Decomposition of Lie automorphisms of upper triangular matrix algebra over commutative rings
Let 2 be an invertible element in the commutative ring \(R\) and let \(T_{n+1}(R)\) be the algebra of \((n+1)\times (n+1)\) upper triangular matrices with entries in \(R\). In the paper under review the authors describe the Lie automorphisms of \(T_{n+1}(R)\), i.e. the \(R\)-module automorphisms \(\varphi\) of \(T_{n+1}(R)\) with the property \(\varphi([x,y])=[\varphi(x),\varphi(y)]\), \(x,y\in T_{n+1}(R)\). The main result is that for \(n\geq 2\) every Lie automorphism \(\varphi\) is a product of four automorphisms, \(\varphi=\omega_{\varepsilon}\eta_c\theta\lambda_d\). Here the construction of the graph automorphism \(\omega_{\varepsilon}\) involves an idempotent \(\varepsilon\) of \(R\); \(\eta_c\) is a central automorphism which is identical on the strictly upper triangular matrices and is of the form \(\eta_c(e_{jj})=e_{jj}+c_je\), \(c_j\in R\), on the diagonal; the inner automorphism \(\theta\) conjugates by a unitriangular matrix; the diagonal automorphism \(\lambda_d\) conjugates by an invertible diagonal matrix. In the final remarks the authors mention that the case of \(2\times 2\) upper triangular matrices is simpler and also give the complete description of the Lie automorphisms.
scientific article; zbMATH DE number 5126021