Pointed Hopf algebras

By N. Andruskiewitsch, Hans Jürgen Schneider. 2001

383 citations recorded by the source corpus.

This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras $A$ by first determining the graded Hopf algebra $\gr A$ associated to the coradical filtration of $A$. The $A_{0}$-coinvariants elements form a braided Hopf algebra $R$ in the category of Yetter-Drinfeld modules over the coradical $A_{0} = \ku \Gamma$, $\Gamma$ the group of group-like elements of $A$, and $\gr A \simeq R # A_{0}$. We call the braiding of the primitive elements of $R$ the infinitesimal braiding of $A$. If this braiding is of Cartan type \cite{AS2}, then it is often possible to determine $R$, to show that $R$ is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that $\gr A \simeq R # \ku \Gamma$. In the last Chapter, as a concrete illustration of the method, we describe explicitly all finite-dimensional pointed Hopf algebras $A$ with abelian group of group-likes $G(A)$ and infinitesimal braiding of type $A_{n}$ (up to some exceptional cases). In other words, we compute all the liftings of type $A_n$; this result is our main new contribution in this paper.

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Citations in the corpus

Finite-dimensional pointed Hopf algebras with alternating groups are trivial (2008, methodology): “Method [ 9 ]; we sketch a proof for the sake of the generic reader.”
Nichols–Woronowicz algebra model for Schubert calculus on Coxeter groups (2004, result): “The Nichols algebras in the Yetter-Drinfeld category over a group have been an object of extensive study, in particular because they are linked with pointed Hopf algebras — see for example the survey [ AS2 ] of Andruskiewitsch and Schneider.”
From racks to pointed Hopf algebras (2002, background): “braided vector space with V = Cx1 ⊕ ···⊕ Cxθ and c(xi⊗ xj) = qijxj⊗xi, 1 ≤ i,j≤ θ. (1) Assume that qij = qfor all i,j. Thus • (Nichols) If q= −1, then B(V) ≃ Λ(V), hence dimB(V) = 2θ. • (Lusztig; see [AS3]) If qis a primitive third root of 1, then dimB(V) = 27 when θ= 2; and dimB(V) = ∞ when θ>2.”
Nichols algebras of group type with many quadratic relations (2010, methodology): “We refer to [4] for an introduction to Yetter-Drinfeld modules and Nichols algebras.”
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).”
Finite dimensional quasi-Hopf algebras with radical of codimension 2 (2003, unknown): “It is our expectation that this theory will be a nontrivial generalization of the deep and beautiful theory of finite dimensional basic (or, equivalently, pointed) Hopf algebras (see [AS] and references therein).”
On finite dimensional Nichols algebras of diagonal type (2017, methodology): “Nichols algebras play a fundamental role in the classification of pointed Hopf algebras, see [20] and Sect.”
On the structure of (co-Frobenius) Hopf algebras (2010, background): “Here # stands for the Radford biproduct or bosonization, see for example [AS2].”
The Weyl groupoid of a Nichols algebra of diagonal type (2004, unknown): “As in [2] define ad σxi(ρ) :=xiρ− (gi.”
Quantum lines over non-cocommutative cosemisimple Hopf algebras (2004, methodology): “Motivated by the work of Nichols, Andruskiewitsch and Schneider started their vigorous joint work on the classification of pointed Hopf algebras; see [2–4,6] (1998– 2002).”
Finite quantum groups over abelian groups of prime exponent (2000, methodology): “For liftings of gr A when R is a Nichols algebra of Cartan type, the quantum Serre relations of connected vertices in general still hold as we show in Theorem 6.8 below; however the powers of the root vectors are not necessarily 0. We should have aNα α = uα ∈ kG(A); the determination of uα when α is”