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On the finite basis property of some solvable varieties of Lie algebras of finite axiomatic rank. (Russian) Zbl 0594.17008

Let \({\mathfrak N}_t\) denote a variety of Lie algebras over a field \(k\) defined by an identity \(x_ 0(\text{ad}\, x_1) \cdots (\text{ad}\, x_t)=0\). This variety consists of all Lie \(k\)-algebras which are nilpotent of class at most \(t\). Denote by \({\mathfrak N}_t {\mathfrak N}_s\) the variety of all Lie algebras which are extensions of \({\mathfrak N}_t\)-algebras by \({\mathfrak N}_s\)-algebras.
Theorem. Let \(V\) be a subvariety in \({\mathfrak N}_t {\mathfrak N}_s\) defined by a set of identities depending on a finite set of variables. If \(| k| \geq t+1\), then \(V\) is finitely based.
In particular, if char k\(=0\), then any subvariety in \({\mathfrak N}_t {\mathfrak N}_1\) is finitely based [A. N. Krasil’nikov, Vestn. Mosk. Univ., Ser. I 1982, No. 2, 34–38 (1982; Zbl 0494.17010)].

MSC:

17B30 Solvable, nilpotent (super)algebras
17B99 Lie algebras and Lie superalgebras

Citations:

Zbl 0494.17010
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