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Amaury Freslon; Moritz Weber
On bi-free De Finetti theorems
Annales mathématiques Blaise Pascal, 23 no. 1 (2016), p. 21-51, doi: 10.5802/ambp.353
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Class. Math.: 46L54, 46L53, 20G42
Mots clés: Quantum groups, free probability, De Finetti theorem

Résumé - Abstract

We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of $n$-freeness.

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