Oriented Closed Polyhedral Maps and the Kitaev Model

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}(\...

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Bibliographic Details
Published in:arXiv.org 2024-06
Main Author: Szlachányi, Kornél
Format: Article
Language:English
Subjects:
Algebra
Combinatorial analysis
Operators (mathematics)
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2302.08027