On the quantum groups and semigroups of maps between noncommutative spaces

Abstract

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.

Quantum metric spaces of quantum maps

“…There are many notions in Topology and Geometry that can be translate into NC language. The notion of quantum family of (quantum) maps, defined by Woronowicz [16] and Sołtan [15] (see also [10,11,12]), conclude from the following fact: "Every map f from X to the set of all maps from Y to Z (or in other word, any family of maps from Y to Z parameterized by f with parameters x in X) can be considered as a mapf : X ×Y → Z defined byf (x, y) = f (x)(y)." A translation of this to noncommutative language is as follows.…”

“…Definition 1.1. ( [10,11,12,15,16] Let B,C be unital C*-algebras. A quantum family of morphisms from B to C (or, a quantum family of maps from QC to QB) is a pair (A, Φ) consisting of a unital C*-algebra A and a unital *-homomorphism Φ : B → C ⊗ A, where ⊗ denotes the spatial tensor product of C*-algebras.…”