The dynamics of Leavitt path algebras
The dynamics of Leavitt path algebras
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DOI：
10.1016/j.jalgebra.2013.03.012
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发表时间：
2012-09
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影响因子：
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通讯作者：
R. Hazrat
中科院分区：
文献类型：
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作者：
R. Hazrat
最近表明，符号动力学中有限型移位的流等价的概念与Leavitt路径代数理论中的Morita理论和Grothendieck群有关（Abrams等人，2011，[4]）。本文证明了有限型移位的共轭性的概念与分次Morita理论密切相关，从而与分次Grothendieck群密切相关。这符合我们在这两个理论中的一般框架：共轭产生流等价，分级森田等价可以提升到森田等价。从有限有向图出发，观察到与E相关联的莱维特路代数的分次Grothendieck群与E相关联的有限型移位的Krieger维数群重合，这提供了莱维特路代数理论与符号动力学之间的联系。已经证明，作为Z[x，x−1]-模的有序分次Grothendieck群（我们称之为分次维数群）完全分类了有单位元的Leavitt路代数（Hazrat，2013，[20]）。通过上述对应，利用符号动力学的结果，我们证明了：对于两个纯无限单有单位元的Leavitt路代数，如果它们的分次维数群同构，则它们同构.
Recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the Morita theory and the Grothendieck group in the theory of Leavitt path algebras (Abrams et al., 2011, [4]). In this paper we show that the notion of the conjugacy of shifts of finite type is closely related to the graded Morita theory and consequently the graded Grothendieck group. This fits into the general framework we have in these two theories: Conjugacy yields the flow equivalence, and the graded Morita equivalence can be lifted to the Morita equivalence. Starting from a finite directed graph, the observation that the graded Grothendieck group of the Leavitt path algebra associated to E coincides with the Krieger dimension group of the shift of finite type associated to E provides a link between the theory of Leavitt path algebras and symbolic dynamics. It has been conjectured that the ordered graded Grothendieck group as Z[x,x−1]-module (we call this the graded dimension group) classifies the unital Leavitt path algebras completely (Hazrat, 2013, [20]). Via the above correspondence, utilising the results from symbolic dynamics, we prove that for two purely infinite simple unital Leavitt path algebras, if their graded dimension groups are isomorphic, then the algebras are isomorphic.