AI 中文总结
本文证明Leavitt路径代数的分级Morita等价等价于强移位等价，并利用符号动力学反例推翻Hazrat的分级分类猜想。
AI 中文摘要
给定两个有限本质邻接矩阵$A$和$B$，Hazrat的分级分类猜想认为，$K_0$群的保序$\mathbb{Z}[x,x^{-1}]$-模同构蕴含$A$和$B$的Leavitt路径代数的分级Morita等价，而带点版本预测当$K_0$群同构额外保持正则模的类时，Leavitt路径代数存在分级同构。对于任意域$k$，我们证明$A$和$B$在$k$上的Leavitt路径代数是分级Morita等价的当且仅当$A$和$B$是强移位等价的。通过引用符号动力学中Kim和Roush的反例，这表明Hazrat的分级分类猜想是错误的。
英文摘要
Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.
Comments18 pages
