Rank-two Milnor idempotents for the multipullback quantum complex projective plane
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2017
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| _version_ | 1866915942406029312 |
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| author | Farsi, Carla Hajac, Piotr M. Maszczyk, Tomasz Zielinski, Bartosz |
| author_facet | Farsi, Carla Hajac, Piotr M. Maszczyk, Tomasz Zielinski, Bartosz |
| contents | The $K_0$-group of the C*-algebra of multipullback quantum complex projective plane is known to be $\mathbb{Z}^3$, with one generator given by the C*-algebra itself, one given by the section module of the noncommutative (dual) tautological line bundle, and one given by the Milnor module associated to a generator of the $K_1$-group of the C*-algebra of Calow-Matthes quantum 3-sphere. Herein we prove that these Milnor modules are isomorphic either to the section module of a noncommutative vector bundle associated to the $SU_q(2)$-prolongation of the Heegaard quantum 5-sphere $S^5_H$ viewed as a $U(1)$-quantum principal bundle, or to a complement of this module in the rank-four free module. Finally, we demonstrate that one of the above Milnor modules always splits into the direct sum of the rank-one free module and a rank-one non-free projective module that is \emph{not} associated with $S^5_H$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1708_04426 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Rank-two Milnor idempotents for the multipullback quantum complex projective plane Farsi, Carla Hajac, Piotr M. Maszczyk, Tomasz Zielinski, Bartosz K-Theory and Homology Operator Algebras The $K_0$-group of the C*-algebra of multipullback quantum complex projective plane is known to be $\mathbb{Z}^3$, with one generator given by the C*-algebra itself, one given by the section module of the noncommutative (dual) tautological line bundle, and one given by the Milnor module associated to a generator of the $K_1$-group of the C*-algebra of Calow-Matthes quantum 3-sphere. Herein we prove that these Milnor modules are isomorphic either to the section module of a noncommutative vector bundle associated to the $SU_q(2)$-prolongation of the Heegaard quantum 5-sphere $S^5_H$ viewed as a $U(1)$-quantum principal bundle, or to a complement of this module in the rank-four free module. Finally, we demonstrate that one of the above Milnor modules always splits into the direct sum of the rank-one free module and a rank-one non-free projective module that is \emph{not} associated with $S^5_H$. |
| title | Rank-two Milnor idempotents for the multipullback quantum complex projective plane |
| topic | K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/1708.04426 |