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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)].

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)
Full Text: DOI
[1] Bartholdi, L.: Branch rings, thinned rings, tree enveloping rings. Israel J. Math. 154, 93-139 (2006) · Zbl 1173.16303
[2] Bartholdi, L., Grigorchuk, R. I.: Lie Methods in Growth of Groups and Groups of Finite Width. London Math. Soc. Lecture Note Ser. 275, Cambridge Univ. Press, Cambridge (2000) · Zbl 1032.20026
[3] Golod, E. S.: On nilalgebras and finitely approximable p-groups. Izv. Akad. Nauk SSSR 28, 273-276 (1964) · Zbl 0215.39202
[4] Lenagan, T. H., Smoktunowicz, A.: An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension. J. Amer. Math. Soc. 20, 989-1001 (2007) · Zbl 1127.16017
[5] Petrogradsky, V. M.: Examples of self-iterating Lie algebras. J. Algebra 302, 881-886 (2006) · Zbl 1109.17008
[6] Petrogradsky, V. M., Shestakov, I. P.: Examples of self-iterating Lie algebras 2. J. Lie Theory, to appear · Zbl 1253.17011
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