Locally finite cycles of linear mappings in countable dimension

TL;DR

This paper classifies cycles of linear maps on countably dimensional vector spaces where the composition is locally finite, extending Kaplansky's invariants to arbitrary cycle lengths and providing basis structures under local nilpotency.

Contribution

It extends Kaplansky's classification of locally nilpotent endomorphisms to cycles of any length on countable-dimensional spaces, and characterizes basis structures under local nilpotency.

Findings

Classified $n$-cycles of linear maps with locally finite composition.

Extended Kaplansky invariants to arbitrary cycle lengths.

Established basis structures for locally nilpotent cycles.

Abstract

Let $n$ be a positive integer. An $n$-cycle of linear mappings is an $n$-tuple $(u_1,\dots,u_n)$ of linear maps $u_1 \in \mathrm{Hom}(U_1,U_2),u_2 \in \mathrm{Hom}(U_2,U_3),\dots,u_n \in \mathrm{Hom}(U_n,U_1)$, where $U_1,\dots,U_n$ are vector spaces over a field. We classify such cycles, up to equivalence, when the spaces $U_1,\dots,U_n$ have countable dimension and the composite $u_n\circ u_{n-1}\circ \cdots \circ u_1$ is locally finite. When $n=1$, this problem amounts to classifying the reduced locally nilpotent endomorphisms of a countable-dimensional vector space up to similarity, and the known solution involves the so-called Kaplansky invariants of $u$. Here, we extend Kaplansky's results to cycles of arbitrary length. As an application, we prove that if $u_n \circ \cdots \circ u_1$ is locally nilpotent and the $U_i$ spaces have countable dimension, then there are bases $\mathbf{B}_1,\dots,\mathbf{B}_n$ of $U_1,\dots,U_n$, respectively, such that, for every $i \in \{1,\dots,n\}$, $u_i$ maps every vector of $\mathbf{B}_i$ either to a vector of $\mathbf{B}_{i+1}$ or to the zero vector of $U_{i+1}$ (where we convene that $U_{n+1}=U_1$ and $\mathbf{B}_{n+1}=\mathbf{B}_1$).