The automorphism group of a structural matrix algebra
Let \(M_ n(K)\) be the algebra of \(n \times n\) matrices over a field \(K\) and denote by \(E^{ij}\) the matrix unit having 1 in place \((i,j)\) and 0's elsewhere. Let \(\rho\) be a reflexive and transitive relation on \(I = \{1,2, \dots, n\}\). The corresponding structural matrix algebra \({\mathcal S} = {\mathcal S}(\rho,K)\) is the subalgebra of \(M_ n(K)\) spanned by the \(E^{ij}\) with \((i,j) \in \rho\) [see \textit{L. Van Wyk}, Commun. Algebra 16, No. 2, 399-419 (1988; Zbl 0641.16014)]. The author determines the automorphisms of \(\mathcal S\) (Theorem C) and derives a simple criterion for all automorphisms to be inner (Theorem D). If a permutation \(\sigma\) on \(I\) induces an automorphism of \(\rho\), it also induces an automorphism \(\widehat {\sigma}\) of \(\mathcal S\) by \(\widehat {\sigma}(E^{ij}) = E^{\sigma(i), \sigma(j)}\); these automorphisms form a group \(\mathcal P\). A function \(g : \rho \to K^*\) is called transitive if \(g(i,j) g(j,k) = g(i,k)\) whenever \((i,j), (j,k) \in \rho\); such a \(g\) induces an automorphism \(g^*\) of \(\mathcal S\) by \(g^* (E^{ij}) = g(i,j) E^{ij}\). A transitive function \(g\) is called trivial if it has the form \(g(i,j) = s(i) s(j)^{-1}\) for some function \(s: I \to K^*\); if \(g\) is trivial, \(g^*\) is an inner automorphism. Theorem \(C\) asserts that \(\text{Aut }{\mathcal S} = (\text{Inn } {\mathcal S} \rtimes {\mathcal G}) \rtimes {\mathcal P}\) for some (explicitly defined) group \(\mathcal G\) of automorphisms of the form \(g^*\). Let \(\overline {\rho}\) denote the largest equivalence relation \(\subseteq \rho\). Theorem D asserts that \(\text{Aut } {\mathcal S} = \text{Inn } {\mathcal S}\) if and only if (a) every transitive function \(g\) is trivial and (b) every automorphism of \(\rho\) maps onto itself every equivalence class of \(\overline {\rho}\). This generalizes the result of \textit{P. Barker} and \textit{T. P. Kezlan} [Current trends in matrix theory, Proc. Conf. Auburn/Ala. 1986, 33-39 (1987; Zbl 0658.16027)] that every automorphism of the algebra of upper triangular matrices is inner.
scientific article; zbMATH DE number 519688