Related papers: Kawada-It\^o-Kelley Theorem for Quantum Semigroups

We introduce the construction of induced corepresentations in the setting of locally compact quantum groups and prove that the resulting induced corepresentations are unitary under some mild integrability condition. We also establish a…

We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study properties. Our approach is slightly different from the one introduced in the thesis of S.~Roy and, unlike the investigations…

In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The aim of this paper is to carry out a careful analysis regarding Haar weight, in relation to general…

Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids.

The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator. While the former aspect is a main focus of current research, the…

A notion of L^2-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its L^2-Betti…

The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how So{\l}tan's quantum Bohr compactification can be…

The Haar measure on some locally compact quantum groups is constructed. The main example we treat is the az+b-group of Woronowicz. We also briefly consider some other examples (like the ax+b-group). We get the first examples of a locally…

The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a L\'{e}vy process on a locally compact group. In particular a definitive…

We prove that a compact quantum group with faithful Haar state which has a faithful action on a compact space must be a Kac algebra, with bounded antipode and the square of the antipode being identity. The main tool in proving this is the…

We develop theory of multiplicity maps for compact quantum groups, as an application, we obtain a complete classification of right coideal $C^*$-algebras of $C(SU_q(2))$ for $q\in [-1,1]\setminus \{0\}$. They are labeled with Dynkin…

Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient…

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable sigma-C*-algebras. We also relate equivariant vector bundles…

We introduce and study a number of invariants of locally compact quantum groups defined by their scaling and modular groups and the spectrum of their modular elements. Focusing mainly on compact quantum groups we consider the question…

Any bipartite quantum state has quasi-probability representations in terms of separable states. For entangled states these quasi-probabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one…

We introduce the notion of commability between locally compact groups, namely the equivalence relation generated by cocompact inclusions and quotients by compact normal subgroups. We give a classification of focal hyperbolic locally compact…

Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general…

We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We…

Let $S$ be a subsemigroup of a second countable locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^*$-algebra $C^*_\delta(S)$ generated by the operators of translation by all elements of $S$ in $L^2(S)$. We show that this…

In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the…