[Show abstract][Hide abstract] 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 $S^{N-1}\subset S^{N-1}_*\subset S^{N-1}_+$, 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: We discuss the half-liberation operation $X\to X^*$, for the algebraic
submanifolds of the unit sphere, $X\subset S^{N-1}_\mathbb C$. 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 discuss Schur-Weyl 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 $S^{N-1}_\mathbb R$, 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 $S^{N-1}_\mathbb R$ appears as increasing union, over
$d\in\{1,...,N\}$, of its "polygonal" versions $S^{N-1,d-1}_\mathbb R=\{x\in
S^{N-1}_\mathbb R|x_{i_0}... x_{i_d}=0,\forall i_0,...,i_d\ {\rm distinct}\}$.
Motivated by general classification questions for the undeformed noncommutative
spheres, smooth or not, we study here the quantum isometries of
$S^{N-1,d-1}_\mathbb R$, and of its various noncommutative analogues, obtained
via liberation and twisting. We discuss as well a complex version of these
results, with $S^{N-1}_\mathbb R$ replaced by the complex sphere
$S^{N-1}_\mathbb C$.
[Show abstract][Hide abstract] ABSTRACT: A noncommutative sphere $S^{N-1}_\times$ is called undeformed if its
associated integration functional $tr:C(S^{N-1}_\times)\to\mathbb C$ has the
trace property $tr(ab)=tr(ba)$. Examples can be obtained by liberating,
twisting, and liberating+twisting the real and complex spheres $S^{N-1}_\mathbb
R,S^{N-1}_\mathbb C$. 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.
Journal of Geometry and Physics 12/2014; 96. DOI:10.1016/j.geomphys.2015.05.006 · 0.87 Impact Factor
[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
half-liberations
$\widetilde{H}_N^\times,\widetilde{K}_N^\times,\widetilde{O}_N^\times,\widetilde{U}_N^\times$
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 "half-liberation" 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 Gale-Berlekamp
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
non-trivial 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 $|q_0|=...=|q_{N-1}|=1$ 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 brute-force 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 1-norm 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 p-norm 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 Diaconis-Shahshahani type variables.
Linear Algebra and its Applications 10/2012; 438(9). DOI:10.1016/j.laa.2013.01.011 · 0.94 Impact Factor
[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 1-norm on O(N). Our
study includes a detailed discussion of the circulant case
($H_{ij}=\gamma_{j-i}$) and of the two-entry case ($H_{ij}\in{x,y}$), with the
construction of several families of examples, and some 1-norm computations.
Open Systems & Information Dynamics 02/2012; 19(04). DOI:10.1142/S1230161212500242 · 0.69 Impact Factor
[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 self-adjoint 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 $\delta=tr(D)$.
Houston journal of mathematics 01/2012; 41(1). · 0.42 Impact Factor
[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 $F=\{\Gamma_U|U\in U_n\}$ 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 non-abelian 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.
Bulletin of the Polish Academy of Sciences Mathematics 12/2011; 60(2). DOI:10.4064/ba60-2-3
[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
Stone-Weierstrass theorem that $C(X)$ is the algebra generated by the last
$n-k$ 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 $X=G/(G\cap U_k^+)$.
Journal of Geometry and Physics 09/2011; 62(6). DOI:10.1016/j.geomphys.2012.02.003 · 0.87 Impact Factor
[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.
Proceedings of the London Mathematical Society 09/2011; 106(5). DOI:10.1112/plms/pds071 · 1.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This is a presentation of recent work on quantum permutation groups, complex
Hadamard matrices, and the connections between them. A long list of problems is
included. We include as well some conjectural statements, about matrix models.
[Show abstract][Hide abstract] ABSTRACT: We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal",
where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated
orthogonal quantum group, in the sense that there is no intermediate compact
quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we
use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some
maximality results for classical groups, obtained by using Lie algebras and
some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.
Communications in Algebra 06/2011; 41(2). DOI:10.1080/00927872.2011.633138 · 0.39 Impact Factor