Moves on k-graphs preserving Morita equivalence
Moves on k-graphs preserving Morita equivalence
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DOI：
10.4153/s0008414x21000055
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发表时间：
2020-06
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影响因子：
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通讯作者：
C. Eckhardt;Kit Fieldhouse;D. Gent;E. Gillaspy;Ian Gonzales;D. Pask
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文献类型：
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作者：
C. Eckhardt;Kit Fieldhouse;D. Gent;E. Gillaspy;Ian Gonzales;D. Pask
摘要 我们启动了将有向图 $C^*$ -代数的几何分类扩展到更高阶图（k 图）的计划，如 Eilers 等人完成的那样。 （2016 年，预印本）。准确地说，我们确定了可以在 k 图 $\Lambda $ 上执行的四个“移动”或修改，这使得其 $C^*$ -代数 $C^*(\Lambda )$ 的 Morita 等价类保持不变。这些移动（内分裂、延迟、汇删除和缩减）受到 Sørensen (Ergodic Th. Dyn. Syst. 33(2013), 1199–1220) 以及 Bates 和 Pask (Ergodic Th. Dyn. Syst. 24(2004), 367-382) 描述的有向图移动的启发。因此，我们对 k 图的看法主要集中在底层的有向图上。因此，我们包含了关于 k 图与其底层有向图之间关系的两个新结果：定理 2.3 和引理 2.9。
Abstract We initiate the program of extending to higher-rank graphs (k-graphs) the geometric classification of directed graph $C^*$ -algebras, as completed in Eilers et al. (2016, Preprint). To be precise, we identify four “moves,” or modifications, one can perform on a k-graph $\Lambda $ , which leave invariant the Morita equivalence class of its $C^*$ -algebra $C^*(\Lambda )$ . These moves—in-splitting, delay, sink deletion, and reduction—are inspired by the moves for directed graphs described by Sørensen (Ergodic Th. Dyn. Syst. 33(2013), 1199–1220) and Bates and Pask (Ergodic Th. Dyn. Syst. 24(2004), 367–382). Because of this, our perspective on k-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a k-graph and its underlying directed graph.