Algebraic quantum transformation groupoids of compact type

Abstract

In this work, we introduce a class of Timmermann's measured multiplier Hopf *-algebroids called algebraic quantum transformation groupoids of compact type. Each object in this class admits a Pontrjagin-like dual called an algebraic quantum transformation groupoid of discrete type. This compact/discrete duality in the framework of algebraic quantum transformation groupoids recover the one between compact and discrete Van Daele's algebraic quantum groups. Among the non-trivial examples of algebraic quantum transformation groupoids of compact type, we give constructions arising from Fell bundles and quantum quotient spaces.