论文标题
量子图：不同的观点，同构和量子自动形态
Quantum graphs: different perspectives, homomorphisms and quantum automorphisms
论文作者
论文摘要
我们对量子图的概念进行了研究，该量子图是通过任意有限维$ c^*$ - 代数$ b $配备了任意忠实国家的。量子图主要作为$ l^2（b）$，量子邻接矩阵或某些$ b'$ $ b'$的操作员双模型的某些操作员的实现。我们提出了一种简单的，纯粹的代数方法，用于证明这些设置之间的等效性，从而在奇特状态设置中恢复现有结果。对于非统治状态，我们的方法自然建议对操作员的定义进行概括，该定义考虑了该州模块化自动形态群体的（某些方面）。此外，我们表明每个这样的``非地面''量子图都对应于满足超对称条件的``tracial''量子图。我们研究了由UCP图引起的量子图的同态（或CP塑性），以及由UCP图构建的量子图密切相关的示例。我们表明，这些结构满足自动双模块性能。我们研究了量子图的量子自动形态，给出了对紧凑型量子组对操作员双模模的作用的定义，并证明该定义之间的等效性以及使用量子邻接矩阵定义的常见概念。我们努力给出一个相对具有独立的基础，说明，希望这对该领域的其他研究人员有用。
We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional $C^*$-algebras $B$ equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on $L^2(B)$, the quantum adjacency matrices, or as certain operator bimodules over $B'$. We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such ``non-tracial'' quantum graphs corresponds to a ``tracial'' quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CP-morphisms) of quantum graphs arising from UCP maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.
