On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems
The analog of the Chern-Gauss-Bonnet theorem is studied for a C∗-dynamical system consisting of a C∗-algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalg...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2016 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2016
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147426 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems / F. Fathizadeh, O. Fathizadeh // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 60 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862537117087825920 |
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| author | Fathizadeh, F. Gabriel, O. |
| author_facet | Fathizadeh, F. Gabriel, O. |
| citation_txt | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems / F. Fathizadeh, O. Fathizadeh // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 60 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The analog of the Chern-Gauss-Bonnet theorem is studied for a C∗-dynamical system consisting of a C∗-algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A⊂A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.
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| first_indexed | 2025-11-24T11:44:39Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147426 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T11:44:39Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
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| spelling | Fathizadeh, F. Gabriel, O. 2019-02-14T18:22:25Z 2019-02-14T18:22:25Z 2016 On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems / F. Fathizadeh, O. Fathizadeh // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 60 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58B34; 47B25; 46L05 DOI:10.3842/SIGMA.2016.016 https://nasplib.isofts.kiev.ua/handle/123456789/147426 The analog of the Chern-Gauss-Bonnet theorem is studied for a C∗-dynamical system consisting of a C∗-algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A⊂A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case. The authors thank the Hausdorf f Research Institute for Mathematics (HIM) for their hospitality
 and support during the trimester program on Noncommutative Geometry and its Applications
 in 2014, where the present work was partially carried out. They also thank the anonymous
 referees for their constructive feedback. Parts of this article were obtained and written while the
 second author was working as a postdoc at the University of Glasgow. He would like to thank
 C. Voigt for enabling his stay in Scotland. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems Article published earlier |
| spellingShingle | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems Fathizadeh, F. Gabriel, O. |
| title | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems |
| title_full | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems |
| title_fullStr | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems |
| title_full_unstemmed | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems |
| title_short | On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems |
| title_sort | on the chern-gauss-bonnet theorem and conformally twisted spectral triples for c*-dynamical systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147426 |
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