The Nichols algebra of a semisimple Yetter-Drinfeld module

By N. Andruskiewitsch, István Heckenberger, Hans Jürgen Schneider. American Journal of Mathematics, 2008

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We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a "reflection" defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig's automorphisms of quantized Kac-Moody algebras to the nilpotent part. As a direct application we complete the classifications of finite-dimensional pointed Hopf algebras over ${\Bbb S}_3$, and of finite-dimensional Nichols algebras over ${\Bbb S}_4$. This theory has led to surprising new results in the classification of finite-dimensional pointed Hopf algebras with a non-abelian group of group-like elements.

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

View Paper: https://www.semanticscholar.org/paper/a20b56ab5f9c707c0d949e67eb8660e39f743a77

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Citations in the corpus

Root systems and Weyl groupoids for Nichols algebras (2008, background): “The following theorem is one of the main results of [2].”
Hopf Algebras and Root Systems (2020, methodology): “This result was ﬁrst obtained in [ AHS10 ] with a diﬀerent proof.”
On finite dimensional Nichols algebras of diagonal type (2017, background): “7) So, we may suppose that the B(Vi )’s are known and try to infer the shape of B(V ) from them and the cross-braidings c|Vi⊗Vj ; this viewpoint leads to a rich combinatorial analysis [5,16,44,48,50,52].”
Finite-dimensional pointed Hopf algebras with alternating groups are trivial (2008, methodology): “roup isomorphic to A 4 ×hσoi. Then, the result follows from Lemma 2.3. In our last Proposition, we apply the technique of the octahedral subrack O introduced in [AF2, Sec. 4], and based in results of [AHS]. Proposition 3.3. Let σ∈Am be of type (1n1,2n2,4n4,σo), with n 4 >0 and σo 6= id , Othe con”
On Nichols algebras associated to simple racks (2010, background): “-grading, that we shall call the principal grading, as opposed to the natural N-grading. Indeed, if X is abelian, then GX ≃ ZcardX and the principal grading coincides with the one considered e. g. in [AHS]. 3.2. The rack cohomology group H2(X,C×). We now state some general facts about the cocycles o”