Lie algebras acting semitransitively
Let \(X\) be a finite-dimensional complex vector space. A set of linear operators \({\mathcal S}\subseteq \mathrm{Hom}\,(X)\) acts semitransitively if for any two nonzero vectors \(x,y\in X\) there exists an operator \(A\in {\mathcal S}\) such that \(Ax=y\) or \(Ay=x\). The main result characterizes Lie algebras of operators acting semitransitively on \(X\). More precisely, the following is shown. A Lie subalgebra \(\mathfrak{g}\subseteq \mathfrak{gl}(X)\) acts semitransitively on \(X\) if and only if there exists a maximal chain \(0=Y_0<Y_1<\dots<Y_n=X\) of \(\mathfrak{g}\)-invariant subspaces such that \(Y_i\backslash Y_{i-1}\) is mapped transitively onto \(Y_i\) for \(i=1,\dots,n\). In addition, every maximal solvable Lie subalgebra of such \(\mathfrak{g}\) also acts semitransitively on \(X\) and consequently, a minimal semitransitive Lie subalgebra of \(\mathfrak{gl}(X)\) is triangularizable. It is further shown that if an irreducible Lie subalgebra \(\mathfrak{g}\subseteq \mathfrak{gl}(X)\) acts semitransitively on \(X\) then its derived algebra \(\mathfrak{s}:=[\mathfrak{g},\mathfrak{g}]\) either equals a special linear Lie algebra \(\mathfrak{sl}(X)\) or (is conjugate to) a symplectic Lie algebra \(\mathfrak{sp}(X)\). Consequently, in the former case, \(\mathfrak{g}\) acts \((n-1)\)-transitively while in the latter, it acts transitively but not \(2\)-transitively. Also, \(\mathfrak{s}\subseteq\mathfrak{g}\subseteq \mathfrak{s}\oplus \mathbb{C} I\). Similar results for semitransitive Jordan algebras or semitransitive associative algebras of operators were obtained earlier and are given in the references.
scientific article; zbMATH DE number 6138291
scientific article