Free subalgebras of Lie algebras close to nilpotent

Belov, A.; Michailov, R.

Информация о цитировании статьи получена из Web of Science, Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 20 апреля 2016 г.

Авторы: Belov A., Michailov R.
Журнал: Groups, Geometry, and Dynamics
Том: 4
Номер: 1
Год издания: 2010
Издательство: European Mathematical Society Publishing House
Местоположение издательства: Switzerland
Первая страница: 15
Последняя страница: 29
DOI: 10.4171/GGD/73
Аннотация: Let an associative algebra $A$ over a commutative ring with 1 be generated by a finite set $S$. Then $A^{\sim}$ stands for the Lie algebra generated by $S$ under linear operations and the commutation $[a,b]=ab-ba$. The authors consider monomial algebras which are automaton algebras in the sense of {\it V.~A.~Ufnarovski\u\i} [Mat.\ Sb. 180, No. 11, 1548--1560 (1989; Zbl 0685.16007)] and show that if such an algebra $A$ has exponential growth, then the Lie algebra $A^{\sim}$ contains a free 2-generated subalgebra (Theorem 2). As an immediate corollary, they find the free 2-generated Lie algebra as a subalgebra in the Lie algebra generated by $x_1,\dots,x_{n+2}$ subject to all relations of the form $[x_{i_1},\dots,x_{i_k}]=0$, where $k\le n$ and $i_s=i_t$ for some $s\ne t$ (Theorem~1). An analogous result is proved in the group setting (Theorem~3). The proof of the main result relies on some interesting observations from combinatorics on words. Reviewer's remark: The paper is seriously spoiled by language inaccuracies and typos.Reviewer: Mikhail Volkov (Ekaterinburg
Добавил в систему: Белов Алексей Яковлевич

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