Quantum families of maps and quantum semigroups on finite quantum spaces

Sołtan, Piotr M.

doi:10.1016/j.geomphys.2008.11.007

Cited by 58 publications

(99 citation statements)

References 8 publications

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“…As we mentioned above, this is a purely algebraic reformulation of the notion of quantum family of all maps [15]. Analogue of Theorem 3.3 of [15], we show that m(B, A) exists if A is a free finite rank K-module. Also we consider some elementary properties of M(?, A) as a functor on the category of algebras.…”

Section: Introductionsupporting

confidence: 57%

“…So ltan [15] and S.L. Woronowicz [21] have introduced the notion of quantum space of all maps between two C*-algebraic NC spaces.…”

Section: Introductionmentioning

confidence: 99%

“…A is a finite dimensional C*-algebra. He also showed ( [15], [17]) that if B = A then C has a canonical C*-bialgebra structure (as in the classical case the space X X is a topological semigroup) and the bialgebra structure of some classes of quantum groups, e.g. quantum permutation groups ( [20], [2]) and quantum isometry groups ( [1]), are the same as of C.…”

Section: Introductionmentioning

confidence: 99%

“…To define this notion they used the C*-dual of the above universal property: For two quantum spaces QA and QB, a quantum space QC together with a *-homomorphism φ : B → A ⊗ C is called the quantum family of all maps from QA to QB if for every *-homomorphism ψ : B → A ⊗ D, between C*-algebras, there is a unique *-homomorphism ψ : C → D satisfying (id A ⊗ψ)φ = ψ. So ltan [15] has showed that such universal C*-algebra C and *-homomorphism φ exist when QA is a finite quantum space i.e. A is a finite 2010 Mathematics Subject Classification.…”

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On the quantum groups and semigroups of maps between noncommutative spaces

Sadr

2017

Czech.Math.J.

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Abstract. We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P.M. So ltan. 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 NC space. As special cases three classes of NC objects are introduced: quantum group of gauge transformations, Pontryagin dual of a quantum group, and Galois-Hopf-algebra of an algebra extension. IntroductionOur work is based on the extension to the noncommutative setting of the following picture. Let X and Y be compact Hausdorff spaces. If S is a Hausdorff space then there is a canonical one-to-one correspondence between continuous maps f : S × X → Y and continuous families f = {f (s, ?)} s∈S of continuous maps from X to Y with parameter space S. Moreover, by Exponential Law of Topology, there is a canonical homeomorphismwhere the mapping spaces are endowed with compact-open topology. Indeed, the family of all continuous maps from X to Y , that is the familyē induced by the evaluation map e : Y X × X → Y defined by e(a, x) = a(x), and the space Y X are completely characterized by the following universal property.For every Hausdorff space S and any continuous map f : S × X → Y there is a unique continuous mapf that makes the following diagram commutative.In the context of NC Geometry and Operator Algebra P.M. So ltan [15] and S.L. Woronowicz [21] have introduced the notion of quantum space of all maps between two C*-algebraic NC spaces. To define this notion they used the C*-dual of the above universal property: For two quantum spaces QA and QB, a quantum space QC together with a *-homomorphism φ : B → A ⊗ C is called the quantum family of all maps from QA to QB if for every *-homomorphism ψ : B → A ⊗ D, between C*-algebras, there is a unique *-homomorphism ψ : C → D satisfying (id A ⊗ψ)φ = ψ. So ltan [15] has showed that such universal C*-algebra C and *-homomorphism φ exist when QA is a finite quantum space i.e. A is a finite 2010 Mathematics Subject Classification. 16T05, 16T10, 16T20, 58B34.

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Section: Introductionsupporting

confidence: 57%

“…So ltan [15] and S.L. Woronowicz [21] have introduced the notion of quantum space of all maps between two C*-algebraic NC spaces.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the quantum groups and semigroups of maps between noncommutative spaces

Sadr

2017

Czech.Math.J.

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“…It is quite natural to formulate a quantum analogue of the above, by considering, in the spirit of Woronowicz and Soltan (see [19] and [13]), 'quantum families of isometries', which can be defined to be a pair (B, α) where B is a (not necessarily commutative) C * -algebra and α :…”

Section: Proofmentioning

confidence: 99%

Quantum Group of Isometries in Classical and Noncommutative Geometry

Goswami

2008

Commun. Math. Phys.

201

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We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the universal object in a bigger category, namely the category of 'quantum families of smooth isometries', defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum isometry group. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in [11] as the universal quantum group of holomorphic isometries of the noncommutative torus.

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Quantum Symmetry Groups and Related Topics

Skalski

2017

Quantum Symmetries

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Quantum families of maps and quantum semigroups on finite quantum spaces

Cited by 58 publications

References 8 publications

On the quantum groups and semigroups of maps between noncommutative spaces

On the quantum groups and semigroups of maps between noncommutative spaces

Quantum Group of Isometries in Classical and Noncommutative Geometry

Quantum Symmetry Groups and Related Topics

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