Editing Quantum group (section)

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In [[mathematics]] and [[theoretical physics]], the term '''quantum group''' denotes one of a few different kinds of [[noncommutative algebra]]s with additional structure. These include Drinfeld–Jimbo type quantum groups (which are [[quasitriangular Hopf algebra]]s), [[Compact quantum group|compact matrix quantum groups]] (which are structures on unital separable [[C*-algebra]]s), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group.

The term "quantum group" first appeared in the theory of [[Integrable system#Quantum integrable systems|quantum integrable systems]], which was then formalized by [[Vladimir Drinfeld]] and [[Michio Jimbo]] as a particular class of [[Hopf algebra]]. The same term is also used for other Hopf algebras that deform or are close to classical [[Lie groups]] or [[Lie algebras]], such as a "bicrossproduct" class of quantum groups introduced by [[Shahn Majid]] a little after the work of Drinfeld and Jimbo.

In Drinfeld's approach, quantum groups arise as [[Hopf algebra]]s depending on an auxiliary parameter ''q'' or ''h'', which become [[universal enveloping algebra]]s of a certain Lie algebra, frequently [[semisimple Lie algebra|semisimple]] or [[affine Lie algebra|affine]], when ''q'' = 1 or ''h'' = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple [[algebraic group]] or a [[compact Lie group]].

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