Automorphism groups of algebras of triangular matrices
The author is interested in the group of automorphisms Aut(A) of some subalgebras A of the algebra \(T_ n\) of all upper triangular \(n\times n\) matrices over an integral domain from the point of view of the Noether- Skolem theorem. If \(t=(t_{ij})\in T_ n\), let G(t) be the graph on \(<n>=\{1,2,...,n\}\) defined in the usual way: (i,j) is an edge of G(t) if and only if \(t_{ij}\neq 0\). For \(S\subseteq T_ n\) are defined two graphs on \(<n>\) as follows: \(G(S)=\cup \{G(t)|\) \(t\in S\}\) and \(Z(S)=G(T_ n)\setminus G(S)\). For \(H=G(T_ n)\) is defined the algebra A(H) generated by the set \(\{t| Z(\{t\})\supseteq H\}\). Moreover let \(P_ A(H)\) denote the set of all mappings \(\phi_{\sigma}: A\to A\), \(t\to P_{\sigma}^{-1}tP_{\sigma}\), where \(P_{\sigma}\) is the permutation matrix defined by the permutation \(\sigma\) under which the complement of H is invariant. The main result can be formulated as follows: Theorem: Let H be a subgraph of \(G(T_ n)\) without loops (i,i) and for which \(Z(A(H))=H\). Then for \(A=A(H)\) the set \(P_ A(H)\) is a subgroup of Aut(A) and Aut(A) is the semidirect product of \(P_ A(H)\) and Inn(A).
scientific article; zbMATH DE number 4116754