Pointed Hopf algebras: a guided tour to the liftings

Angiono, Iván; Iglesias, Agustín García

doi:10.15446/recolma.v53nsupl.83958

Cited by 8 publications

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References 27 publications

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“…For the last part, we notice that in general it is not known whether an arbitrary R = ⊕ n≥0 R n arising from the standard filtration is coradically graded, or whether the subalgebra R of R generated by Z = R 1 is a Nichols algebra, albeit B(Z) is a quotient of R. Assuming that R is finite-dimensional over L basic with abelian group, we show that R is a Nichols algebra in Subsection 3.4; the proof relies strongly on the description of liftings of Nichols algebras over abelian groups [AnG,AnG2].…”

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confidence: 85%

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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On Nichols algebras over basic Hopf algebras

Andruskiewitsch

Angiono

2020

Math. Z.

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“…Following Cartier (see [3]), one uses the term hyperalgebra for any connected cocommutative bialgebra. The existence of the antipode is automatic for such bialgebras (see [4, 2.2.8]), so they are in fact Hopf algebras.…”

Section: Cocommutative Hopf Algebrasmentioning

confidence: 99%

“…Proposition 2. 3 Suppose that an associative algebra A over a field F satisfies a nontrivial polynomial identity of degree d. Then for any q ∈ F, A satisfies a nontrivial identity of the form f q (x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Delta Sets and Polynomial Identities in Pointed Hopf Algebras

Bahturin

Witherspoon

2021

Algebr Represent Theor

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“…Let H be a (finite-dimensional) Hopf algebra over a field F. Suppose that H acts on a unital algebra A. This means that there is a unital left H-module structure on A written by (h, a) → h * a such that h * 1 = ε(h)1 and h * (ab) = (h (1) * a)(h (2) * b) where h ∈ H, a, b ∈ A and ∆(h) = h (1) ⊗ h (2) . If • is another action of H on A, we say that the actions are isomorphic if there is an algebra automorphism ϕ : A → A such that h • ϕ(a) = ϕ(h * a) for any h ∈ H and any a ∈ A.…”

Section: Introduction S1mentioning

confidence: 99%

“…In Section 2, we recall the classification of gradings by abelian groups on the matrix algebras. In the case where the group G of group-likes of a Hopf algebra H is semisimple, the action of G is equivalent to the grading by the dual group G. Since any finite-dimensional pointed Hopf algebra H with abelian group G of group likes is generated by the group like and skew-primitive elements (see the survey [2], to determine the action of H we need to know the action of skew-primitive elements on G-graded algebras. Some information about this is given in Proposition 3.1.…”

Section: Introduction S1mentioning

confidence: 99%