Weingarten integration over noncommutative homogeneous spaces

Banica, Teodor

Annales mathématiques Blaise Pascal, Tome 24 (2017) no. 2, pp. 195-224.

On présente une extension de la formule d’intégration de Weingarten, pour les espaces homogènes non commutatifs, vérifiant des hypothèses « d’aisance » adéquates. Les espaces qu’on considère sont des variétés algebriques non commutatives, généralisant les espaces du type $X=G/H\subset {ℂ}^N$, avec $H\subset G\subset U_N$ étant des sous-groupes du groupe unitaire, vérifiant certaines conditions d’uniformité. On traite d’abord les questions d’axiomatisation, ensuite on établit la formule de Weingarten, et on finit avec quelques conséquences probabilistes.

We discuss an extension of the Weingarten formula, to the case of noncommutative homogeneous spaces, under suitable “easiness” assumptions. The spaces that we consider are noncommutative algebraic manifolds, generalizing the spaces of type $X=G/H\subset {ℂ}^N$, with $H\subset G\subset U_N$ being subgroups of the unitary group, subject to certain uniformity conditions. We discuss various axiomatization issues, then we establish the Weingarten formula, and we derive some probabilistic consequences.

Classification : 46L51, 14A22, 60B15
Keywords: Noncommutative manifold, Weingarten integration
Mot clés : Variété non commutative, Integration de Weingarten

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