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Teodor Banica

Publications and source records attributed to Teodor Banica.

At least 19 recordsLinked to original sources
arXiv2018

Higher orbitals of quizzy quantum group actions

The hyperoctahedral group $H_N$ is known to have two natural liberations: the "good" one $H_N^+$, which is the quantum symmetry group of $N$ segments, and the "bad" one $\bar{O}_N$, which is the quantum symmetry group of the $N$-hypercube. We study here this phenomenon, in the general "quizzy" framework, which covers the various liberations and twists of $H_N,O_N$. Our results include: (1) an interpretation of the embedding $\bar{O}_N\subset S_{2^N}^+$, as corresponding to the antisymmetric representation of $O_N$, (2) a study of the liberations of $H_N$, notably with the result $ =O_N^+$, and (3) a comparison of the $k$-orbitals for the inclusions $H_N\subset H_N^+$ and $H_N\subset\bar{O}_N$, for $k\in\mathbb N$ small.

math.QA↗
arXiv2018

Rigidity questions for real half-classical manifolds

Let $X$ be a noncommutative real compact algebraic manifold, in the sense that $C(X)=C^*(x_1,\ldots,x_N|x_i=x_i^*,f_α(x_1,\ldots,x_N)=0)$, with $f_α\in\mathbb R $. Associated to $X$ are its classical version $X^\times$, obtained via the relations $ab=ba$, and its half-classical version $X^*$, obtained via the relations $abc=cba$. We discuss here some general questions regarding the inclusions $X^\times\subset X^*\subset X$, and notably the comparison of the corresponding quantum isometry groups. Our main results concern the half-classical case, $X=X^*$, and more specifically, the case of the submanifolds $X\subset S^{N-1}_{\mathbb R,*}$.

math.QA↗
arXiv2018

Block-modified Wishart matrices: the easy case

Associated to any complex Wishart matrix $W$ of parameters $(dn,dm)$ and any linear map $φ:M_n(\mathbb C)\to M_n(\mathbb C)$ is the "block-modified" matrix $\tilde{W}=(id\otimesφ)W$. Following some previous work with Nechita, we study here the asymptotic $*$-distribution of $\tilde{W}$, in the $d\to\infty$ limit, in the case where the modification map $φ$ is "easy", or more generally super-easy, in the quantum algebra/representation theory sense. Under suitable assumptions on $φ$ we obtain in this way a compound free Poisson law.

math.PR↗
arXiv2018

Quantum groups, from a functional analysis perspective

It is well-known that any compact Lie group appears as closed subgroup of a unitary group, $G\subset U_N$. The unitary group $U_N$ has a free analogue $U_N^+$, and the study of the closed quantum subgroups $G\subset U_N^+$ is a problem of general interest. We review here the basic tools for dealing with such quantum groups, with all the needed preliminaries included, and we discuss as well a number of more advanced topics.

math.QA↗
arXiv2018

Quasi-flat representations of uniform groups and quantum groups

Given a discrete group $Γ= $ and a number $K\in\mathbb N$, a unitary representation $ρ:Γ\to U_K$ is called quasi-flat when the eigenvalues of each $ρ(g_i)\in U_K$ are uniformly distributed among the $K$-th roots of unity. The quasi-flat representations of $Γ$ form altogether a parametric matrix model $π:Γ\to C(X,U_K)$. We compute here the universal model space $X$ for various classes of discrete groups, notably with results in the case where $Γ$ is metabelian. We are particularly interested in the case where $X$ is a union of compact homogeneous spaces, and where the induced representation $\tildeπ:C^*(Γ)\to C(X,U_K)$ is stationary in the sense that it commutes with the Haar functionals. We present several positive and negative results on this subject. We also discuss similar questions for the discrete quantum groups, proving a stationarity result for the discrete dual of the twisted orthogonal group $O_2^{-1}$.

math.QA↗
arXiv2018

The planar algebra of a fixed point subfactor

We consider inclusions of type $(P\otimes A)^G\subset(P\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\subset B$. In the case $[A,B]=0$, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form $P(A\subset B)^G$, with $G$ acting in some natural sense on the bipartite graph algebra $P(A\subset B)$.

math.OA↗
arXiv2018

Modelling questions for quantum permutations

Given a quantum permutation group $G\subset S_N^+$, with orbits having the same size $K$, we construct a universal matrix model $π:C(G)\to M_K(C(X))$, having the property that the images of the standard coordinates $u_{ij}\in C(G)$ are projections of rank $\leq 1$. Our conjecture is that this model is inner faithful under suitable algebraic assumptions, and is in addition stationary under suitable analytic assumptions. We prove this conjecture for the classical groups, and for several key families of group duals.

math.OA↗
arXiv2018

Higher transitive quantum groups: theory and models

We investigate the notion of $k$-transitivity for the quantum permutation groups $G\subset S_N^+$, with a brief review of the known $k=1,2$ results, and with a study of what happens at $k\geq3$. We discuss then matrix modelling questions for the algebras $C(G)$, notably by introducing the related notions of double and triple flat matrix model. At the level of the examples, our main results concern the quantum groups coming from the complex Hadamard matrices, and from the Weyl matrices.

math.QA↗
arXiv2017

Tannakian duality for affine homogeneous spaces

Associated to any closed quantum subgroup $G\subset U_N^+$ and any index set $I\subset\{1,\ldots,N\}$ is a certain homogeneous space $X_{G,I}\subset S^{N-1}_{\mathbb C,+}$, called affine homogeneous space. We discuss here the abstract axiomatization of the algebraic manifolds $X\subset S^{N-1}_{\mathbb C,+}$ which can appear in this way, by using Tannakian duality methods.

math.QA↗
arXiv2017

The free unitary compact quantum group

The free analogues of $U(n)$ in Woronowicz's compact quantum group theory are the quantum groups $\{A_u(F)|F\in GL(n,\mathbb C)\}$ introduced by Van Daele and Wang. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If $F\bar{F}\in\mathbb R I_n$ we find an embedding $A_u(F)_{red}\subset C(\mathbb T)*_{red}A_o(F)$, where $A_o(F)$ is the deformation of $SU(2)$ that we previously studied. We use the representation theory and Powers' method for showing that the reduced algebras $A_u(F)_{red}$ are simple, with at most one trace.

math.QA↗
arXiv2017

Maximal torus theory for compact quantum groups

Associated to any compact quantum group $G\subset U_N^+$ is a canonical family of group dual subgroups $\widehatΓ_Q\subset G$, parametrized by unitaries $Q\in U_N$, playing the role of "maximal tori" for $G$. We present here a series of conjectures, relating the various algebraic and analytic properties of $G$ to those of the family $\{\widehatΓ_Q|Q\in U_N\}$.

math.QA↗
arXiv2017

Unitary easy quantum groups: geometric aspects

We discuss the classification problem for the unitary easy quantum groups, under strong axioms, of noncommutative geometric nature. Our main results concern the intermediate easy quantum groups $O_N\subset G\subset U_N^+$. To any such quantum group we associate its Schur-Weyl twist $\bar{G}$, two noncommutative spheres $S,\bar{S}$, a noncommutative torus $T$, and a quantum reflection group $K$. Studying $(S,\bar{S},T,K,G,\bar{G})$ leads then to some natural axioms, which can be used in order to investigate $G$ itself. We prove that the main examples are covered by our formalism, and we conjecture that in what concerns the case $U_N\subset G\subset U_N^+$, our axioms should restrict the list of known examples.

math.QA↗
arXiv2017

Isolated partial Hadamard matrices, and related topics

We study the isolated partial Hadamard matrices, under the assumption that the entries are roots of unity, or more generally, under the assumption that the combinatorics comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the master Hadamard matrices, and the McNulty-Weigert construction. We discuss then the notion of isolation in some related contexts, of the magic unitary matrices, and of the quantum permutation groups, with a number of conjectures on the subject.

math.CO↗
arXiv2017

Thoma type results for discrete quantum groups

Thoma's theorem states that a group algebra $C^*(Γ)$ is of type I if and only if $Γ$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group $Γ$, we have a stationary model of type $π:C^*(Γ)\to M_F(C(L))$, with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\widehatΓ\subset S_N^+$, by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank $\leq1$.

math.QA↗
arXiv2017

Complex analogues of the half-classical geometry

Under very strong axioms, there is precisely one real noncommutative geometry between the classical one and the free one, namely the half-classical one, coming from the relations $abc=cba$. We discuss here the complex analogues of this geometry, notably with a study of the geometry coming from the commutation relations between all the variables $\{ab^*,a^*b\}$, that we believe to be the "correct" one.

math.QA↗
arXiv2017

Complex Hadamard matrices with noncommutative entries

We axiomatize and study the matrices of type $H\in M_N(A)$, having unitary entries, $H_{ij}\in U(A)$, and whose rows and columns are subject to orthogonality type conditions. Here $A$ can be any $C^*$-algebra, for instance $A=\mathbb C$, where we obtain the usual complex Hadamard matrices, or $A=C(X)$, where we obtain the continuous families of complex Hadamard matrices. Our formalism allows the construction of a quantum permutation group $G\subset S_N^+$, whose structure and computation is discussed here.

math.QA↗
arXiv2017

Regularity questions for complex Hadamard matrices

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.

math.CO↗