arXiv subjects

Teodor Banica

Publications and source records attributed to Teodor Banica.

At least 37 records · Page 2Linked to original sources

arXiv2017

Super-easy quantum groups: definition and examples

We investigate the "two-parameter" quantum symmetry groups that we previously constructed with Skalski, with the conclusion that some of these quantum groups, namely those without singletons, are "super-easy" in a suitable sense, that we axiomatize here. Our formalism covers as well the symplectic group $Sp_n$ and its free version $Sp_n^+$, and some other interesting examples. Finally, we address the general problem of classifying the super-easy quantum groups, and we make a few comments on it.

math.QA↗

arXiv2017

Weingarten integration over noncommutative homogeneous spaces

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 $X=G/H\subset\mathbb C^N$, 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.

math.QA↗

arXiv2017

Almost Hadamard matrices with complex entries

We discuss an extension of the almost Hadamard matrix formalism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices.

math.CO↗

arXiv2017

Matrix models for noncommutative algebraic manifolds

We discuss the notion of matrix model, $π:C(X)\to M_K(C(T))$, for algebraic submanifolds of the free complex sphere, $X\subset S^{N-1}_{\mathbb C,+}$. When $K\in\mathbb N$ is fixed there is a universal such model, which factorizes as $π:C(X)\to C(X^{(K)})\subset M_K(C(T))$. We have $X^{(1)}=X_{class}$ and, under a mild assumption, inclusions $X^{(1)}\subset X^{(2)}\subset X^{(3)}\subset\ldots\subset X$. Our main results concern $X^{(2)},X^{(3)},X^{(4)},\ldots$, their relation with various half-classical versions of $X$, and lead to the construction of families of higher half-liberations of the complex spheres and of the unitary groups, all having faithful matrix models.

math.QA↗

arXiv2017

Deformed Fourier models with formal parameters

The deformed Fourier matrices $H=F_M\otimes_QF_N$, with $Q\in\mathbb T^{MN}$, produce a matrix model $C(S_{MN}^+)\to M_{MN}(C(\mathbb T^{MN}))$. When $Q\in\mathbb T^{MN}$ is generic, the corresponding fiber can be investigated via algebraic techniques, and the main character law is asymptotically free Poisson. We present here an alternative point of view on these questions, using formal parameters instead of generic parameters, and analytic tools.

math.QA↗

arXiv2016

Quantum groups from stationary matrix models

We study the quantum groups appearing via models $C(G)\subset M_K(C(X))$ which are "stationary", in the sense that the Haar integration over $G$ is the functional $tr\otimes\int_X$. Our results include a number of generalities, notably with a substantial list of examples, and a detailed discussion in the quantum permutation group case.

math.QA↗

arXiv2016

Flat matrix models for quantum permutation groups

We study the matrix models $π:C(S_N^+)\to M_N(C(X))$ which are flat, in the sense that the standard generators of $C(S_N^+)$ are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at $N=4$, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of $C(S_N^+)$, inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

math.OA↗

arXiv2016

Quantum isometries, noncommutative spheres, and related integrals

The sphere $S^{N-1}_\mathbb R$ has a half-liberated analogue $S^{N-1}_{\mathbb R,*}$, and a free analogue $S^{N-1}_{\mathbb R,+}$. This is a presentation of the construction and main properties of these noncommutative spheres, $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}$, and of their quantum isometry groups.

math.OA↗

arXiv2016

Liberation theory for noncommutative homogeneous spaces

We discuss the liberation question, in the homogeneous space setting. Our first series of results concerns the axiomatization and classification of the families of compact quantum groups $G=(G_N)$ which are "uniform", in a suitable sense. We study then the quotient spaces of type $X=(G_M\times G_N)/(G_L\times G_{M-L}\times G_{N-L})$, and the liberation operation for them, with a number of algebraic and probabilistic results.

math.OA↗

arXiv2015

Uniqueness results for noncommutative spheres and projective spaces

It is known that, under strong combinatorial axioms, $O_N\subset O_N^*\subset O_N^+$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}$, the noncommutative projective spaces $P^{N-1}_\mathbb R\subset P^{N-1}_\mathbb C\subset P^{N-1}_+$, and the projective orthogonal quantum groups $PO_N\subset PO_N^*\subset PO_N^+$.

math.OA↗

arXiv2015

Half-liberated manifolds, and their quantum isometries

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.

math.OA↗

arXiv2015

The algebraic structure of quantum partial isometries

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.

math.OA↗

arXiv2015

Liberations and twists of real and complex spheres

We study the 10 noncommutative spheres obtained by liberating, twisting, and liberating+twisting the real and complex spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$. At the axiomatic level, we show that, under very strong axioms, these 10 spheres are the only ones. Our main results concern the computation of the quantum isometry groups of these 10 spheres, taken in an affine real/complex sense. We formulate as well a proposal for an extended formalism, comprising 18 spheres.

math.OA↗

arXiv2015

Quantum isometries of noncommutative polygonal spheres

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 $S^{N-1}_\mathbb C$.

math.OA↗

arXiv2015

A duality principle for noncommutative cubes and spheres

We discuss a general duality principle, between noncommutative analogues of the standard cube $\mathbb Z_2^N$, and nonocommutative analogues of the standard sphere $S^{N-1}_\mathbb R$. This duality is by construction of algebraic geometric nature, and conjecturally connects the corresponding quantum isometry groups, taken in an affine sense.

math.OA↗

arXiv2015

Random walk questions for linear quantum groups

We study the discrete quantum groups $Γ$ whose group algebra has an inner faithful representation of type $π:C^*(Γ)\to M_K(\mathbb C)$. Such a representation can be thought of as coming from an embedding $Γ\subset U_K$. Our main result, concerning a certain class of examples of such quantum groups, is an asymptotic convergence theorem for the random walk on $Γ$. The proof uses various algebraic and probabilistic techniques.

math.OA↗

arXiv2014

The Gale-Berlekamp game for Hadamard matrices

Given an Hadamard matrix $H\in M_N(\pm1)$ we consider the function $φ:\mathbb Z_2^N\times\mathbb Z_2^N\to\mathbb Z$ given by $φ(a,b)=\sum_{ij}a_ib_jH_{ij}$, which sums the entries of the various conjugates of $H$, obtained by switching signs on rows and columns. Our claim is that $φ$, or just its probabilistic distribution $μ\in\mathcal P(\mathbb Z)$, that we call "glow" of the matrix, should encode important information about $H$. We present here a number of results and conjectures in this direction, notably with a general decomposition result for $μ$.

math.CO↗

arXiv2014

The Gale-Berlekamp game for complex Hadamard matrices

Associated to a complex Hadamard matrix $H\in M_N(\mathbb C)$ is the complex probability measure $μ\in\mathcal P(\mathbb C)$ describing the distribution of $φ(a,b)= $, 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) $μ$ 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.

math.CO↗