Regular Hom-algebras admitting a multiplicative basis
type: Item
instance of: scholarly article
title: Regular Hom-algebras admitting a multiplicative basis
zbMATH Open document ID: 1498.17009
published in: Georgian Mathematical Journal
publication date: 2021-08-05T00:00:00Z
review text: On one hand, Hom-type algebras generalize common algebras. Concretely, they are given on linear spaces by multiplications twisted by linear maps. Initially there exists certain freedom on where to twist, but is of interest to twist the identities for the product implied by the linear map in a strategic way (some possibilities are mentioned in [\textit{Y. Frégier} and \textit{A. Gohr}, J. Gen. Lie Theory Appl. 4, Article ID G101001, 16 p. (2010; Zbl 1281.17002)]). Many authors study Hom-type algebras motivated by their applications in Physics because in this context arose the definition. In the other hand, the concept of multiplicative basis appears in the study of physical problems and there are many works about algebras with a multiplicative basis (see for instance [\textit{R. Bautista} et al., Invent. Math. 81, 217--285 (1985; Zbl 0575.16012); \textit{C. De La Mora} and \textit{P. J. Wojciechowski}, Linear Algebra Appl. 419, No. 2--3, 287--298 (2006; Zbl 1114.15011); \textit{H. Kupisch} and \textit{J. Waschbüsch}, Math. Z. 186, 401--405 (1984; Zbl 0549.16013); \textit{A. Makhlouf} and \textit{S. Silvestrov}, J. Algebra Appl. 9, No. 4, 553--589 (2010; Zbl 1259.16041)]). In this work the author studies the structure of a Hom-algebra \(\mathfrak{H}\) (respect with the linear map \(\alpha\)) admitting a multiplicative basis \(\mathcal{B}=\{e_i\}_{i\in I}\) with no restrictions on the identities for the product. By using connections techniques (that explicitally involves \(\alpha\)) it is defined an equivalence relation in \(I\). Theorem 2.12 shows that if \(\mathfrak{H}\) is \textit{regular} (i.e., \(\alpha\) is an algebra automorphism) it decomposes as the direct sum of well-described ideals. Additionally, in Theorem 3.3 the minimality of \(\mathfrak{H}\) is characterized in terms of the multiplicative basis. Finally, in Theorem 3.4 is proved that if \(\mathcal{B}\) is a basis of division the previous decomposition of \(\mathfrak{H}\) is actually by means of the fami…
Mathematics Subject Classification ID: 17A60
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17D30
zbMATH DE Number: 7379137
zbMATH Keywords: Hom-algebra
zbMATH Keywords: multiplicative basis
zbMATH Keywords: infinite dimensional algebra
zbMATH Keywords: structure
author: Antonio J. Calderón Martín
reviewed by: José M. Sánchez-Delgado
MaRDI profile type: Publication
full work available at URL: gmj-2020-2068
OpenAlex ID: W3045435648
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DOI: 10.1515/GMJ-2020-2068
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description: scientific article; zbMATH DE number 7379137