On Hopf algebras with triangular
                    decomposition

Vay, Cristian

doi:10.1090/conm/728/14662

Cited by 4 publications

(7 citation statements)

References 23 publications

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“…Eventually, we will be more explicit in the successive subsections according to our convenience. For more details we refer the reader to [6,9,27].…”

Section: The General Frameworkmentioning

confidence: 99%

Simple modules of small quantum groups at dihedral groups

García,

Vay

2024

Doc. Math.

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Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups D 4t with t 3. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.

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“…Eventually, we will be more explicit in the successive subsections according to our convenience. For more details we refer the reader to [6,9,27].…”

Section: The General Frameworkmentioning

confidence: 99%

Simple modules of small quantum groups at dihedral groups

García,

Vay

2024

Doc. Math.

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“…This is the adjoint algebra in the sense of [Sh19] in the relative version [LW2,Mo22]. For highest-weight theory in the context of diagonal Nichols algberas see [Vay19].…”

Section: Splitting Statementsmentioning

confidence: 99%

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

Creutzig

Ridout

Rupert

2023

Commun. Math. Phys.

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Let U be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize U under the assumption that there exists a commutative algebra A in U with certain properties: Let C be the category of local A-modules in U and B the category of A-modules in U, which are in our set-up usually much simpler categories than U. Then we can characterize U as a relative Drinfeld center Z C (B) and B as representations of a certain Hopf algebra inside C.In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that U is abelian equivalent to the category of modules of some quantum group U q or some generalization thereof, and if C is braided equivalent to a category of graded vector spaces, and if A has a certain form, then we already obtain a braided tensor equivalence between U and Rep(U q ).A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra A and the corresponding category C are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra M(p) and the unrolled small quantum groups of sl 2 at 2p-th root of unity. Another new example is the Kazhdan-Lusztig correspondence between the affine vertex algebra of gl 1|1 and an unrolled quantum group of gl 1|1 . * this should be q = exp πi r ∨ (k+h ∨ ) with h ∨ the dual Coxeter number of g and r ∨ the lacing number of the even subalgebra.

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“…We will consider the graded Dmodules as graded modules over D ≤0 , D ≥0 and D(S 3 ) by restricting the action. In particular, the following is a direct consequence of (g) above and it is a particular case of [27,Proposition 5.2].…”

Section: Gradedmentioning

confidence: 99%

“…We summarize all the information needed for our work. We follow the notation and conventions of [24,26,27]. Most of the properties which we will list hold for any finite dimensional Nichols algebra over a finite dimensional semisimple Hopf algebra.…”

Section: Preliminariesmentioning

confidence: 99%

“…Indeed, the simple modules over a semisimple Lie algebra are parametrized by the weights of the Cartan subalgebra, while the tensor products of simple modules are described by the Clebsch-Gordon formula. Moreover, (1) holds for Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over finite-dimensional Hopf algebras, see for instance [1,5,17,24,27]. Notice that a Taft algebra can be presented as a bosonization of the quantum line k x | x n = 0 over the cyclic group of order n, the first example of a finite-dimensional Nichols algebra.…”

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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On Hopf algebras with triangular decomposition

Cited by 4 publications

References 23 publications

Simple modules of small quantum groups at dihedral groups

Simple modules of small quantum groups at dihedral groups

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

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

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