Open AccessPushouts of extensions of groupoids by bundles of abelian groups
Author(s) -
Marius Ionescu,
Alex Kumjian,
Jean Renault,
Aidan Sims,
Dana P. Williams
Publication year - 2021
Publication title -
new zealand journal of mathematics
Language(s) - English
Resource type - Journals
eISSN - 1179-4984
pISSN - 1171-6096
DOI - 10.53733/136
Subject(s) - mathematics , abelian group , isomorphism (crystallography) , pure mathematics , double groupoid , cartesian product , algebra over a field , dual (grammatical number) , crossed product , action (physics) , extension (predicate logic) , discrete mathematics , computer science , art , chemistry , physics , literature , quantum mechanics , crystal structure , programming language , crystallography
We analyse extensions $\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.