De-Equivariantization of Hopf Algebras

Angiono, Iván; Galíndo, César; Pereira, Mariana

doi:10.1007/s10468-012-9392-9

Cited by 9 publications

(15 citation statements)

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“…For each simple root α there is an automorphism B α of u q so that the B α together give an action of the braid group B(Γ) on u q [20]. 2 Now for each µ ∈ Φ + we take…”

Section: The Small Quantum Group Belavin-drinfeld Triples and Assocmentioning

confidence: 99%

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Small quantum groups associated to Belavin-Drinfeld triples

Negron

2018

Trans. Amer. Math. Soc.

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Abstract. For a simple Lie algebra g of type A, D, E we show that any Belavin-Drinfeld triple on the Dynkin diagram of g produces a collection of Drinfeld twists for Lusztig's small quantum group uq(g). These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e. new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in g associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of uq(g) which express a known algebraic group action on its category of representations, and pose a subsequent question regarding the classification of all twists.

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“…For each simple root α there is an automorphism B α of u q so that the B α together give an action of the braid group B(Γ) on u q [20]. 2 Now for each µ ∈ Φ + we take…”

Section: The Small Quantum Group Belavin-drinfeld Triples and Assocmentioning

confidence: 99%

“…According now to[3, Thm. 2.8] and[2, Prop. 4.1] we have a tensor equivalence between the de-equivariantization corep (O q (Θ)) Θ and rep(u q (g)).…”

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confidence: 99%

Small quantum groups associated to Belavin-Drinfeld triples

Negron

2018

Trans. Amer. Math. Soc.

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“…By [5] we may assume that H is coradically graded. By [4,Proposition 3.3], C is the category of comodules over a finite-dimensional coradically graded coquasi-Hopf algebra with trivializable 3-cocycle.…”

Section: 2mentioning

confidence: 99%

“…In a previous paper [4] written jointly with M. Pereira, we studied deequivariantizations of Hopf algebras by applying Tannakian techniques. We explicitly constructed a coquasi-bialgebra such that its tensor category of comodules realizes the de-equivariantization of a Hopf algebra, [4,Theorem 2.8].…”

Section: Introductionmentioning

confidence: 99%

“…Their ideas for the construction of coquasi-Hopf algebras are different to our techniques, mainly they do not use the concept of de-equivariantization. Instead, they find an specific representative for the cohomology class of a trivializable 3-cocycle and an specific trivialization, suitable for the construction of a pointed Hopf algebra of the form H := B(V )#kΓ such that the coquasi-Hopf algebra B(V )#k ω Λ is a quotient of H. These examples correspond to the general construction in [4].…”

Section: Introductionmentioning

confidence: 99%

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Pointed finite tensor categories over abelian groups

Angiono

Galíndo

2017

Int. J. Math.

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Abstract. We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of a finite dimensional pointed Hopf algebras over an abelian group.

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Autoequivalences of Tensor Categories Attached to Quantum Groups at Roots of 1

Davydov

Etingof

Nikshych

2018

Progress in Mathematics

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We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group u q (g), where q is a root of unity.To the memory of Bertram Kostant IntroductionLet k be an algebraically closed field of characteristic zero. Let G be a simple algebraic group over k, and let g = Lie(G) be its Lie algebra. Let q be a root of unity of odd order coprime to 3 if G is of type G 2 , and coprime to the determinant of the Cartan matrix of G. Let u q (g) be Lusztig's small quantum group attached to g [Lu1]. Then u q (g) is a quasitriangular Hopf algebra, so the category of its finite dimensional representationsRep u q (g) is a finite braided tensor category [EGNO]. One of the main goals of this paper is to compute the Picard group Pic(Rep u q (g)) of this category, i.e., the group of equivalence classes of invertible Rep u q (g)-module categories. Picard groups of braided tensor categories and, in particular, Brauer-Picard groups of tensor categories play a crucial role in classification of graded extensions [ENO] and also appear as symmetry groups of three-dimensional topological field theories [FS]. It is known that Pic(Rep u q (g)) is isomorphic to the group Aut br (Rep u q (g)) of braided autoequivalences of Rep u q (g) [DN, ENO]. We show under some restrictions on q that Aut br (Rep u q (g)) is isomorphic to the group Aut(g) of automorphisms of g, i.e., Aut br (Rep u q (g)) = Γ ⋉ G ad , where G ad is the adjoint group of G and Γ = Γ g is the automorphism group of the Dynkin diagram of g. Namely, we prove this when the order ℓ of q is sufficiently large and also for classicalMoreover, we show that Rep u q (g) has only two braidings (the standard one and its reverse) and deduce that any tensor autoequivalence of Rep u q (g) is necessarily braided.Thus, the group of tensor autoequivalences (also known as the group of biGalois objects) of Rep u q (g) is isomorphic to Γ ⋉ G ad . This generalizes the result of Bichon [Bi2], who proved this fact for g = sl 2 .Date: March 21, 2017. 1 2 ALEXEI DAVYDOV, PAVEL ETINGOF, AND DMITRI NIKSHYCHWe also consider the braided tensor category O q (G) − comod of finite dimensional comodules over the function algebra O q (G), which is the G-equivariantization of Rep u q (g).We show that every braided autoequivalence of O q (G) − comod comes from a Dynkin diagram automorphism if ℓ is sufficiently large, and prove a similar result in the non-braided case. This generalizes a result of Neshveyev and Tuset [NT1, NT2], who proved this when q is not a root of unity. We also show this for the classical groups SL N , Sp N , SO N if ℓ > N.As a tool, we introduce the notion of a finitely dominated tensor category. We show that the category of comodules over a finitely presented Hopf algebra is finitely dominated and prove that tensor autoequivalences of a finitely dominated category that preserve a tensor generator form an algebraic group. While this theory plays an auxiliary role in our paper, it may be of independent interest. We expect t...

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De-Equivariantization of Hopf Algebras

Cited by 9 publications

References 20 publications

Small quantum groups associated to Belavin-Drinfeld triples

Small quantum groups associated to Belavin-Drinfeld triples

Pointed finite tensor categories over abelian groups

Autoequivalences of Tensor Categories Attached to Quantum Groups at Roots of 1

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