A study of Galois objects for algebraic quantum groups

Abstract

We supplement the study of Galois theory for algebraic quantum groups started in the paper 'Galois Theory for Multiplier Hopf Algebras with Integrals' by A. Van Daele and Y.H. Zhang. We examine the structure of the Galois objects: algebras equipped with a Galois coaction such that only the scalars are coinvariants. We show how their structure is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We also show how to reflect the quantum group across the Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.

32 pages

References in corpus (2)

• Multiplier Hopf algebras imbedded in C${}^{\ast }$-algebraic quantum groups
• Quantum torsors with fewer axioms

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• I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type
• Multiplier Hopf monoids
• A correspondence between homogeneous and Galois coactions of Hopf algebras
• Actions of measured quantum groupoids on a finite basis
• On cocycle twisting of compact quantum groups