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 TannakäKrein formalism which further strengthens Manin's approach to symmetry and
moduli-objects in noncommutative geometry.
