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00:00:58 1 Intuitive meaning
00:01:38 2 Drinfeld–Jimbo type quantum groups
00:03:05 2.1 Representation theory
00:15:16 2.1.1 Case 1: iq/i is not a root of unity
00:18:32 2.1.2 Case 2: iq/i is a root of unity
00:18:39 2.2 Quasitriangularity
00:19:57 2.2.1 Case 1: iq/i is not a root of unity
00:23:02 2.2.2 Case 2: iq/i is a root of unity
00:23:26 2.3 Quantum groups at iq/i = 0
00:28:25 2.4 Description and classification by root-systems and Dynkin diagrams
00:29:53 3 Compact matrix quantum groups
00:30:02 4 Bicrossproduct quantum groups
00:30:12 5 See also
00:32:48 6 Notes
00:32:57 7 References
00:33:18 Description and classification by root-systems and Dynkin diagrams
00:35:44 Compact matrix quantum groups
00:37:05 f(xy) for all f ∈ C(G), and for all x, y ∈ G (where (f ⊗ g)(x, y)
00:43:47 δij for all i, j). Furthermore, a representation v, is called unitary if the matrix for v is unitary (or equivalently, if κ(vij)
00:46:06 α ⊗ α − γ ⊗ γ*, ∆(γ)
00:46:24 α*, κ(γ)
00:46:38 −μγ*, κ(α*)
00:48:53 α ⊗ α − μβ ⊗ β*, Δ(β)
00:49:11 α*, κ(β)
00:49:25 −μβ*, κ(α*)
00:49:56 1, then SUμ(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).
00:50:07 Bicrossproduct quantum groups
00:52:29 See also



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SUMMARY
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In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebras 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" first appeared in the theory of 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 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. It has become fashionable to introduce the adjective quantum in cases where quantum groups act on objects. For example, there are quantum planes and quantum Grassmannians.