Le langage des espaces et des groupes quantiques. (The language of quantum spaces and quantum groups)
type: Item
instance of: scholarly article
title: Le langage des espaces et des groupes quantiques. (The language of quantum spaces and quantum groups)
zbMATH Open document ID: 0783.17007
DOI: 10.1007/BF02096770
author: Georges Maltsiniotis
published in: Communications in Mathematical Physics
publication date: 1993-05-16T00:00:00Z
review text: A well known difficulty in non-commutative geometry is the following one: On a non-commutative algebra, there does not seem to exist a canonical choice of a differential structure (calculus of differential forms) coinciding in the special case of commutative algebras with the classical ones (given in a natural way by the underlying space). The author takes the point of view that, at least in the context of quantum groups and quantum spaces, the proper objects of a differential geometric study are not the usual quantum groups and spaces but graded differential algebras built over them. [Compare also \textit{Yu. I. Manin}, Notes on quantum groups and quantum de Rham complexes, Teor. Mat. Fiz. 92, No. 3, 425-450 (1992).] The category of quantum spaces is defined as the opposite category to the category of graded differential algebras generated by the elements of degree zero. The main concern of the paper is to introduce the typical notions of the theory of quantum groups in this category. After recalling some well-known facts about graded differential algebras the category of quantum spaces is defined. There are quantum subspaces, products of quantum spaces, quantum spaces of finite type. An important special case of a quantum space is a quantum cone (a bigraded differential algebra, the second gradation stemming from a gradation of the zeroth order (with respect to the first gradation) component). Then the notion of quantum monoid is introduced, which generalizes the notion of bialgebra. A special case are matrix quantum monoids. There exists the quantum monoid of endomorphisms of a quantum cone which is uniquely determined by some universality property. Quantum groups are defined as quantum monoids equipped with an antipode, which again means a generalization of the usual notion of antipode. For every matrix quantum monoid, there is an associated quantum group, which is a matrix quantum monoid under some additional assumption (the quantum monoid is Cramer). This last…
reviewed by: R. Matthes
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 165686
zbMATH Keywords: non-commutative geometry
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum spaces
zbMATH Keywords: graded differential algebras
zbMATH Keywords: quantum monoid
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02096770
OpenAlex ID: W2039852384
description: scientific article; zbMATH DE number 165686