COMPUTING ASSOCIATED PROJECTIVE MODULES USING THE MILNOR CLUTCHING

Speaker(s)

PIOTR M. HAJAC

Affiliation

IMPAN, Warszawa, Poland

Language of the talk

English

Date

Jan. 7, 2026, 5:15 p.m.

Information about the event

IMPAN - Room 405 & zoom

Title in Polish

COMPUTING ASSOCIATED PROJECTIVE MODULES USING THE MILNOR CLUTCHING

Seminar

North Atlantic Noncommutative Geometry Seminar

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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.)

 

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