Nilpotent linear spaces in M5(ℂ)

The structure of maximal nilpotent linear subspaces of [Formula: see text], and more generally of [Formula: see text] for any field [Formula: see text] with at least three elements, is now well established. In this work we extend the analysis to dimension five. The main result of this paper consists of the description, up to similarity, of all two-dimensional nilpotent linear subspaces of [Formula: see text]. We expect that these results will allow us to describe every maximal nilpotent linear subspace of [Formula: see text]. We think that this classification is central to the solution of the classical problem of Albert about finite-dimensional power-associative nilalgebras with dimension [Formula: see text]. In addition, we provide new and interesting examples of irreducible and maximal subspaces of [Formula: see text] nilpotent matrices.