Invariant Subspaces of Nilpotent Linear Operators. I

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 $TV\to V$ a linear operator with $T^n=0$ for some $n$, and such that $T(U)\subseteq U$. 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 sufficient 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 indecomposables 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.

55 pages, minor modification in (0.1.3), to appear in: Journal fuer die reine und angewandte Mathematik

References in corpus (1)

• Invariant Subspaces of Nilpotent Linear Operators. I

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