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In the last decade we have witnessed the birth of a fascinating new mathematical theory. It is often called by algebraists the theory of quantum groups and by topologists quantum topology. These terms, however, seem to be too restrictive and do not convey the breadth of this new domain which is closely related to the theory of von Neumann algebras, the theory of Hopf algebras, the theory of representations of semisimple Lie algebras, the topology of knots, etc. The most spectacular achievements in this theory are centered around quantum groups and invariants of knots and 3-dimensional manifolds.
The whole theory has been, to a great extent, inspired by ideas that arose in theoretical physics. Among the relevant areas of physics are the theory of exactly solvable models of statistical mechanics, the quantum inverse scattering method, the quantum theory of angular momentum, 2-dimensional conformal field theory, etc. The development of this subject shows once more that physics and mathematics intercommunicate and influence each other to the profit of both disciplines.
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Vladimir G. Turaev, Indiana University, Bloomington, USA.
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