Finite Quantum Groups and Cartan Matrices

By N. Andruskiewitsch, Hans Jürgen Schneider. 2000

257 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/03e4a483b39dbe5e7d27c3f2c86b3218a37d9945

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Citations in the corpus

The Weyl groupoid of a Nichols algebra of diagonal type (2004, methodology): “There exist classification results of Rosso [10, Theorem 21] and Andruskiewitsch and Schneider [1, Theorem 1.1] on Nichols algebras of Cartan type with finite Gel’fand–Kirillov dimension (F3) and finite dimension (F1), respectively.”
Pointed hopf algebras of dimension 32 (2000, background): “an odd prime number), solving thus by the negative the tenth Kaplansky conjecture. In [D] there is a classiﬁcation of pointed Hopf algebras of dimension pn with coradical of dimension pn−1. In [AS1], [AS2], [AS3] there are classiﬁcations of pointed Hopf algebras of dimension p3, p4 and a large numbe”
On finite dimensional Nichols algebras of diagonal type (2017, background): “(a) [19,44] If A is of finite type, then Δq+ = ΔA +.”
Finite quantum groups over abelian groups of prime exponent (2000, methodology): “This result was independently obtained by Musson [28] using different methods starting from our previous article [5].”
The Nichols algebra of a semisimple Yetter-Drinfeld module (2008, background): “[AS00] , Finite quantum groups and Cartan matrices.”
On Nichols algebras associated to simple racks (2010, background): “H. Let us consider ﬁrst the case when G is abelian. A braided vector space (V,c) is of diagonal type if V has a basis (vi)1≤i≤n such that c(vi ⊗ vj) = qijvj ⊗ vi, where the qij’s are non-zero scalars [AS1]. The class of braided vector spaces of diagonal type corresponds to the class of pointed Hopf ”
Triangular Hopf algebras with the Chevalley property (2000, methodology): “We recall that the Lifting method [AS1, AS2 ] attaches to H several invariants:”
Abelian and non-abelian second cohomologies of quantized enveloping algebras (2007, methodology): “Under the same assumption as in [13], Kassel and Schneider [18] proved the result in (1).”
Nichols algebras of diagonal type (2011, background): “der particular assumptions: ⊲ [L] for the positive part of quantized enveloping algebras of semisimple Lie algebras and small quantum groups, using the full representation theory of quantum groups; ⊲ [AS2] for braidings of Cartan type; ⊲ [A1] for braidings of standard type; ⊲ [Y] for the positive pa”
Braided symmetric and exterior algebras (2005, background): “Another, yet unrelated approach, based on Woronowicz’s quantum differential calculus associates to V an algebra B(V ) (which is sometimes referred to as ”quantum symmetric algebra” or Nichols-Woronowicz algebra, see e.g., [ 2 , 12, 18, 20]).”
Classification of arithmetic root systems (2006, background): “However note that under some hypotheses all examples are deformations of the upper triangular part of a semisimple Lie algebra [20] [5].”
Finite Weyl groupoids (2010, background): “operties of Nichols algebras of diagonal type. The Weyl groupoid provides simpliﬁcation, generalization, and uniﬁcation of related ﬁniteness results by Rosso [Ros98] and Andruskiewitsch and Schneider [AS00]. A very general deﬁnition of the Weyl groupoid of a Nichols algebra and the necessary structu”