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 -freeness.
Mots clés : Quantum groups, free probability, De Finetti theorem
@article{AMBP_2016__23_1_21_0,
author = {Freslon, Amaury and Weber, Moritz},
title = {On bi-free {De} {Finetti} theorems},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {21--51},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {23},
number = {1},
year = {2016},
doi = {10.5802/ambp.353},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/ambp.353/}
}
TY - JOUR AU - Freslon, Amaury AU - Weber, Moritz TI - On bi-free De Finetti theorems JO - Annales mathématiques Blaise Pascal PY - 2016 SP - 21 EP - 51 VL - 23 IS - 1 PB - Annales mathématiques Blaise Pascal UR - http://archive.numdam.org/articles/10.5802/ambp.353/ DO - 10.5802/ambp.353 LA - en ID - AMBP_2016__23_1_21_0 ER -
%0 Journal Article %A Freslon, Amaury %A Weber, Moritz %T On bi-free De Finetti theorems %J Annales mathématiques Blaise Pascal %D 2016 %P 21-51 %V 23 %N 1 %I Annales mathématiques Blaise Pascal %U http://archive.numdam.org/articles/10.5802/ambp.353/ %R 10.5802/ambp.353 %G en %F AMBP_2016__23_1_21_0
Freslon, Amaury; Weber, Moritz. On bi-free De Finetti theorems. Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 1, pp. 21-51. doi : 10.5802/ambp.353. http://archive.numdam.org/articles/10.5802/ambp.353/
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