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 Σ 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 (Σ)* of Σ as being the S...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2024 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2024
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/212259 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Oriented Closed Polyhedral Maps and the Kitaev Model. Kornél Szlachányi. SIGMA 20 (2024), 048, 55 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862684276364935168 |
|---|---|
| author | Szlachányi, Kornél |
| author_facet | Szlachányi, Kornél |
| citation_txt | Oriented Closed Polyhedral Maps and the Kitaev Model. Kornél Szlachányi. SIGMA 20 (2024), 048, 55 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes Σ 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 (Σ)* of Σ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on (Σ)*, 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.
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| first_indexed | 2026-03-17T09:43:27Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212259 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T09:43:27Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Szlachányi, Kornél 2026-02-03T07:56:17Z 2024 Oriented Closed Polyhedral Maps and the Kitaev Model. Kornél Szlachányi. SIGMA 20 (2024), 048, 55 pages 1815-0659 2020 Mathematics Subject Classification: 05E99; 16T05; 81T25 arXiv:2302.08027 https://nasplib.isofts.kiev.ua/handle/123456789/212259 https://doi.org/10.3842/SIGMA.2024.048 A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes Σ 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 (Σ)* of Σ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on (Σ)*, 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. The author wishes to thank the anonymous referees for their thorough reading of the manuscript and for their valuable comments, which considerably improved the final version of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Oriented Closed Polyhedral Maps and the Kitaev Model Article published earlier |
| spellingShingle | Oriented Closed Polyhedral Maps and the Kitaev Model Szlachányi, Kornél |
| title | Oriented Closed Polyhedral Maps and the Kitaev Model |
| title_full | Oriented Closed Polyhedral Maps and the Kitaev Model |
| title_fullStr | Oriented Closed Polyhedral Maps and the Kitaev Model |
| title_full_unstemmed | Oriented Closed Polyhedral Maps and the Kitaev Model |
| title_short | Oriented Closed Polyhedral Maps and the Kitaev Model |
| title_sort | oriented closed polyhedral maps and the kitaev model |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212259 |
| work_keys_str_mv | AT szlachanyikornel orientedclosedpolyhedralmapsandthekitaevmodel |