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非可換多様体の例

トポロジ ( 換幾 ) みとしては しやすい C * -algebra コンパクト Hausdorff scheme 換環 いれば しかしながら , するために してしまうと イメ がつかみづらくなる

もちろん されている Connes Landi [ CL01 ] Connes Dubois Violette [ CDV02 ] などである Euclid については , Pirkovskii [ Pir ] , Kupriyanov Vitale [ KV15 ] , Dubois-Violette Landi [ DVL ] などがある

  • noncommutative Euclidean space

Euclid であるが , にも がある Passer [ Pas16 ] Natsume-Olsen sphere [ NO97 ] ばれるものに Borsuk-Ulam している

Connes Dubois-Violette , [ CDV08 ] , noncommutative 3 について しく 調 べている その Bellon [ Bel ] である S 1 ,S 2 ,S 3 quantum version については Dabrowski [ Dab03 Dab06 ] やすい Plazas [ Pla08 ] では ラス われている

  • noncommutative sphere
  • noncommutative torus

調 べられているのが だろうか D’Andrea Dabrowski Landi quantum projective plane K -theory などを [ DDL08 ] 調 べている Noncommutative (quantum) projective space としては , Hajac Kaygun Zielinski [ HKZ12 ] Buachalla [ ÓB12 ] がある をその めて えているは , D’Andrea Landi [ DL13 ] である Brzezinski Fairfax [ BF12 ] quantum weighted projective space して いる

  • noncommutative projective space
  • noncommutative weighted projective space

Grassmann quantum version についてもいくつかの があるら しい Lakshmibai Reshetikhin [ LR91 ] など Chakraborty Sundar [ CS11 ] によると , Stiefel Podkolzin Vainerman [ PV99 ] より されたようである Chakraborty らは , その K -theory して いる

  • noncommutative Grassmannian manifold
  • noncommutative Stiefel manifold

References

[Bel]     Marc Bellon. Lectures on the three–dimensional non–commutative spheres, arXiv:0710.4434 .

[BF12]     Tomasz Brzeziński and Simon A. Fairfax. Quantum teardrops. Comm. Math. Phys. , 316(1):151–170, 2012, arXiv:1107.1417 .

[CDV02]     Alain Connes and Michel Dubois-Violette. Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Comm. Math. Phys. , 230(3):539–579, 2002, http://dx.doi.org/10.1007/s00220-002-0715-2 .

[CDV08]     Alain Connes and Michel Dubois-Violette. Noncommutative finite dimensional manifolds. II. Moduli space and structure of noncommutative 3-spheres. Comm. Math. Phys. , 281(1):23–127, 2008, arXiv:math/0511337 .

[CL01]     Alain Connes and Giovanni Landi. Noncommutative manifolds, the instanton algebra and isospectral deformations. Comm. Math. Phys. , 221(1):141–159, 2001, http://dx.doi.org/10.1007/PL00005571 .

[CS11]     Partha Sarathi Chakraborty and S. Sundar. K -groups of the quantum homogeneous space su q ( n ) ∕su q ( n - 2). Pacific J. Math. , 252(2):275–292, 2011, arXiv:1006.1742 .

[Dab03]     Ludwik Dabrowski. The garden of quantum spheres. In Noncommutative geometry and quantum groups (Warsaw, 2001) , volume 61 of Banach Center Publ. , pages 37–48. Polish Acad. Sci., Warsaw, 2003, arXiv:math/0212264 .

[Dab06]     Ludwik Dabrowski. Geometry of quantum spheres. J. Geom. Phys. , 56(1):86–107, 2006, arXiv:math/0501240 .

[DDL08]     Francesco D’Andrea, Ludwik D    browski, and Giovanni Landi. The noncommutative geometry of the quantum projective plane. Rev. Math. Phys. , 20(8):979–1006, 2008, arXiv:0712.3401 .

[DL13]     Francesco D’Andrea and Giovanni Landi. Geometry of quantum projective spaces. In Noncommutative geometry and physics. 3 , volume 1 of Keio COE Lect. Ser. Math. Sci. , pages 373–416. World Sci. Publ., Hackensack, NJ, 2013, arXiv:1203.0621 .

[DVL]     Michel Dubois-Violette and Giovanni Landi. Noncommutative Euclidean spaces, arXiv:1801.03410 .

[HKZ12]     Piotr M. Hajac, Atabey Kaygun, and Bartosz Zieliński. Quantum complex projective spaces from Toeplitz cubes. J. Noncommut. Geom. , 6(3):603–621, 2012, arXiv:1008.0673 .

[KV15]     V. G. Kupriyanov and P. Vitale. Noncommutative d via closed star product. J. High Energy Phys. , (8):024, front matter+24, 2015, arXiv:1502.06544 .

[LR91]     V. Lakshmibai and N. Reshetikhin. Quantum deformations of SL n ∕B and its Schubert varieties. In Special functions (Okayama, 1990) , ICM-90 Satell. Conf. Proc., pages 149–168. Springer, Tokyo, 1991.

[NO97]     T. Natsume and C. L. Olsen. Toeplitz operators on noncommutative spheres and an index theorem. Indiana Univ. Math. J. , 46(4):1055–1112, 1997, http://dx.doi.org/10.1512/iumj.1997.46.1152 .

[ÓB12]     Réamonn Ó Buachalla. Quantum bundle description of quantum projective spaces. Comm. Math. Phys. , 316(2):345–373, 2012, arXiv:1105.1768 .

[Pas16]     Benjamin W. Passer. A noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres. J. Operator Theory , 75(2):337–366, 2016, arXiv:1503.01822 .

[Pir]     A. Yu. Pirkovskii. Quantum polydisk, quantum ball, and a q -analog of Poincaré’s theorem, arXiv:1311.0309 .

[Pla08]     Jorge Plazas. Examples of noncommutative manifolds: complex tori and spherical manifolds. In An invitation to noncommutative geometry , pages 419–445. World Sci. Publ., Hackensack, NJ, 2008, arXiv:math/0703849 .

[PV99]     G. B. Podkolzin and L. I. Vainerman. Quantum Stiefel manifold and double cosets of quantum unitary group. Pacific J. Math. , 188(1):179–199, 1999, http://dx.doi.org/10.2140/pjm.1999.188.179 .