Editing Quantum group (section)

===Description and classification by root-systems and Dynkin diagrams===
There has been considerable progress in describing finite quotients of quantum groups such as the above ''U<sub>q</sub>''('''g''') for ''q<sup>n</sup>'' = 1; one usually considers the class of '''pointed''' [[Hopf algebras]], meaning that all simple left or right comodules are 1-dimensional and thus the sum of all its simple subcoalgebras forms a group algebra called the '''coradical''':

* In 2002 H.-J. Schneider and N. Andruskiewitsch <ref>Andruskiewitsch, Schneider: Pointed Hopf algebras, New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.</ref> 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 ''U<sub>q</sub>''('''g''') decompose into ''E''′s (Borel part), dual ''F''′s and ''K''′s (Cartan algebra) just like ordinary [[Semisimple Lie algebra]]s:
::<math>\left(\mathfrak{B}(V)\otimes k[\mathbf{Z}^n]\otimes\mathfrak{B}(V^*)\right)^\sigma</math>
: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 cocycle 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]] <math>\mathfrak{B}(V)</math> of the braided vectorspace. [[File:Dynkin4A3lift.png|thumb|generalized Dynkin diagram for a pointed Hopf algebra linking four A3 copies]]

* A crucial ingredient was I. Heckenberger's [[Nichols algebra|classification of finite Nichols algebras]] for abelian groups in terms of generalized [[Dynkin diagram]]s.<ref>Heckenberger: Nichols algebras of diagonal type and arithmetic root systems, Habilitation thesis 2005.</ref> When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dynkin diagram).
[[File:Dynkin Diagram Triangle.jpg|thumb|A rank 3 Dynkin diagram associated to a finite-dimensional Nichols algebra]]

* Meanwhile, Schneider and Heckenberger<ref>Heckenberger, Schneider: Root system and Weyl gruppoid for Nichols algebras, 2008.</ref> have generally proven the existence of an '''arithmetic''' [[root system]] also in the nonabelian case, generating a [[Poincaré–Birkhoff–Witt theorem|PBW basis]] as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used<ref>Heckenberger, Schneider: Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl grupoid, 2009.</ref> on specific cases  ''U<sub>q</sub>''('''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'''.