Related papers: Subgroups of Quantum Groups
The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for…
We show that finite index quantum subgroups of a discrete quantum group are induced from finite index quantum subgroups of the unimodularization. As an application, we classify all finite index quantum subgroups of free products of the…
We introduce the notion of self-similarity for compact quantum groups. For a finite set $X$, we introduce a $C^*$-algebra $\mathbb{A}_X$, which is the quantum automorphism group of the infinite homogeneous rooted tree $X^*$. Self-similar…
Q-system completion can be thought of as a notion of higher idempotent completion of C*-2-categories. We introduce a notion of quantum bi-elements, and study Q-system completion in the context of compact quantum groups. We relate our notion…
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum…
Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the…
We have written down a set of notes on compact quantum groups from which all the different aspects can be learned in an easy way and such that a lot of insight can be obtained without too much effort. Compact quantum groups have been…
We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum…
We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and…
A rigorous general definition of quantum probability is given, which is valid for elementary events and for composite events, for operationally testable measurements as well as for inconclusive measurements, and also for non-commuting…
The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of…
We review the literature about reaching agreement in quantum networks, also called quantum consensus. After a brief introduction to the key feature of quantum computing, allowing the reader with no quantum theory background to have minimal…
The purpose of this paper is to present the notion of quotient of supergroups in different categories using the unified treatment of the functor of points and to examine some physically interesting examples.
This is an introduction to the group field theory approach to quantum gravity, with emphasis on motivations and basic formalism, more than on recent results; we elaborate on the various ingredients, both conceptual and formal, of the…
Motivated by generalizing Szemer\'edi's theorem, we the elements in a discrete quantum group fixing a sequence of finite subsets and prove that the set of these elements is a quantum subgroup. Using this we obtain a version of mean ergodic…
We complete the classification of quantum subgroups of $SL_q(2)$ with $q$ a root of unity of arbitrary order, that is, Hopf algebra quotients of the quantum function algebras $\mathcal{O}_{q} (SL_2(\mathbb{C}))$.
We study the (compact) quantum subgroups of the compact quantum group $SU_{-1}(3)$: we show that any non-classical such quantum subgroup is a twist of a compact subgroup of SU(3) or is isomorphic to a quantum subgroup of $U_{-1}(2)$.
We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups. Several new quantum groups constructed by Banica, Curran and Speicher since the…
Group field theory is a background-independent approach to quantum gravity whose starting point is the definition of a quantum field theory on an auxiliary group manifold (not interpreted as spacetime, but rather as the finite-dimensional…
A unified framework for different formulations of quantum theoery is introduced specifying what is meant by a quantum mechanical theory in general.