Verma and simple modules for quantum groups at non-abelian groups

Pogorelsky, Barbara; Vay, Cristian

doi:10.1016/j.aim.2016.06.019

Cited by 13 publications

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“…Proof. The isomorphism follows as in [33,Lemma 7]. We sketch the proof and leave the details for the reader.…”

Section: And the Identity Yxmentioning

confidence: 98%

“…We sketch the proof and leave the details for the reader. First, we have to check that (B(V )#K) * op ≃ B(V )#K * op similar to [33,Lemma 5]. Here, we consider V = V * as the Yetter-Drinfeld module over K * op with action and coaction defined by f · y, x = f, S −1 (x (−1) ) y, x (0) and y, h · x = y (−1) , h y (0) , x for all f ∈ K * op , h ∈ H, x ∈ V and y ∈ V .…”

Section: And the Identity Yxmentioning

confidence: 99%

“…In this setting, the Drinfeld double of H 0 plays the role of the Cartan subalgebra and their simple modules are the corresponding weights (unlike Lie theory, they are not necessarily one-dimensional). 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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…In this great generality, a character formula is not expected and we should restrict ourselves, for instance, to certain families of Nichols algebras. We refer the reader to [3,13,22,33] to see how the simple modules look like in some explicit examples.…”

Section: Introductionmentioning

confidence: 99%

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On Hopf algebras with triangular decomposition

Vay¹

2019

Tensor Categories and Hopf Algebras

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In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with triangular decomposition. Finally, we extend to these Hopf algebras the main results of [39] regarding projective modules over Drinfeld doubles of bosonizations of Nichols algebras and groups.2010 Mathematics Subject Classification. Primary 16T05, 20G05, 16P10. Partially supported by CONICET, Secyt (UNC), FONCyT PICT 2016-3957 and MathAm-Sud project GR2HOPF.We are interested in graded quotient Hopf algebras of U (H,R) (V ) admitting a triangular decomposition with H in the middle. The main example is the following. Let B(V ) and B(V ) be the Nichols algebras of V and V , with defining ideals J (V ) and J (V ), respectively. We defineProposition 4.2. u (H,R) (V ) is a graded Hopf algebra quotient of U (H,R) (V ) and B(V )⊗H⊗B(V ) −→ u (H,R) (V ) is a triangular decomposition. Also, the Hopf subalgebra generated by V and H, resp. V and H, is isomorphic toProof. We only have to prove the triangular decomposition. A direct computation shows that the composition of the braidings c V,V : V ⊗V → V ⊗V and c V ,V : V ⊗V → V ⊗V is the identity. Then, the Nichols algebra B(V ⊕V ) is isomorphic to B(V )⊗B(V ) as vector spaces by [16, Theorem 2.2Since the generators of J do not belong to this ideal, we have that u (H,R) (V ) ≃ T (V ⊕ V )#H J (V ), J (V ) J ≃ B(V )⊗H⊗B(V ) as vector spaces. 4.1. Examples.

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“…Proof. The isomorphism follows as in [33,Lemma 7]. We sketch the proof and leave the details for the reader.…”

Section: And the Identity Yxmentioning

confidence: 98%

Section: And the Identity Yxmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Hopf algebras with triangular decomposition

Vay¹

2019

Tensor Categories and Hopf Algebras

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On Projective Modules Over Finite Quantum Groups

Vay

2017

Transformation Groups

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Abstract. Let D be the Drinfeld double of the bosonization B(V )#kG of a finite-dimensional Nichols algebra B(V ) over a finite group G. It is known that the simple D-modules are parametrized by the simple modules over D(G), the Drinfeld double of G. This parametrization can be obtained by considering the head L(λ) of the Verma module M(λ) for every simple D(G)-module λ.In the present work, we show that the projective D-modules are filtered by Verma modules and the BGG Reciprocity [P(µ) :holds for the projective cover P(µ) of L(µ). We use graded characters to proof the BGG Reciprocity and obtain a graded version of it. As a by-product we show that a Verma module is simple if and only if it is projective. We also describe the tensor product between projective modules.

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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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Verma and simple modules for quantum groups at non-abelian groups

Cited by 13 publications

References 11 publications

On Hopf algebras with triangular decomposition

On Hopf algebras with triangular decomposition

On Projective Modules Over Finite Quantum Groups

On Nichols algebras over basic Hopf algebras

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