On the classification of finite-dimensional pointed Hopf algebras

By N. Andruskiewitsch, Hans Jürgen Schneider. 2005

366 citations recorded by the source corpus.

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elementsG.A/ is abelian such that all prime divisors of the order of G.A/ are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f9f2abe74e2e3c8eef83aee3fb84ba3649b90eb7

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Citations in the corpus

Abelian and non-abelian second cohomologies of quantized enveloping algebras (2007, methodology): “Proof. (1) We see from [ 3, Theorem 2.6 ] that those xN”
On finite dimensional Nichols algebras of diagonal type (2017, methodology): “Clearly, A0 is a subalgebra iff A0 = A[0]; in this case the method outlined below was introduced in [18,20], see also [21], the extension being proposed in [11].”
On finite-dimensional Hopf algebras (2014, methodology): “This was achieved in [15] for diagonal 439 braidings of Cartan type An, with explicit formulae, and in [16] for diagonal braidings of 440 finite Cartan type, with recursive formulae.”
Hopf Algebras and Root Systems (2020, neutral): “Several years after their discovery, general classiﬁcation results for pointed Hopf algebras were obtained ([ AS02 ]; [ AS04 ], [ AA08 ], [ AS10 ] depending on [ Ros98 ], [ Kha99 ], [ Hec06 ], [ Hec08 ]).”
Root systems and Weyl groupoids for Nichols algebras (2008, result): “These results allowed one to complete the classification of a large class of finite-dimensional pointed Hopf algebras with abelian group of group-like elements [6].”