arXiv subjects

Moritz Weber

Publications and source records attributed to Moritz Weber.

At least 37 records · Page 2Linked to original sources

arXiv2019

Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

math.QA↗

arXiv2019

Models of quantum permutations

For $N\ge 4$ we present a series of *-homomorphisms $φ_n:C(S_N^+)\rightarrow B_n$ where $S_N^+$ is the quantum permutation group. They are not necessarily representations of the quantum group $S_N^+$ but they yield good operator algebraic models of quantum permutation matrices. The C*-algebras $B_n$ allow the construction of an inverse limit $B_{\infty}$ which defines a compact matrix quantum group $S_N\subsetneq G\subseteq S_N^+$. We know $G=S_N^+$ for $N=4,5$ from Banica's work, but we have to leave open the case $N\ge 6$.

math.OA↗

arXiv2019

Intertwiner spaces of quantum group subrepresentations

We consider compact matrix quantum groups whose $N$-dimensional fundamental representation decomposes into an $(N-1)$-dimensional and a one-dimensional subrepresentation. Even if we know that the compact matrix quantum group associated to this $(N-1)$-dimensional subrepresentation is isomorphic to the given $N$-dimensional one, it is a priori not clear how the intertwiner spaces transform under this isomorphism. In the context of so-called easy and non-easy quantum groups, we are able to define a transformation of linear combinations of partitions and we explicitly describe the transformation of intertwiner spaces. As a side effect, this enables us to produce many new examples of non-easy quantum groups being isomorphic to easy quantum groups as compact quantum groups but not as compact matrix quantum groups.

math.QA↗

arXiv2019

Categories of Two-Colored Pair Partitions, Part II: Categories Indexed by Semigroups

Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to "half-liberated easy" interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.

math.CO↗

arXiv2018

Categories of Two-Colored Pair Partitions, Part I: Categories Indexed by Cyclic Groups

We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however crossings are allowed. There are two families of such categories, the first of which is indexed by cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka-Krein result, the categories in the two families correspond to easy quantum groups interpolating the classical unitary group and Wang's free unitary quantum group. In fact, they are all half-liberated in some sense and our results imply that there are many more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside.

math.CO↗

arXiv2018

Factorization of Frieze Patterns

In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the negative by giving counterexamples. We conclude that even under stronger assumptions, there is no canonical decomposition.

math.CO↗

arXiv2018

Quantum Symmetries of Graph C*-Algebras

The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

math.OA↗

arXiv2018

On the representation theory of partition (easy) quantum groups

Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S.L. Woronowicz. In the case of easy quantum groups, the intertwiner spaces are given by the combinatorics of partitions, see the inital work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group $S_N^+$, the free orthogonal quantum group $O_N^+$ as well as the hyperoctahedral quantum group $H_N^+$.

math.QA↗

arXiv2018

Partition quantum spaces

We propose a definition of partition quantum spaces. They are given by universal $C^*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first $d$ columns of easy quantum groups. However, we define them as universal $C^*$-algebras rather than as $C^*$-subalgebras of easy quantum groups. We also investigate the minimal number $d$ needed to recover an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, $d$ takes the values one or two.

math.OA↗

arXiv2017

Partition C*-algebras II - Links to Compact Matrix Quantum Groups

In a recent article, we gave a definition of partition C*-algebras. These are universal C*-algebras based on algebraic relations which are induced from partitions of sets. In this follow up article, we show that often we can associate a Hopf algebra structure to partition C*-algebras, and also a compact matrix quantum group structure. This follows the lines of Banica and Speicher's approach to quantum groups; however, we access them in a more algebraic way circumventing Tannaka-Krein duality. We give criteria when these quantum groups are quantum subgroups of Wang's free orthogonal quantum group. As a consequence, we see that even if we start with (generalized) categories of partitions which do not contain the pair partitions, in many cases we do not go beyond the class of Banica-Speicher quantum groups (aka easy quantum groups). However, we also discuss possible non-unitary Banica-Speicher quantum groups.

math.OA↗

arXiv2017

Partition C*-algebras

We give a definition of partition C*-algebras: To any partition of a finite set, we assign algebraic relations for a matrix of generators of a universal C*-algebra. We then prove how certain relations may be deduced from others and we explain a partition calculus for simplifying such computations. This article is a small note for C*-algebraists having no background in compact quantum groups, although our partition C*-algebras are motivated from those underlying Banica-Speicher quantum groups (also called easy quantum groups). We list many open questions about partition C*-algebras that may be tackled by purely C*-algebraic means, ranging from ideal structures and representations on Hilbert spaces to K-theory and isomorphism questions. In a follow up article, we deal with the quantum algebraic structure associated to partition C*-algebras.

math.OA↗

arXiv2016

Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

math.OA↗

arXiv2016

Quantum groups based on spatial partitions

We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions -- which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang's free orthogonal quantum group $O_n^+$ which do not contain the symmetric group $S_n$; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of certain categories of partitions which do neither contain the pair partition nor the identity partition.

math.QA↗

arXiv2016

Quantum groups with partial commutation relations

We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the orthogonal group: They are partially commutative in a way such that they do not interpolate between the classical and the free quantum versions of the orthogonal group. Likewise we define non-interpolating, partially commutative quantum versions of the symmetric group recovering Bichon's quantum automorphism groups of graphs. They fit with the mixture of classical and free independence as recently defined by Speicher and Wysoczanski (rediscovering $Λ$-freeness of Mlotkowski), due to some weakened version of a de Finetti theorem.

math.QA↗

arXiv2015

Unitary Easy Quantum Groups: the free case and the group case

Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of ($C^*$-algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group $O_n^+$. We now give a definition of unitary easy quantum groups using colored partitions to tackle the problem of finding quantum subgroups of $U_n^+$. In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups. We also study the notion of easy groups.

math.QA↗

arXiv2015

The classification of tensor categories of two-colored noncrossing partitions

Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These so called categories of partitions are exactly the tensor categories being used in the theory of Banica and Speicher's orthogonal easy quantum groups. In our approach, we additionally allow a coloring of the points. This serves as the basis for the introduction of unitary easy quantum groups, which is done in a separate article. The present article however is purely combinatorial. We find all categories of two-colored noncrossing partitions. For doing so, we extract certain parameters with values in the natural numbers specifying the colorization of the categories on a global as well as on a local level. It turns out that there are ten series of categories, each indexed by one or two parameters from the natural numbers, plus two additional categories. This is just the beginning of the classification of categories of two-colored partitions and we point out open problems at the end of the article.

math.CO↗

arXiv2015

Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension

We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the generators does not have atoms. Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.

math.OA↗

arXiv2015

On bi-free De Finetti theorems

We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of n-freeness.

math.PR↗