On Hopf algebras over quantum subgroups

García, Gastón Andrés; Giraldi, João Matheus Jury

doi:10.1016/j.jpaa.2018.04.018

Cited by 22 publications

(28 citation statements)

References 8 publications

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“…Proof. The proof is completely analogous to that of [GG16,Lemma 2.12]. Now we describe the module structure of P(k χ j ) for j ∈ I 0,5 by Lemmas 3.4 (3) & 3.8.…”

Section: Proofmentioning

confidence: 92%

“…This work contributes to the classification of finite-dimensional Hopf algebras over k without the dual Chevalley property, that is, the coradical is not a subalgebra. Until now, there are few classification results on such Hopf algebras with non-pointed duals, some exceptions being [GG16,HX17].…”

Section: Introductionmentioning

confidence: 99%

“…The present paper is a sequel to [AC13,GG16,HX16]. In [HX16], the authors fixed a Hopf algebra C (See Definition 2.4) and constructed some Hopf algebras of dimension 72 without the dual Chevalley property but left some questions about Nichols algebras unsolved.…”

Section: Introductionmentioning

confidence: 99%

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On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property

Xiong

2019

Communications in Algebra

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Let k be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over k whose Hopf coradical is isomorphic to the unique 12dimensional Hopf algebra C without the dual Chevalley property, such that the diagrams are strictly graded and the corresponding infinitesimal braidings are indecomposable objects in C C YD. In particular, we obtain new Nichols algebras of dimension 18 and 36 and two families of new Hopf algebras of dimension 216. , and as an algebra is generated by the elements a, b, g, x satisfying the relations in C cop , the relations in A op cop 1 and ag = ga, ax + ξ −2 xa = λ −1 θξ −2 (ba 3 − gb), bg = −gb, bx + ξ −2 xb = θξ −2 (a 4 − ga). The projective class ring and representation type of the Drinfeld double DWe study the representation type of D and the projective class ring of D, which is a subring of the Green ring. We refer to [ARS95] for the representation theory.

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“…Proof. The proof is completely analogous to that of [GG16,Lemma 2.12]. Now we describe the module structure of P(k χ j ) for j ∈ I 0,5 by Lemmas 3.4 (3) & 3.8.…”

Section: Proofmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property

Xiong

2019

Communications in Algebra

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“…The classification of the finite-dimensional Nichols algebras B(Z) with Z ∈ L L YD was addressed in [GG]. If dim B(Z) < ∞, then Z should be semisimple by [GG,Theorem 4.5] and the classification is achieved assuming that Z is simple [GG, Theorem A]. The classification might be concluded as an application of the previous result.…”

Section: Now the Assignment Degmentioning

confidence: 99%

On Nichols algebras over basic Hopf algebras

Andruskiewitsch

Angiono

2020

Math. Z.

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This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field k of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter-Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand-Kirillov dimension.Contents 1 NICOLÁS ANDRUSKIEWITSCH AND IVÁN ANGIONO 5.1. A block and a point, weak interaction 37 5.2. A block and a point, mild interaction 41 References 43Since both F and G preserve dimensions and Nichols algebras, the proof of Theorem 1.1 is reduced to the following Claim:Claim A follows from Proposition 2.10, valid for any finite-dimensional Hopf algebra H. Observe that the Claim itself does not provide directly new finite-dimensional Hopf algebras as, cf. the proof of Proposition 2.10,But the Hopf algebras of the form B(Z)#L are new, except for the small L mentioned above.Observe that the Nichols algebras B(Z) bear a Weyl groupoid since Z is semisimple by [AHS, HS2]; these Weyl groupoids were studied in [CL].Theorem 1.2. Let L be a basic Hopf algebra, G and V as above, such that G is abelian.This is a drastic simplification, since most of the times L L YD ≃ K K YD is wild. When G is abelian, Theorems 1.1 and 1.2 together with [H2] reduce the complete classification of finite-dimensional Nichols algebras in L L YD to a computational problem. Analogously to the proof of Theorem 1.1, Theorem 1.2 boils down to Claim B. Assume that G is a finite abelian group. If Z ∈ K K YD has dim B(Z) < ∞, then Z is semisimple.Claim B is proved as Theorem 3.9. We point out that Theorems 1.1 and 1.2 generalize, and were motivated by, [GG, Theorem 4.5].1.3. The main result. In Subsection 3.4 we prove that the diagram R of a finite-dimensional Hopf algebra H whose Hopf coradical is basic is necessarily a Nichols algebra, under suitable hypothesis. Together with the results in §1.2, this rounds up the following statement. Theorem 1.3. Let L be a basic finite-dimensional Hopf algebra such that G = Hom alg (L, k) is an abelian group. Let H be a Hopf algebra with H [0] ≃ L, so that gr H ≃ R#L. Then the following are equivalent: Thus the braided bosonization of a Nichols algebra does not need to be a Nichols algebra. Compare with [U3, Thm. 4.3.1]. Proof. Let J = B(M )#A(V ). The filtration of M (λ) arising from its grading induces a filtration of M , and thus a Hopf algebra filtration on J. By (3.3), the first two terms of this filtration are J (0) = k#k#kG ≃ kG, J (1) = k#k#kG ⊕ k#V #kG ⊕ λ#k#kG. (3.8) Hence J 0 = kG by [Mo, Lemma 5.3.4] and J is pointed. Let V ∈ kG kG YD be the infinitesimal braiding of J; by (3.8) there is a monomorphismby (the same proof as) Lemma 3.4, cf. Remark 3.5. Now the proof follows as the one of Lemma 3.11. Indeed R#A(V ) is pointed by [Mo, Lemma 5.3.4], with coradical kG. Let V be the infinitesimal braiding of J. ThenLet J ′ be the suba...

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“…(c) Semisimple (but not simple) Yetter-Drinfeld modules [5,[48][49][50], and references therein. (d) Yetter-Drinfeld modules over Hopf algebras that are not group algebras, see for example [4,13,14,43,54,55].…”

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confidence: 99%

On finite dimensional Nichols algebras of diagonal type

2017

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This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59-124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164: 175-188, 2006; Heckenberger and Yamane in Math Z 259:255-276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) Communicated by Efim Zelmanov.

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On Hopf algebras over quantum subgroups

Cited by 22 publications

References 8 publications

On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property

On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property

On Nichols algebras over basic Hopf algebras

On finite dimensional Nichols algebras of diagonal type

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