page_title: Quantum group
section_title: Description and classification by root-systems and Dynkin diagrams
hierarchical_section_title: Quantum group / Drinfeld–Jimbo type quantum groups / Description and classification by root-systems and Dynkin diagrams
caption_attribution_description: English: A dynkin diagram describing a pointed Hopf algebra with 4 linked borel parts of root system type each A3
context_page_description: In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups, compact matrix quantum groups, and bicrossproduct quantum groups.
  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.
context_section_description: There has been considerable progress in describing finite quotients of quantum groups such as the above Uq(g) for qⁿ = 1; one usually considers the class of pointed Hopf algebras, meaning that all subcoideals are 1-dimensional and thus there sum form a group called coradical:
  In 2002 H.-J. Schneider and N. Andruskiewitsch  finished their classification of pointed Hopf algebras with an abelian co-radical group (excluding primes 2, 3, 5, 7), especially as the above finite quotients of Uq(g) decompose into E′s (Borel part), dual F′s and K′s (Cartan algebra) just like ordinary Semisimple Lie algebras:
  Here, as in the classical theory V is a braided vector space of dimension n spanned by the E′s, and σ (a so-called cocylce twist) creates the nontrivial linking between E′s and F′s. Note that in contrast to classical theory, more than two linked components may appear. The role of the quantum Borel algebra is taken by a Nichols algebra  of the braided vectorspace. 
  A crucial ingredient was I. Heckenberger's classification of finite Nichols algebras for abelian groups in terms of generalized Dynkin diagrams. When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dankin diagram).
  Meanwhile, Schneider and Heckenberger have generally proven the existence of an arithmetic root system also in the nonabelian case, generating a PBW basis as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used on specific cases  Uq(g) and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of the Weyl group of the Lie algebra g.
caption: generalized Dynkin diagram for a pointed Hopf algebra linking four A3 copies
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