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Some examples of nil Lie algebras. (English) Zbl 1144.17013
The reviewer constructed an example of a finitely generated restricted Lie algebra of polynomial growth with a nil $$p$$-mapping over the field of characteristic $$p=2$$ [see V. M. Petrogradsky, J. Algebra 302, No. 2, 881–886 (2006; Zbl 1109.17008)].
Now, the authors extend that example to arbitrary positive characteristic $$p$$. Let $$p$$ be an arbitrary prime. Consider the truncated polynomial ring $$R=K[t_i\mid i=0,1,2,\dots ]/ (t_i^p\mid i=0,1,2,\dots)$$. Let $$\partial_i={\partial\over {\partial t_i}}$$, $$i\geq 0$$, denote the respective derivations. Consider the operators
\begin{aligned} v_1&= \partial_1+t_0^{p-1}(\partial_2+t_1^{p-1}(\partial_3 +t_2^{p-1}( \partial_4+\cdots))),\\ v_2&= \partial_2+t_1^{p-1}(\partial_3+t_2^{p-1}(\partial_4+\cdots)). \end{aligned}
Let $$L\subset \operatorname{Der} R$$ be the restricted Lie algebra generated by these derivations. In particular, the authors establish the following properties of this algebra. First, $$L$$ has polynomial growth. Second, $$L$$ has a nil-$$p$$-mapping.
These restricted Lie algebras are natural analogues of the Grigorchuk and Gupta-Sidki groups, [see R. I. Grigorchuk, Funct. Anal. Appl. 14, 41–43 (1980; Zbl 0595.20029), N. Gupta and S. Sidki, Math. Z. 182, 385–388 (1983; Zbl 0513.20024)].

MSC:
 17B50 Modular Lie (super)algebras 17B65 Infinite-dimensional Lie (super)algebras 16P90 Growth rate, Gelfand-Kirillov dimension 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 16S32 Rings of differential operators (associative algebraic aspects)
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References:
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