#### Bulg. J. Phys. vol.35 no.s1 (2008), pp. 460-465

#### Piecewise Principal and Piecewise Trivial Comodule Algebras

R. Matthes

go back^{}*Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa 00-682, Poland*Abstract.This note is a short review of recent joint work with P.M. Hajac, U. Krähmer and B. Zieliński [12]. We propose the notions of a piecewise principal or, more specifically, piecewise trivial comodule algebra as analogues of the notion of a locally trivial principal bundle in the setting of noncommutative geometry. We show that the global comodule algebra of a piecewise principal comodule algebra is always principal, and that there is an interpretation of the notion of covering we use in terms of flabby sheaves over a certain universal finite space. Thus piecewise principal comodule algebras are principal extensions in the sense of [4], and piecewise trivial comodule algebras give rise to locally trivial quantum principal bundles in the sense of [16] (with flabby sheaves, and smash products as local models). Moreover, piecewise trivial comodule algebras include the notions of locally trivial quantum principal bundles of [6] and [8].