Lie algebra. (English. Russian original) Zbl 1115.17300

Russ. Math. 42, No. 8, 41-45 (1998); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1998, No. 8, 43-47 (1998).
From the text: The characteristic of the basic field within the whole article is assumed to be zero. As usual, we denote by \({\mathcal A}\) the variety of the Abelian Lie algebras, \({\mathcal N}_c\) stands for the variety of nilpotent algebras with their nilpotence degree not exceeding \(c\). Then \({\mathcal N}_c{\mathcal A}\) is the variety of the Lie algebras, whose commutants are nilpotent with a nilpotence degree not exceeding \(c\). The latter variety is defined by the identity \[ (x_0y_0) (x_1y_1) (x_2y_2)\cdots (x_cy_c)= 0. \tag{1} \] Let us consider two identities which for \(m\geq c\) are consequences of identity (1): \[ (x_0y_0) (xy)^m= 0, \tag{2} \]
\[ \sum_{p\in S_m,\,q\in S_m} (-1)^p (-1)^q (x_0y_0) (x_{p(1)}y_{q(1)}) (x_{p(2)}y_{q(2)})\cdots (x_{p(m)}y_{q(m)})= 0. \tag{3} \] In the present article we investigate the question whether identity (1) follows from identities (2) and (3). Let us note that each separate identity either (2) or (3), does not imply nilpotence of the commutant even for small values of \(m\), additional condition of decidability of degree 3, and strong conditions for the variety growth.

MSC:

17B01 Identities, free Lie (super)algebras
17B30 Solvable, nilpotent (super)algebras
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