Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913072104341504 |
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| author | D'Andrea, Francesco Hajac, Piotr M. Maszczyk, Tomasz Zieliński, Bartosz |
| author_facet | D'Andrea, Francesco Hajac, Piotr M. Maszczyk, Tomasz Zieliński, Bartosz |
| contents | We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_08304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes D'Andrea, Francesco Hajac, Piotr M. Maszczyk, Tomasz Zieliński, Bartosz K-Theory and Homology Operator Algebras Quantum Algebra We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case. |
| title | Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes |
| topic | K-Theory and Homology Operator Algebras Quantum Algebra |
| url | https://arxiv.org/abs/2512.08304 |