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This textbook presents the second edition of Manin's celebrated
1988 Montreal lectures, which influenced a new generation of
researchers in algebra to take up the study of Hopf algebras and
quantum groups. In this expanded write-up of those lectures, Manin
systematically develops an approach to quantum groups as symmetry
objects in noncommutative geometry in contrast to the more
deformation-oriented approach due to Faddeev, Drinfeld, and others.
This new edition contains an extra chapter by Theo Raedschelders
and Michel Van den Bergh, surveying recent work that focuses on the
representation theory of a number of bi- and Hopf algebras that
were first introduced in Manin's lectures, and have since gained a
lot of attention. Emphasis is placed on the Tannaka-Krein
formalism, which further strengthens Manin's approach to symmetry
and moduli-objects in noncommutative geometry.
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