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

TL;DR

This paper applies Milnor's K-theory construction to quantum projective planes, explicitly computing K-theory classes and revealing positive cone phenomena unique to quantum cases.

Contribution

It introduces an explicit formula for K_1-classes of modules over quantum projective planes using Milnor idempotents, linking K-theory with noncommutative geometry.

Findings

K_0$-generators are explicitly determined for quantum projective planes.

All K_0$-generators lie in the positive cone, a quantum-specific phenomenon.

Explicit homotopies relate K-theory classes to elementary projections in Toeplitz algebra.

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.