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Oriented Closed Polyhedral Maps and the Kitaev Model
Kornél Szlachányi Wigner Research Centre for Physics, Budapest, Hungary
Аннотация:
A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $\Sigma$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(\Sigma)^*$ of $\Sigma$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(\Sigma)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate “topological invariance” of states created by ribbon operators.
Ключевые слова:
Hopf algebra; polyhedral map; quantum double; ribbon operator; topological invariance
Поступила: 7 апреля 2023 г.; в окончательном варианте 14 мая 2024 г.; опубликована 8 июня 2024 г.
Образец цитирования:
Kornél Szlachányi, “Oriented Closed Polyhedral Maps and the Kitaev Model”, SIGMA, 20 (2024), 048, 55 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma2050 https://www.mathnet.ru/rus/sigma/v20/p48
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