arXiv subjects

Moritz Weber

Publications and source records attributed to Moritz Weber.

43 records · Page 3Linked to original sources

arXiv2015

Absence of algebraic relations and of zero divisors under the assumption of finite non-microstates free Fisher information

We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables in an appropriate sense exclude algebraic relations for the generators. Moreover, under the assumption of finite non-microstates free Fisher information, 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.

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arXiv2014

A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups

We present a link between easy quantum groups, discrete groups and combinatorics. By this, we infer new connections between quantum isometry groups, reflection groups, varieties of groups and the combinatorics of partitions. More precisely, we consider easy quantum groups and find a relation to subgroups of the infinite free product $\mathbb Z_2^{*\infty}$ of $\mathbb Z_2=\mathbb Z/2\mathbb Z$. We obtain a link with reflection groups and thus with varieties of groups, which yields a statement on the complexity of the class of easy quantum groups on the one hand, and a "quantum invariant" for varieties of groups on the other hand. Moreover, we reveal a triangular relationship between easy quantum groups, categories of partitions and discrete groups (reflection groups). As a by-product, we obtain a large number of new quantum isometry groups.

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arXiv2014

Easy quantum groups and quantum subgroups of a semi-direct product quantum group

We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy quantum groups between the free orthogonal quantum group and the permutation group. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for easy quantum groups in the scope of this article.

math.QA↗

arXiv2013

The full classification of orthogonal easy quantum groups

In 1987, Woronowicz gave a definition of compact matrix quantum groups generalizing compact Lie groups in the setting of noncommutative geometry. About twenty years later, Banica and Speicher isolated a class of compact matrix quantum groups with an intrinsic combinatorial structure. These so called easy quantum groups are determined by categories of partitions. They have been proven useful in order to understand various aspects of quantum groups, in particular linked with Voiculescu's free probability theory. Furthermore, they exhibit a way to find examples of compact quantum groups besides q-deformations and quantum isometry groups. These characteristics naturally motivated attempts to fully classify them. This is completed in the present article.

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arXiv2013

On the classification of easy quantum groups

In 2009, Banica and Speicher began to study the compact quantum subgroups of the free orthogonal quantum group containing the symmetric group S_n. They focused on those whose intertwiner spaces are induced by some partitions. These so-called easy quantum groups have a deep connection to combinatorics. We continue their work on classifying these objects introducing some new examples of easy quantum groups. In particular, we show that the six easy groups O_n, S_n, H_n, B_n, S_n' and B_n' split into seven cases on the side of free easy quantum groups. Also, we give a complete classification in the half-liberated case.

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arXiv2013

The combinatorics of an algebraic class of easy quantum groups

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the orthogonal group as well as Wang's quantum permutation group and his free orthogonal quantum group. In this article, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another article.

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arXiv2012

On C*-Algebras Generated by Isometries with Twisted Commutation Relations

In the theory of C*-algebras, interesting noncommutative structures arise as deformations of the tensor product. For instance, the rotation algebra may be seen as a scalar twist deformation of the tensor product of the functions on the circle with itself. We deform the tensor product of two Toeplitz algebras in the same way, introducing the universal C*-algebra generated by two isometries u and v such that uv=e^{it}vu and u*v=e^{-it}vu*, for a fixed real parameter t. Since the second relation implies the first one, we also consider the universal C*-algebra generated by two isometries u and v with the weaker relation uv=e^{it}vu. Such a "weaker case" does not exist in the case of unitaries, and it turns out to be much more interesting than the twisted "tensor product case" of two Toeplitz algebras. We show that the C*-algebra in the "tensor product case" is nuclear, whereas in the "weaker case" it is not even exact. Also, we compute the K-groups and we obtain K_0 = Z and K_1 = 0 for both C*-algebras. This answers a question raised by Murphy in 1994 concerning the K-theory of the C*-algebra associated to N^2.

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