Regular Hom-algebras admitting a multiplicative basis
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 family of its minimal ideals. We highlight that in this paper it is considered a little bit more general concept of \textit{multiplicative} basis than the usual one (see [\textit{R. Bautista} et al., Invent. Math. 81, 217--285 (1985; Zbl 0575.16012); \textit{V. Bovdi}, Algebr. Represent. Theory 6, No. 3, 353--368 (2003; Zbl 1042.20002); \textit{V. Bovdi} et al., Algebr. Represent. Theory 18, No. 2, 297--306 (2015; Zbl 1331.16021); \textit{H. Kupisch} and \textit{J. Waschbüsch}, Math. Z. 186, 401--405 (1984; Zbl 0549.16013)] for example).
scientific article; zbMATH DE number 7379137
scientific article