On balanced systems of idempotents
type: Item
instance of: scholarly article
title: On balanced systems of idempotents
zbMATH Open document ID: 1022.16018
published in: Sbornik: Mathematics
publication date: 2002-01-03T00:00:00Z
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 1681713
zbMATH Keywords: idempotents
zbMATH Keywords: balanced systems
zbMATH Keywords: matrix algebras
zbMATH Keywords: balanced bases
zbMATH Keywords: orthogonal decompositions
zbMATH Keywords: regular representations
MaRDI profile type: Publication
full work available at URL: sm2001v192n04abeh000558
OpenAlex ID: W2050624487
DOI: 10.1070/SM2001V192N04ABEH000558
author: D. N. Ivanov
review text: The notion of `balanced \(t\)-systems' in complex semisimple finite-dimensional associative algebras was introduced by the author in his previous papers [Mat. Sb. 189, No. 12, 83-102 (1998; Zbl 0937.16033) and Mat. Sb. 191, No. 4, 67-90 (2000; see the preceding review Zbl 1022.16017)]. In this paper, he considers `balanced bases' of the algebra \(M_n(\mathbb{C})\) of complex \(n\times n\)-matrices, that is, balanced \(2\)-systems \(\mathcal E\) that form a linear basis of \(M_n(\mathbb{C})\). He constructs balanced bases for \(M_{p^n}(\mathbb{C})\) with \(p\) any odd prime. His construction utilizes the same special basis of \(M_{p^n}(\mathbb{C})\) that was used by \textit{A. I. Kostrikin, I. A. Kostrikin}, and \textit{V. A. Ufnarovskij} [in Tr. Mat. Inst. Steklova 158, 105-120 (1981; Zbl 0526.17003)] to construct orthogonal decompositions of the corresponding Lie algebra (into a direct sum of Cartan subalgebras that are pairwise orthogonal with respect to the Killing form). In the case of \(M_{2^n}(\mathbb{C})\), the author establishes non-existence of balanced bases of a certain kind.
reviewed by: Pham Huu Tiep
description: scientific article; zbMATH DE number 1681713