arXiv subjects

Franck Lesieur

Publications and source records attributed to Franck Lesieur.

2 recordsLinked to original sources

arXiv2007

Measured quantum groupoids

In this article, we give a definition for measured quantum groupoids. We want to get objects with duality extending both quantum groups and groupoids. We base ourselves on J. Kustermans and S. Vaes' works about locally compact quantum groups that we generalize thanks to formalism introduced by M. Enock and J.M. Vallin in the case of inclusion of von Neumann algebras. From a structure of Hopf-bimodule with left and right invariant operator-valued weights, we define a fundamental pseudo-multiplicative unitary. To get a satisfying duality in the general case, we assume the existence of an antipode given by its polar decomposition. This theory is illustrated with many examples among others inclusion of von Neumann algebras (M. Enock) and a sub family of measured quantum groupoids with easier axiomatic.

math.OA↗

arXiv2004

Measured quantum groupoids

In this article, part of the author's thesis, we propose a definition for measured quantum groupoid. The aim is the construction of objects with duality including both quantum groups and groupoids. We base ourselves on J. Kustermans and S. Vaes' works about locally compact quantum groups that we generalize thanks to formalism introduced by M. Enock and J.M. Vallin in the case of inclusion of von Neumann algebras. From a structure of Hopf-bimodule with left and right invariant operator-valued weights, we define a fundamental pseudo-multiplicative unitary. We introduce the notion of quasi-invariant weight on the basis and, then, we construct an antipode with polar decomposition, a coinvolution, a scaling group, a modulus and a scaling operator. This theory is illustrated with different examples. Duality of measured quantum groupoids will be discussed in a forthcoming article.

math.OA↗