On bi-free De Finetti theorems
Annales Mathématiques Blaise Pascal, Tome 23 (2016) no. 1, pp. 21-51.

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.

Publié le :
DOI : https://doi.org/10.5802/ambp.353
Classification : 46L54,  46L53,  20G42
Mots clés : Quantum groups, free probability, De Finetti theorem
@article{AMBP_2016__23_1_21_0,
author = {Amaury Freslon and Moritz Weber},
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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.353/}
}
Amaury Freslon; Moritz Weber. On bi-free De Finetti theorems. Annales Mathématiques Blaise Pascal, Tome 23 (2016) no. 1, pp. 21-51. doi : 10.5802/ambp.353. https://ambp.centre-mersenne.org/articles/10.5802/ambp.353/

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