Computing associated projective modules using the Milnor clutching

Abstract

Principal bundles can be viewed as strongly monoidal functors from the finite-dimensional representation category of a structure group to the category of associated vector bundles. Much in the same way, principal comodule algebras can be characterized as comodule algebras inducing a strongly monodoidal functor from the finite-dimensional corepresentation category of a structure Hopf algebra to the category of associated bimodules. Principal comodule algebras enjoy fully fledged cyclic-homology Chern-Weil theory that can be used to compute idempotents of the associated finitely generated projective modules and classify their K-theory classes. The goal of this talk is to present a new method of computing such idempotents using the celebrated Milnor's connecting homomorphism in K-theory. Although one can always try to compute these idempotents using strong connections, such calculations can be very complicated. I will show that, in the case of piecewise-cleft principal comodule algebras, we have an alternative way to compute the idempotents using the Milnor clutching construction. The new method will be demonstrated in the setting of the K-theory of quantum complex projective planes. (Based on joint work w F. D'Andrea, T. Maszczyk and B. Zieliński.)