Abstract

Let k be a field. We consider triples (V, U, T), where V is a finite dimensional k-space, U a subspace of V and T : V → V a linear operator with Tⁿ = 0 for some n, and such that T(U) U. Thus, T is a nilpotent operator on V, and U is an invariant subspace with respect to T. We will discuss the question whether it is possible to classify these triples. These triples (V, U, T) are the objects of a category with the Krull-Remak-Schmidt property, thus it will be suffcient to deal with indecomposable triples. Obviously, the classification problem depends on n, and it will turn out that the decisive case is n = 6. For n < 6, there are only finitely many isomorphism classes of indecomposable triples, whereas for n > 6 we deal with what is called “wild” representation type, so no complete classification can be expected. For n = 6, we will exhibit a complete description of all the indecomposable triples.