Editing Quantum group (section)

== Intuitive meaning ==
The discovery of quantum groups was quite unexpected since it was known for a long time that [[Compact group|compact groups]] and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a [[group algebra of a topological group|group algebra]] or a [[universal enveloping algebra]], then a group algebra or enveloping algebra can be "deformed", although the deformation will no longer remain a group algebra or enveloping algebra. More precisely, deformation can be accomplished within the category of [[Hopf algebra]]s that are not required to be either [[commutative]] or [[cocommutative]]. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the [[noncommutative geometry]] of [[Alain Connes]]. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum [[Yang–Baxter equation]] and [[quantum inverse scattering method]] developed by the Leningrad School ([[Ludwig Faddeev]], [[Leon Takhtajan]], [[Evgeny Sklyanin]], [[Nicolai Reshetikhin]] and [[Vladimir Korepin]]) and related work by the Japanese School.<ref>{{citation|first=Christian|last=Schwiebert|title=Generalized quantum inverse scattering|year=1994|arxiv=hep-th/9412237v3|bibcode = 1994hep.th...12237S|pages=12237 }}</ref> The intuition behind the second, [[bicrossproduct]], class of quantum groups was  different and came from the search for self-dual objects as an approach to [[quantum gravity]].<ref>{{citation|first=Shahn|last=Majid|title=Hopf algebras for physics at the Planck scale|year=1988|journal=Classical and Quantum Gravity|volume= 5|pages=1587–1607|doi=10.1088/0264-9381/5/12/010|bibcode=1988CQGra...5.1587M|issue=12|citeseerx=10.1.1.125.6178}}</ref>