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Le langage des espaces et des groupes quantiques. (The language of quantum spaces and quantum groups). (French) Zbl 0783.17007
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 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 construction is a generalization of an analogous construction of Yu. I. Manin [Quantum groups and non-commutative geometry (Montreal, 1988; Zbl 0724.17006)], which itself is a generalization of the usual algebraic procedure of associating to a semigroup the group of its invertible elements.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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