In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra. There is no single, all-encompassing definition, but instead a family of broadly similar objects.
The term "quantum group" often denotes a kind of noncommutative algebra with additional structure that first appeared in the theory of quantum integrable systems, and which was then formalized by Vladimir Drinfel'd 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 algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or 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.
Just as groups often appear as symmetries, quantum groups act on many other mathematical objects and it has become fashionable to introduce the adjective quantum in such cases; for example there are quantum planes and quantum Grassmannians.
Read more about Quantum Group: Intuitive Meaning, Drinfel'd-Jimbo Type Quantum Groups, Compact Matrix Quantum Groups, Bicrossproduct Quantum Groups
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