Publications (75)54.04 Total impact
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ABSTRACT: It is known that, under strong axioms, $O_N\subset O_N^*\subset O_N^+$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N1}\subset S^{N1}_*\subset S^{N1}_+$, the noncommutative projective spaces $P^N_\mathbb R\subset P^N_\mathbb C\subset P^N_+$, and the projective orthogonal quantum groups $PO_N\subset PO_N^*\subset PO_N^+$.  [Show abstract] [Hide abstract]
ABSTRACT: The glow of an Hadamard matrix H ∈ MN(C) is the probability measure μ ∈〉(ℂ) describing the distribution of ϕ(a,b) = 〈a,Hb〉, where a,b ∈ TN are random. We prove that ϕ/N becomes complex Gaussian with N →∞, and that the universality holds as well at order 2. In the case of a Fourier matrix, FG ∈MN(ℂ) with G = N , the universality holds up to order 4, and the fluctuations are encoded by certain subtle integrals, which appear in connection with several Hadamardrelated questions. In the Walsh matrix case, G = ℤn2 , we conjecture that the glow is polynomial in N = 2n.  [Show abstract] [Hide abstract]
ABSTRACT: We discuss the halfliberation operation $X\to X^*$, for the algebraic submanifolds of the unit sphere, $X\subset S^{N1}_\mathbb C$. There are several ways of constructing this correspondence, and we take them into account. Our main results concern the computation of the affine quantum isometry group $G^+(X^*)$, for the sphere itself.  [Show abstract] [Hide abstract]
ABSTRACT: We study the discrete quantum groups $\Gamma$ whose group algebra has an inner faithful representation of type $\pi:C^*(\Gamma)\to M_K(\mathbb C)$. Such a representation can be thought of as coming from an embedding $\Gamma\subset U_K$. Our main result, concerning a certain class of examples of such quantum groups, is an asymptotic convergence theorem for the random walk on $\Gamma$. The proof uses various algebraic and probabilistic techniques.  [Show abstract] [Hide abstract]
ABSTRACT: We discuss SchurWeyl duality, in the quantum isometry group framework. Our main result is a general duality principle, between noncommutative analogues of the standard cube $\mathbb Z_2^N$, and nonocommutative analogues of the standard sphere $S^{N1}_\mathbb R$, which connects the corresponding quantum isometry groups. As a consequence, we show that all the intermediate easy quantum groups $H_N\subset G\subset O_N^+$ naturally appear as quantum isometry groups. We discuss as well a number of unitary extensions of these results.  [Show abstract] [Hide abstract]
ABSTRACT: The real sphere $S^{N1}_\mathbb R$ appears as increasing union, over $d\in\{1,...,N\}$, of its "polygonal" versions $S^{N1,d1}_\mathbb R=\{x\in S^{N1}_\mathbb Rx_{i_0}... x_{i_d}=0,\forall i_0,...,i_d\ {\rm distinct}\}$. Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of $S^{N1,d1}_\mathbb R$, and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with $S^{N1}_\mathbb R$ replaced by the complex sphere $S^{N1}_\mathbb C$.  [Show abstract] [Hide abstract]
ABSTRACT: A noncommutative sphere $S^{N1}_\times$ is called undeformed if its associated integration functional $tr:C(S^{N1}_\times)\to\mathbb C$ has the trace property $tr(ab)=tr(ba)$. Examples can be obtained by liberating, twisting, and liberating+twisting the real and complex spheres $S^{N1}_\mathbb R,S^{N1}_\mathbb C$. We show here that, under very strong axioms, there are exactly 10 such spheres, and we compute the associated quantum isometry groups. We formulate as well a proposal for an extended formalism, comprising 18 spheres.  [Show abstract] [Hide abstract]
ABSTRACT: The partial isometries of $\mathbb R^N,\mathbb C^N$ form compact semigroups $\widetilde{O}_N,\widetilde{U}_N$. We discuss here the liberation question for these semigroups, and for their discrete versions $\widetilde{H}_N,\widetilde{K}_N$. Our main results concern the construction of halfliberations $\widetilde{H}_N^\times,\widetilde{K}_N^\times,\widetilde{O}_N^\times,\widetilde{U}_N^\times$ and of liberations $\widetilde{H}_N^+,\widetilde{K}_N^+,\widetilde{O}_N^+,\widetilde{U}_N^+$. We include a detailed algebraic and probabilistic study of all these objects, justifying our "halfliberation" and "liberation" claims.  [Show abstract] [Hide abstract]
ABSTRACT: Associated to a complex Hadamard matrix $H\in M_N(\mathbb C)$ is the complex probability measure $\mu\in\mathcal P(\mathbb C)$ describing the distribution of $\varphi(a,b)=<a,Hb>$, where $a,b\in\mathbb T^N$ are random. This measure is called "glow" of the matrix, due to the analogy with the GaleBerlekamp switching game, where $H,a,b$ are real. We prove here that: (1) $\mu$ becomes complex Gaussian in the $N\to\infty$ limit, (2) the universality holds as well at order 2, (3) the order 3 term seems to be quite interesting, particularly for the master Hadamard matrices, (4) in the Fourier matrix case, some of the higher order terms control counting problems for circulant Hadamard matrices.  [Show abstract] [Hide abstract]
ABSTRACT: We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest nontrivial case is M=3, and we obtain here several results, notably with a classification at N=7. We discuss as well the potential applications of the M=3 results to various $M=N$ questions.  [Show abstract] [Hide abstract]
ABSTRACT: The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. The singularity at a point $H\in C_N$ is described by a filtration of cones $T^\times_HC_N\subset T^\circ_HC_N\subset T_HC_N\subset\widetilde{T}_HC_N$, coming from the trivial, affine, smooth and first order deformations. We study here these cones in the case where $H=F_N$ is the Fourier matrix, $(w^{ij})$ with $w=e^{2\pi i/N}$, our main result being a simple description of $\widetilde{T}_HC_N$. As a consequence, the rationality conjecture $dim_\mathbb R(\widetilde{T}_HC_N)=dim_\mathbb Q(\widetilde{T}_HC_N\cap M_N(\mathbb Q))$ holds at $H=F_N$.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $q_0=...=q_{N1}=1$ the quantity $\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the bruteforce minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here these questions, with some results and conjectures.  [Show abstract] [Hide abstract]
ABSTRACT: In our previous work, we introduced the following relaxation of the Hadamard property: a square matrix $H\in M_N(\mathbb R)$ is called "almost Hadamard" if $U=H/\sqrt{N}$ is orthogonal, and locally maximizes the 1norm on O(N). We review our previous results, notably with the formulation of a new question, regarding the circulant and symmetric case. We discuss then an extension of the almost Hadamard matrix formalism, by making use of the pnorm on O(N), with $p\in[1,\infty]{2}$, with a number of theoretical results on the subject, and the formulation of some open problems.  [Show abstract] [Hide abstract]
ABSTRACT: The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. We have $C_N=M_N(\mathbb T)\cap\sqrt{N}U_N$, and following Tadej and \.Zyczkowski we investigate here the computation of the enveloping tangent space $\widetilde{T}_HC_N=T_HM_N(\mathbb T)\cap T_H\sqrt{N}U_N$, and notably of its dimension $d(H)=\dim(\widetilde{T}_HC_N)$, called undephased defect of $H$. Our main result is an explicit formula for the defect of the Fourier matrix $F_G$ associated to an arbitrary finite abelian group $G=\mathbb Z_{N_1}\times...\times\mathbb Z_{N_r}$. We also comment on the general question "does the associated quantum permutation group see the defect", with a probabilistic speculation involving DiaconisShahshahani type variables.  [Show abstract] [Hide abstract]
ABSTRACT: We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1norm on O(N). Our study includes a detailed discussion of the circulant case ($H_{ij}=\gamma_{ji}$) and of the twoentry case ($H_{ij}\in{x,y}$), with the construction of several families of examples, and some 1norm computations.  [Show abstract] [Hide abstract]
ABSTRACT: We study the random matrices of type $\tilde{W}=(id\otimes\varphi)W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $\varphi:M_n(\mathbb C)\to M_n(\mathbb C)$ is a selfadjoint linear map. We prove that, under suitable assumptions, we have the $d\to\infty$ eigenvalue distribution formula $\delta m\tilde{W}\sim\pi_{mn\rho}\boxtimes\nu$, where $\rho$ is the law of $\varphi$, viewed as a square matrix, $\pi$ is the free Poisson law, $\nu$ is the law of $D=\varphi(1)$, and $\delta=tr(D)$.  [Show abstract] [Hide abstract]
ABSTRACT: We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\hat{\Lambda}\subset G$. In the matrix case $G\subset U_n^+$ the embedding condition is equivalent to having a quotient map $\Gamma_U\to\Lambda$, where $F=\{\Gamma_UU\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon's classification of group dual subgroups $\hat{\Lambda}\subset S_n^+$. These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have nonabelian group dual isometries.  [Show abstract] [Hide abstract]
ABSTRACT: We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $\pi:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Ces\`aro limit of the linear functional $\phi=tr\circ\pi$. We discuss then some consequences of this result, notably to inner linearity questions.  [Show abstract] [Hide abstract]
ABSTRACT: Given a quantum subgroup $G\subset U_n$ and a number $k\leq n$ we can form the homogeneous space $X=G/(G\cap U_k)$, and it follows from the StoneWeierstrass theorem that $C(X)$ is the algebra generated by the last $nk$ rows of coordinates on $G$. In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type $X=G/(G\cap U_k^+)$.  [Show abstract] [Hide abstract]
ABSTRACT: Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length.
Publication Stats
1k  Citations  
54.04  Total Impact Points  
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Institutions

20082015

Université de CergyPontoise
95001 CEDEX, IledeFrance, France


20062009

University of Toulouse
Tolosa de Llenguadoc, MidiPyrénées, France


20022008

Paul Sabatier University  Toulouse III
 Département de Mathématiques
Tolosa de Llenguadoc, MidiPyrénées, France


2007

Claude Bernard University Lyon 1
Villeurbanne, RhôneAlpes, France


19701998

University of California, Berkeley
 Department of Mathematics
Berkeley, California, United States


1997

French National Centre for Scientific Research
Lutetia Parisorum, ÎledeFrance, France
