Noncommutative Differential Geometry of Generalized Weyl Algebras
Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p;q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p;q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particu...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2016 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2016
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147755 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Noncommutative Differential Geometry of Generalized Weyl Algebras / T. Brzeziński // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862540398299185152 |
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| author | Brzeziński, T. |
| author_facet | Brzeziński, T. |
| citation_txt | Noncommutative Differential Geometry of Generalized Weyl Algebras / T. Brzeziński // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p;q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p;q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p;q) are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the integral space coincides with the order of the defining polynomial p(z). It is proven that the restriction of these first-order differential calculi to the calculi on B(p;q) is isomorphic to the direct sum of degree 2 and degree −2 components of A(p;q). A Dirac operator for B(p;q) is constructed from a (strong) connection with respect to this differential calculus on the (free) spinor bimodule defined as the direct sum of degree 1 and degree −1 components of A(p;q). The real structure of KO-dimension two for this Dirac operator is also described.
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| first_indexed | 2025-11-24T16:09:03Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147755 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T16:09:03Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
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| spelling | Brzeziński, T. 2019-02-15T19:11:48Z 2019-02-15T19:11:48Z 2016 Noncommutative Differential Geometry of Generalized Weyl Algebras / T. Brzeziński // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16S38; 58B32; 58B34 DOI:10.3842/SIGMA.2016.059 https://nasplib.isofts.kiev.ua/handle/123456789/147755 Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p;q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p;q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p;q) are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the integral space coincides with the order of the defining polynomial p(z). It is proven that the restriction of these first-order differential calculi to the calculi on B(p;q) is isomorphic to the direct sum of degree 2 and degree −2 components of A(p;q). A Dirac operator for B(p;q) is constructed from a (strong) connection with respect to this differential calculus on the (free) spinor bimodule defined as the direct sum of degree 1 and degree −1 components of A(p;q). The real structure of KO-dimension two for this Dirac operator is also described. The author would like to express his gratitude to the referees for many helpful and detailed
 comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Noncommutative Differential Geometry of Generalized Weyl Algebras Article published earlier |
| spellingShingle | Noncommutative Differential Geometry of Generalized Weyl Algebras Brzeziński, T. |
| title | Noncommutative Differential Geometry of Generalized Weyl Algebras |
| title_full | Noncommutative Differential Geometry of Generalized Weyl Algebras |
| title_fullStr | Noncommutative Differential Geometry of Generalized Weyl Algebras |
| title_full_unstemmed | Noncommutative Differential Geometry of Generalized Weyl Algebras |
| title_short | Noncommutative Differential Geometry of Generalized Weyl Algebras |
| title_sort | noncommutative differential geometry of generalized weyl algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147755 |
| work_keys_str_mv | AT brzezinskit noncommutativedifferentialgeometryofgeneralizedweylalgebras |