On the quantum groups and semigroups of maps between noncommutative spaces

TL;DR

This paper introduces algebraic frameworks for quantum families of maps and automorphisms between noncommutative spaces, leading to new quantum groups related to finite spaces and their symmetries.

Contribution

It develops algebraic notions of quantum families of maps and automorphisms, constructing new classes of Hopf-algebras representing quantum symmetries of noncommutative spaces.

Findings

Constructed quantum groups of all maps from finite spaces to quantum groups.

Defined quantum groups of automorphisms of finite noncommutative spaces.

Introduced special classes: gauge transformations, Pontryagin duals, and Galois-Hopf-algebras.

Abstract

We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P.M. Soltan. 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.