Nilpotent linear transformations and the solvability of power-associative nilalgebras
type: Item
instance of: scholarly article
title: Nilpotent linear transformations and the solvability of power-associative nilalgebras
zbMATH Open document ID: 1066.17001
author: Ivan Correa
author: Irvin Roy Hentzel
author: Pedro Pablo Julca
author: Luiz Antonio Peresi
published in: Linear Algebra and its Applications
publication date: 2005-02-22T00:00:00Z
review text: In 1948, \textit{A. A. Albert} [Trans. Am. Math. Soc. 64, 552--593 (1948; Zbl 0033.15402)] conjectured that the finite-dimensional commutative power-associative nilalgebras over a field are nilpotent. Suttles [Notices Am. Math. Soc. 19, A-566 (1972)] gave a counterexample of dimension \(5\) which is, nevertheless, solvable. Albert's problem was then restated as follows: Is every finite dimensional commutative power-associative nilalgebra over a field solvable? The paper under review is devoted to solve in the affirmative this question for dimension \(7\), and for dimension \(n\) and nilindex \(n-1\) or \(n-2\), provided that the characteristic of the ground field is zero or sufficiently large. Results by \textit{M. Gerstenhaber} [Duke Math. J. 27, 21-31 (1960; Zbl 0213.04101)] are instrumental in the proofs.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17A05
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 2136670
zbMATH Keywords: Nilpotent linear transformations
zbMATH Keywords: commutative power-associative nilalgebra
zbMATH Keywords: Albert's problem
zbMATH Keywords: solvable
zbMATH Keywords: nilpotent
MaRDI profile type: Publication
full work available at URL: j.laa.2004.08.007
OpenAlex ID: W2044796542
cites work: Power-associative rings
cites work: On solvability of noncommutative power-associative nilalgebras
cites work: On the solvability of the commutative power-associative nilalgebras of dimension 6.
cites work: On the solvability of the five-dimensional commutative power-associative nilalgebras
cites work: On a class of commutative power-associative nilalgebras
cites work: On Nilalgebras and Linear Varieties of Nilpotent Matrices, I
cites work: On nilalgebras and linear varieties of nilpotent matrices. II
cites work: On nilalgebras and linear varieties of nilpotent matrices. III
cites work: On Commutative Power-Associative Nilalgebras of Low Dimension
cites work: Q5522244
DOI: 10.1016/J.LAA.2004.08.007
description: scientific article; zbMATH DE number 2136670