On Projective Modules Over Finite Quantum Groups

Vay, Cristian

doi:10.1007/s00031-017-9469-y

Cited by 4 publications

(21 citation statements)

References 29 publications

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“…The small quantum groups and the Drinfeld doubles of bosonizations of Nichols algebras can be obtained via this procedure. Finally, we extend to this wider class of Hopf algebras some results of [39], regarding projective modules over Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over groups. These results rely fundamentally on properties of the Nichols algebra.…”

Section: Introductionmentioning

confidence: 85%

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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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Section: Introductionmentioning

confidence: 85%

“…[39, Lemma 1]). For all λ ∈ Λ, M(λ) is the projective cover of Inf B − H (λ) and the injective hull ofInf B − H (λ V λ) in B − G. Proof.…”

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confidence: 99%

On Hopf algebras with triangular decomposition

Vay¹

2019

Tensor Categories and Hopf Algebras

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“…We give more details in §2 and in the appendix. By [24,Theorem 6] and [26,Corollary 17], L(λ) is projective if and only if λ ∈ Λ sp := {(e, −), (σ, +), (τ, 1), (τ, 2)}. In conclusion, question (2) does not hold in this example.…”

Section: Introductionmentioning

confidence: 87%

“…In fact, the graded characters of the simple modules form a Z[t, t −1 ]-basis of the Grothendieck ring of the category of graded D-modules [26,Theorem 9]. However, two simple modules could have identical ungraded character as for instance L(e, ρ) and L(τ, 0), see Remark 2.6.…”

Section: Introductionmentioning

confidence: 99%

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On the Representation Theory of the Drinfeld Double of the Fomin-Kirillov Algebra FK3$\mathcal {FK}_{3}$

Pogorelsky

Vay

2018

Algebr Represent Theor

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Let D be the Drinfeld double of F K 3 #kS 3 . The simple D-modules were described in [24]. In the present work, we describe the indecomposable summands of the tensor products between them. We classify the extensions of the simple modules and show that D is of wild representation type. We also investigate the projective modules and their tensor products. 2000 Mathematics Subject Classification.16W30. C. V. was partially supported by CONICET, Secyt (UNC), FONCyT PICT 2016-3957, Programa de Cooperación MINCyT-FWO, MathAmSud project GR2HOPF and ESCALA Docente AUGM.The remainder simple modules generate a single block of the category of D-modules because they are composition factors of an indecomposable module, the Verma module of (σ, −) [24, Theorem 7]. In Section 3, we compute the extensions between these simple modules and show that D is of wild representation type. We draw the separated quiver of D in Figure 1.The major effort of our work is in describing the indecomposable summands of the tensor products of simple modules.Theorem 1.1. Let D be the Drinfeld double of FK 3 #kS 3 . Given λ, µ ∈ Λ, the indecomposable summands of the tensor product L(λ)⊗L(µ) are described in Propositions 4.1, 4.3, 4.7, 4.9, 4.10, 5.5 and 5.6.The outcome of the above is resumed in Table 2. We find out new indecomposable modules A, B and C which are not either simple or projective. We schematize them in Figures 2, 3 and 4, respectively. If one of the factors is projective, the tensor product is also projective and then we can use results from [26] in order to describe its direct summands. However, we do not have enough space in the table to write them except when both factors are projective. In this case, Ind(λ · µ) is the induced module D⊗ D(S3) (λ⊗µ) which is not necessary indecomposable. The cells under the diagonal are empty because D is quasitriangular and hence the tensor product is commutative. We do not include L(ε) in the table because it is the unit object.

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Highest weight theory for finite-dimensional graded algebras with triangular decomposition

Bellamy

Thiel

2018

Advances in Mathematics

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We consider self-injective finite-dimensional graded algebras admitting a triangular decomposition. In the preceding paper [7] we have shown that the graded module category of such an algebra is a highest weight category and has tilting objects in the sense of Ringel. In this paper we focus on the degree zero part of the algebra, the core of the algebra. We show that the core captures essentially all relevant information about the graded representation theory. Using tilting theory, we show that the core is cellular. We then describe a canonical construction of a highest weight cover, in the sense of Rouquier, of this cellular algebra using a finite subquotient of the highest weight category. Thus, beginning with a self-injective graded algebra admitting a triangular decomposition, we canonically construct a quasi-hereditary algebra which encodes key information, such as the graded multiplicities, of the original algebra. Our results are general and apply to a wide variety of examples, including restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras. We expect that the cell modules and quasi-hereditary algebras introduced here will provide a new way of understanding these important examples.into graded subalgebras given by the multiplication map, where we assume that − is concentrated in negative degree, in degree zero, and + in positive degree.There are a variety of examples of such algebras:(1) Restricted enveloping algebras (g );The proof of this theorem is a modification of very recent work of Andersen-Stroppel-Tubbenhauer [3], who prove this in the case of quantum groups. Unfortunately, one essential ingredient in loc. cit. is the notion of weight spaces, which is

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On Projective Modules Over Finite Quantum Groups

Cited by 4 publications

References 29 publications

On Hopf algebras with triangular decomposition

On Hopf algebras with triangular decomposition

On the Representation Theory of the Drinfeld Double of the Fomin-Kirillov Algebra FK3$\mathcal {FK}_{3}$

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

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