Cyclic Homology and Quantum Orbits
A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homo...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147108 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Cyclic Homology and Quantum Orbits / T. Maszczyk, S. Sütlü // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 40 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728501049688064 |
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| author | Maszczyk, T. Sütlü, S. |
| author_facet | Maszczyk, T. Sütlü, S. |
| citation_txt | Cyclic Homology and Quantum Orbits / T. Maszczyk, S. Sütlü // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 40 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed.
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| first_indexed | 2025-12-07T19:09:28Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147108 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:09:28Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
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| spelling | Maszczyk, T. Sütlü, S. 2019-02-13T16:47:03Z 2019-02-13T16:47:03Z 2015 Cyclic Homology and Quantum Orbits / T. Maszczyk, S. Sütlü // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 19D55; 57T15; 06A15; 46A20 DOI:10.3842/SIGMA.2015.041 https://nasplib.isofts.kiev.ua/handle/123456789/147108 A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed. The authors would like to thank the anonymous referees for their constructive comments improving
 the paper. The paper was partially supported by the NCN grant 2011/01/B/ST1/06474.
 S. Sutlu would like to thank his former PhD advisor B. Rangipour for drawing his attention to the
 homology of the coalgebra-Galois extensions, Institut des Hautes Etudes Scientifiques (IHES) ´
 for the hospitality provided during part of this work, and finally the organizers of the conference
 “From Poisson Brackets to Universal Quantum Symmetries”, held at IMPAN, Warsaw, for the
 stimulating environment provided. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cyclic Homology and Quantum Orbits Article published earlier |
| spellingShingle | Cyclic Homology and Quantum Orbits Maszczyk, T. Sütlü, S. |
| title | Cyclic Homology and Quantum Orbits |
| title_full | Cyclic Homology and Quantum Orbits |
| title_fullStr | Cyclic Homology and Quantum Orbits |
| title_full_unstemmed | Cyclic Homology and Quantum Orbits |
| title_short | Cyclic Homology and Quantum Orbits |
| title_sort | cyclic homology and quantum orbits |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147108 |
| work_keys_str_mv | AT maszczykt cyclichomologyandquantumorbits AT sutlus cyclichomologyandquantumorbits |