Lengths of cyclic algebras and commutative subalgebras of quaternion matrices
type: Item
instance of: scholarly article
title: Lengths of cyclic algebras and commutative subalgebras of quaternion matrices
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 2021-09-02T00:00:00Z
review text: Given an associative unital algebra \(A\) of finite dimension over a field \(F\) and a finite set of generators \(S\) of \(A\) over \(F\), there exists a natural number \(k\) for which \(A=\sum_{i=0}^k \sum_{b_1,\dots,b_i \in S} F b_1 \cdot \dots \cdot b_i\). The minimal such \(k\) is the length of \(A\) with respect to \(S\). The general length of \(A\) is the supremum of its length with respect to \(S\) where \(S\) ranges over all finite sets of generators. For example, when \(A\) is a cyclic central simple algebra of degree \(n\), one can take \(S\) to be the set consisting of the two standard generators \(x\) and \(y\) (where \(F[x]/F\) is the cyclic extension with \(\sigma\) generating its Galois group, and \(yxy^{-1}=\sigma(x)\)), and then the length of \(A\) with respect to \(S\) is exactly \(2(n-1)\), which means its general length is at most \(2n-2\). The author uses this observation and other known results in the literature to show that the general length of \(A\) is exactly \(2(n-1)\) for \(n\leq 5\). The author also studies commutative subalgebras of the matrix algebra over \(F\), and extends some of the results to the matrix algebra over \(\mathbb{H}\), Hamilton's division ring of real quaternions.
reviewed by: Adam Chapman
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 7390153
zbMATH Keywords: cyclic algebra
zbMATH Keywords: division algebra
zbMATH Keywords: length of an algebra
zbMATH Keywords: matrix algebra
zbMATH Keywords: real quaternions
MaRDI profile type: Publication
full work available at URL: j.laa.2021.07.010
OpenAlex ID: W3185685598
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DOI: 10.1016/J.LAA.2021.07.010
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description: scientific article; zbMATH DE number 7390153