On \(F\)-algebras of algebraic matrices over a subfield \(F\) of the center of a division ring
The author considers the structure of irreducible subalgebras of \(n\)-by-\(n\) matrices over a fixed division ring \(D\). Suppose \(F\subseteq D\) is a division subring, contained in the center of \(D\). The main result classifies irreducible \(F\)-subalgebras in \(M_n(D)\) that consist of \(F\)-algebraic matrices. It is proven that, modulo similarity, such an algebra equals \(M_{n/r}({\mathcal D}_r)\) for some integer \(r\mid n\), and some irreducible division subring \({\mathcal D}_r\subset M_r(D)\). Application to irreducibility and to simultaneous triangularizability of subalgebras, consisting of \(F\)-algebraic matrices, are also given. In case of commutative \(D\), the main result, for instance, implies that (i) irreducible \(F\)-subalgebra of \(M_n(D)\) equals \(M_n(F)\) modulo similarity, provided the spectrum of any its element is contained in \(F\), and (ii) that an irreducible semigroup \({\mathcal S}\subset M_n(D)\) lies in \(M_n(F)\) modulo similarity, provided that \(\{0\}\not=\) Tr\(\,{\mathcal S}\subseteq F\), and \(F\) enjoys certain algebraic property. Reviewer's remark: A more general problem than (i) asks whether the irreducible \textit{semigroup} lies in \(M_n(F)\), modulo similarity, provided the spectrum of any its element is contained in \(F\). This was already answered positively for some special fields \(F\). Besides the references listed in the article, \(p\)-adic fields were treated in a paper by \textit{J. Bernik} [Pac. J. Math. 208, No. 2, 201--207 (2003; Zbl 1057.16013)]. The reviewer was informed that J. Bernik recently managed to give a positive answer for arbitrary fields \(F\).
scientific article; zbMATH DE number 5077998
scientific article