Abstract
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⊂ℂ N , with H⊂G⊂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.
Highlights
Given a compact group action G X, assumed to be transitive, we have X = G/H, where H = {g ∈ G | gx0 = x0} is the stabilizer of a given point x0 ∈ X
: C(X) ⊂ C(G) → C, and can be explicitely computed provided that we know how to integrate over G, for instance via a Weingarten type formula
We recall that any compact quantum group G has a Haar integration functional : C(G) → C, having the following invariance properties:
Summary
: C(X) ⊂ C(G) → C, and can be explicitely computed provided that we know how to integrate over G, for instance via a Weingarten type formula. An extension to spaces of type GN /GN−M , with M ≤ N , and with G = (GN ) subject to some suitable uniformity assumptions (“easiness”) was discussed in [5]. The common feature of these spaces X = G/H is that they are “easy”, in the sense that one can explicitely integrate on them, via a Weingarten type formula. The purpose of the present paper is to provide an axiomatic framework for such spaces, to advance at the level of the general theory, and to enlarge the class of known examples. The paper is organized as follows: Sections 1-2 are preliminary sections, in Sections 3-4 we restrict the attention to the affine space case, in Sections 5-6 we discuss some basic examples, and in Sections 7-8 we focus on the easy space case and we discuss a number of probabilistic aspects
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