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Quantum Groups and Non-commutative Differential Geometry

In: Mathematical Physics X

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  • Yu. I. Manin

    (Steklov Math. Institute)

Abstract

Recently it became common to identify the notion of a quantum group with that of a Hopf algebra. However, this does not quite agree with the experience gained in classical group theory. In fact, classically, Hopf algebras arise in the following framework. One starts with a category S of “spaces” (finite sets, schemes, differential manifolds, topological spaces ...) and a linear functor ℱ on it, covariant or contravariant. The functor must satisfy certain formal properties, in particular, transform direct products into tensor products. The values of this functor upon group objects in S will then be Hopf algebras, commutative in the contravariant case, cocommutative in the covariant case.

Suggested Citation

  • Yu. I. Manin, 1992. "Quantum Groups and Non-commutative Differential Geometry," Springer Books, in: Konrad Schmüdgen (ed.), Mathematical Physics X, pages 113-122, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-77303-7_9
    DOI: 10.1007/978-3-642-77303-7_9
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