On Nichols algebras over basic Hopf algebras

Abstract

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...

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

“…Remark 4.21. The authors in [AA18] gave a characterization of finite-dimensional Nichols algebras over basic Hopf algebras. In particular, as stated in [AA18,Example 2.14], the Nichols algebras in Theorem A can be recovered (up to isomorphism) in a similar way.…”

On Hopf algebras with triangular decomposition

“…T (V ) or B(V ). The bosonization B#H is the Hopf algebra with underlying vector space B⊗H, and multiplication and comultiplication given by (2) for all x,x ∈ B and h,h ∈ H; with ∆ B (x) = x (1) ⊗x (2) .…”

“…where r denotes another copy of R. The element R is called the R-matrix of H. It turns out that R is invertible and R −1 = S(R (1) )⊗R (2)…”

“…Then R (l) and R (r) are finite-dimensional Hopf subalgebras of H and (R (l) ) * cop −→ R (r) , p → p(R (1) )R (2) , is an isomorphism of Hopf algebras [35, Proposition 2]. Let ψ : R (r) −→ (R (l) ) * cop be its inverse map.…”

“…Different proofs were given for H being the bosonization of a finite-dimensional Nichols algebra over a finite abelian group by Heckenberger-Yamane [18] and by Pogorelsky and the author [33] for general finite groups. Andruskiewitsch-Angiono [2] recently extended the proof of [33] to bosonizations over any finite-dimensional Hopf algebra. In this great generality, a character formula is not expected and we should restrict ourselves, for instance, to certain families of Nichols algebras.…”