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Kostrikin, Α. I.

doi:10.1007/978-3-642-74324-5

Cited by 51 publications

(36 citation statements)

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“…This is the result of A. I. Kostrikin and E. I. Zelmanov [9]. Another proof via super-words pointed out here was obtained by J. Backeline and independently by A. D. Chanyshev [7].…”

Section: =2supporting

confidence: 71%

Free subalgebras of Lie algebras close to nilpotent

Belov¹,

Mikhailov²

2009

Groups Geom. Dyn.

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Abstract. We prove that for every automata algebra of exponential growth the associated Lie algebra contains a free subalgebra. For n 1, let L nC2 be a Lie algebra with generators x 1 ; : : : ; x nC2 and the following relations: for k Ä n, any commutator (with any arrangement of brackets) of length k which consists of fewer than k different symbols from fx 1 ; : : : ; x nC2 g is zero. As an application of this result about automata algebras, we prove that L nC2 contains a free subalgebra for every n 1. We also prove the similar result about groups defined by commutator relations. Let G nC2 be a group with n C 2 generators y 1 ; : : : ; y nC2 and the following relations: for k Ä n, any left-normalized commutator of length k which consists of fewer than k different symbols from fy 1 ; : : : ; y nC2 g is trivial. Then the group G nC2 contains a 2-generated free subgroup.The main technical tool is combinatorics of words, namely combinatorics of periodical sequences and period switching.

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“…This is the result of A. I. Kostrikin and E. I. Zelmanov [9]. Another proof via super-words pointed out here was obtained by J. Backeline and independently by A. D. Chanyshev [7].…”

Section: =2supporting

confidence: 71%

Free subalgebras of Lie algebras close to nilpotent

Belov¹,

Mikhailov²

2009

Groups Geom. Dyn.

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“…We show that the subalgebra C generated by M and d is the whole algebra 7 . The root elements x −α 3 , x − α 2 +α 4 , and…”

Section: The Minimal Number Of Extremal Generators For Lie Algebras Omentioning

confidence: 99%

“…8 6 Type E 7 . Let us denote by α 1 α 2 α 3 α 4 α 5 α 6 α 7 the simple roots of 7 . Take the subalgebra M of type D 4 in 7 exactly as in 6 .…”

Section: The Minimal Number Of Extremal Generators For Lie Algebras Omentioning

confidence: 99%

“…To generate the whole algebra 7 , consider the extremal element d = exp x α 2 +α 3 +α 4 +α 5 +α 6 +α 7 exp x − α 1 +α 3 exp x − α 1 +α 2 +α 3 +2α 4 +α 5 +α 6 exp x α 3 +α 4 +α 5 +α 6 exp x − α 1 +α 2 +α 3 +2α 4 +2α 5 +α 6 +α 7 x α 1 = x α 1 + x α 4 +α 5 +α 6 + x α 1 +α 3 +α 4 +α 5 +α 6 + x α 2 +α 4 +α 5 +α 6 +α 7 + x α 1 +α 2 +α 3 +α 4 +α 5 +α 6 +α 7 − x α 1 +α 2 +2α 3 +3α 4 +2α 5 +α 6…”

Section: The Minimal Number Of Extremal Generators For Lie Algebras Ounclassified

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Lie Algebras Generated by Extremal Elements

Cohen¹,

Steinbach²,

Ushirobira³

et al. 2001

Journal of Algebra

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We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals) over a field of characteristic distinct from 2. There is an associative bilinear form on such a Lie algebra; we study its connections with the Killing form. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal numbers of extremal generators for the Lie algebras of type A n n ≥ 1 , B n n ≥ 3 , C n n ≥ 2 , D n n ≥ 4 , E n n = 6 7 8 , F 4 and G 2 are shown to be n + 1, n + 1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

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“…Kostrikin's book [23]). The Lie ring L is ~graded L = L1 ~9 L2 ~9 ... and generated by L1 as a Lie ring.…”

Section: Introductionmentioning

confidence: 97%