Module maps over locally compact quantum groups

Zhiguo Hu; Matthias Neufang; Zhong-Jin Ruan

Displaying similar documents to “Module maps over locally compact quantum groups”

Quantum SU(2) and the Baum-Connes conjecture

Christian Voigt (2012)

Banach Center Publications

Similarity:

We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $S U_{q} (2)$ of Woronowicz. As an illustration of this result we determine the K-groups of quantum automorphism groups of simple matrix algebras.

Contractible quantum Arens-Michael algebras

Nina V. Volosova (2010)

Banach Center Publications

Similarity:

We consider quantum analogues of locally convex spaces in terms of the non-coordinate approach. We introduce the notions of a quantum Arens-Michael algebra and a quantum polynormed module, and also quantum versions of projectivity and contractibility. We prove that a quantum Arens-Michael algebra is contractible if and only if it is completely isomorphic to a Cartesian product of full matrix C*-algebras. Similar results in the framework of traditional (non-quantum) approach are established,...

Quantum 4-sphere: the infinitesimal approach

F. Bonechi, M. Tarlini, N. Ciccoli (2003)

Banach Center Publications

Similarity:

We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to $_{q} (S U (2))$ .

Free dynamical quantum groups and the dynamical quantum group $S U_{Q}^{d y n} (2)$

Thomas Timmermann (2012)

Banach Center Publications

Similarity:

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical $S U_{q} (2)$ studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

The geometric reductivity of the quantum group $S L_{q} (2)$

Michał Kępa, Andrzej Tyc (2011)

Colloquium Mathematicae

Similarity:

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants $A^{G}$ is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple,...

Quantum detailed balance conditions with time reversal: the finite-dimensional case

Franco Fagnola, Veronica Umanità (2011)

Banach Center Publications

Similarity:

We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely $t r (ρ^{1 / 2} x ρ_{t}^{1 / 2} (y)) = t r (ρ^{1 / 2} θ y * θ ρ_{t}^{1 / 2} (θ x * θ))$ for all x,y ∈ and t ≥ 0. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying...

Examples of quantum braided groups

Hlavatý, Ladislav

Similarity:

Summary: The author gives the defining relations of a new type of bialgebras that generalize both the quantum groups and braided groups as well as the quantum supergroups. The relations of the algebras are determined by a pair of matrices $(R, Z)$ that solve a system of Yang-Baxter-type equations. The matrix coproduct and counit are of standard matrix form, however, the multiplication in the tensor product of the algebra is defined by virtue of the braiding map given by the matrix $Z$ . Besides...

On the problem of classifying simple compact quantum groups

Shuzhou Wang (2012)

Banach Center Publications

Similarity:

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.

Problems in the theory of quantum groups

Shuzhou Wang (1997)

Banach Center Publications

Similarity:

This is a collection of open problems in the theory of quantum groups. Emphasis is given to problems in the analytic aspects of the subject.

An idempotent for a Jordanian quantum complex sphere

Bartosz Zieliński (2003)

Banach Center Publications

Similarity:

A new Jordanian quantum complex 4-sphere together with an instanton-type idempotent is obtained as a suspension of the Jordanian quantum group $S L_{h} (2)$ .

Anyonic Groups

Shahn Majid (1992)

Recherche Coopérative sur Programme n°25

Similarity:

Quantum spaces: notes and comments on a lecture by S. L. Woronowicz

R. Budzyński, W. Kondracki (1995)

Banach Center Publications

Similarity:

Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors

Sungjin Ra, Hakho Hong (2024)

Applications of Mathematics

Similarity:

This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in $ℝ^{3}$ , we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions...

When is a quantum space not a group?

Piotr Mikołaj Sołtan (2010)

Banach Center Publications

Similarity:

We give a survey of techniques from quantum group theory which can be used to show that some quantum spaces (objects of the category dual to the category of C*-algebras) do not admit any quantum group structure. We also provide a number of examples which include some very well known quantum spaces. Our tools include several purely quantum group theoretical results as well as study of existence of characters and traces on C*-algebras describing the considered quantum spaces as well as...

The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

Shuangjian Guo, Shengxiang Wang (2016)

Colloquium Mathematicae

Similarity:

Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and $B = A^{c o H}$ . Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map $β : A \otimes_{B} A \to A \otimes H$ , $a \otimes_{B} b \mapsto S^{- 1} (b_{[1]}) α (b_{[0] [- 1]}) \otimes β^{- 1} (a) β (b_{[0] [0]})$ , is surjective, we prove that the induction functor $A \otimes_{B} - : ̃ {(_{k})}_{B} \to_{A}^{H}$ is an equivalence of categories.

On the quantum groups and semigroups of maps between noncommutative spaces

Maysam Maysami Sadr (2017)

Czechoslovak Mathematical Journal

Similarity:

We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC...

Quantum permutation groups: a survey

Teodor Banica, Julien Bichon, Benoît Collins (2007)

Banach Center Publications

Similarity:

This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on...

Quantum isometry group for spectral triples with real structure.

Goswami, Debashish (2010)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Similarity:

On two quantum versions of the detailed balance condition

Franco Fagnola, Veronica Umanità (2010)

Banach Center Publications

Similarity:

Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form $⟨ x, y ⟩_{s} : = t r (ρ^{1 - s} x * ρ^{s} y)$ (s ∈ [0,1])...