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- W4290802771 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 $u_ncirc u_{n-1}circ cdots circ u_1$ 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 $mathbf{B}_{n+1}=mathbf{B}_1$)." @default.
- W4290802771 created "2022-08-12" @default.
- W4290802771 creator A5016953630 @default.
- W4290802771 date "2022-08-09" @default.
- W4290802771 modified "2023-09-24" @default.
- W4290802771 title "Locally finite cycles of linear mappings in countable dimension" @default.
- W4290802771 doi "https://doi.org/10.48550/arxiv.2208.04915" @default.
- W4290802771 hasPublicationYear "2022" @default.
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