论文标题
可数维度的线性映射的局部有限循环
Locally finite cycles of linear mappings in countable dimension
论文作者
论文摘要
让$ n $成为一个积极的整数。线性映射的$ n $ cycle是$ n $ -tuple $（u_1，\ dots，u_n）$的线性映射映射$ u_1 \ in \ mathrm {homrm {hom}（u_1，u_2，u_2），u_2 \ u_2 \ u_2 \ in \ mathrm {hom}（hom} \ mathrm {hom}（u_n，u_1）$，其中$ u_1，\ dots，u_n $是字段上的向量空间。我们对此类周期进行分类，直至等效，当Spaces $ U_1，\ DOTS，U_N $具有可计数尺寸，并且复合$ U_N \ Circ u_ {n-1} \ Crocd \ CD \ CDOTS \ CIRC U_1 $是本地有限的。 当$ n = 1 $时，此问题等同于对可计数矢量空间的局部尼尔氏剂内态减少进行分类，并且已知的解决方案涉及所谓的$ u $的Kaplansky不变性。在这里，我们将Kaplansky的结果扩展到任意长度的周期。作为一个应用程序，我们证明，如果$ u_n \ circ \ cdots \ circ u_1 $是本地nilpotent，并且$ u_i $空间具有可计数的维度，则有$ \ mathbf {b} _1，\ dots，\ dots，\ dots，\ mathbf {b} _n $ of $ u_1，$ i_n $ i_， \ {1，\ dots，n \} $，$ u_i $映射$ \ Mathbf {b} _i $的每个向量都要$ \ Mathbf {b} _ {i+1} $的向量，或者to $ u_+1} $ and $ u_ {i+1} $} $ \ mathbf {b} _ {n+1} = \ mathbf {b} _1 $）。
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 $\mathbf{B}_{n+1}=\mathbf{B}_1$).
