Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

Andruskiewitsch, Nicolás; Carnovale, Giovanna; García, Gastón Andrés

doi:10.1007/s00229-020-01248-5

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…We show that this map is injective. Since dim B(𝑀) = 2 6 , it is enough to verify that the set (5.5) linearly generates 𝑅. Let 𝐽 denote the subspace spanned by (5.5) in 𝑅.…”

Section: Type 𝛼mentioning

confidence: 99%

“…The study now naturally branches into two directions: Nichols algebras of rank 1, meaning of irreducible Yetter–Drinfeld modules, and Nichols algebras of rank

&gt;1

composed of the former via root system theory. In rank 1, more finite‐dimensional examples of Nichols algebras were discovered in [7, 29, 33]; later, the research concentrated on successfully ruling out finite‐dimensional Nichols algebras over simple groups, see [6] and the references there.…”

Section: Introductionmentioning

confidence: 99%

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Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

Angiono,

Lentner,

Sanmarco

2023

Proceedings of London Math Soc

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We construct finite‐dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite‐dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch–Schneider. Our starting point is the classification of finite‐dimensional Nichols algebras over nonabelian groups by Heckenberger–Vendramin, which consist of low‐rank exceptions and large‐rank families. We prove that the large‐rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie‐theoretic descriptions of the large‐rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.

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“…We show that this map is injective. Since dim B(𝑀) = 2 6 , it is enough to verify that the set (5.5) linearly generates 𝑅. Let 𝐽 denote the subspace spanned by (5.5) in 𝑅.…”

Section: Type 𝛼mentioning

confidence: 99%

“…The study now naturally branches into two directions: Nichols algebras of rank 1, meaning of irreducible Yetter–Drinfeld modules, and Nichols algebras of rank

&gt;1

Section: Introductionmentioning

confidence: 99%

Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

Angiono,

Lentner,

Sanmarco

2023

Proceedings of London Math Soc

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“…Let G be a group, and C the field of all complex numbers. One problem is to find all the Nichols algebras B(V ) with finite dimension for any V ∈ G G YD, the Yetter-Drinfeld modules over the group algebra G. The cases when G is a finite simple group were studied in [4,8,9,10,11,12]. For the second problem, great progress was acheived when G is an abelian group, see [5,6].…”

Section: Introductionmentioning

confidence: 99%

Finite GK-Dimensional Nichols Algebras Over the Infinite Dihedral Group

Zhang

2023

Algebr Represent Theor

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We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over Q8,the quaternion group . We find all the irreducible Yetter-Drinfeld modules V over Q8, and determine which Nichols algebras B(V ) of V are finite GK-dimensional, and which Nichols algebras B(V ) of V are finite dimensional. Contents 1. Introduction 1 2. Preliminaries 2 2.1. Notations 2 2.2. Yetter-Drinfeld modules and Nichols algebras 2 2.3. The quaternion group Q 8 3 2.4. Gelfand-Kirillov dimension 4 3. The Nichols algebras B(O 1 , ρ) 4 4. The Nichols algebras B(O x , φ) 6 5. The Nichols algebras B(O x 2 , ρ) 8 6. The Nichols algebras B(O y , φ) 10 7. The Nichols algebras B(O xy , φ) 12 8. Acknowledgments 14 References 14

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