The invention of quantum groups is one of the outstanding
achievements of mathematical physics and mathematics in the late
twentieth century. The birth of the new theory and its rapid
development are results of a strong interrelation between
mathematics and physics. Quantu~ groups arose in the work of L.D.
Faddeev and the Leningrad school on the inverse scattering method
in order to solve integrable models. The algebra Uq(sh) appeared
first in 1981 in a paper by P.P. Kulish and N.Yu. Reshetikhin on
the study of integrable XYZ models with highest spin. Its Hopf
algebra structure was discovered later by E.K. Sklyanin. A major
event was the discovery by V.G. Drinfeld and M. Jimbo around 1985
of a class of Hopf algebras which can be considered as
one-parameter deforma- tions of universal enveloping algebras of
semisimple complex Lie algebras. These Hopf algebras will be called
Drinfeld-Jimbo algebras in this book. Al- most simultaneously, S.L.
Woronowicz invented the quantum group SUq(2) and developed his
theory of compact quantum matrix groups. An algebraic approach to
quantized coordinate algebras was given about this time by Yu.I.
Manin.
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