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arXiv · 2512.08304

Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes

Francesco D'Andrea·Piotr M. Hajac·Tomasz Maszczyk·Bartosz Zieliński
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Abstract

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.

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Francesco D'Andrea, Piotr M. Hajac, Tomasz Maszczyk, Bartosz Zieliński. 2025-12-09. Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes. https://arxiv.org/abs/2512.08304

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