Small quantum groups associated to Belavin-Drinfeld triples

Negron, Cris

doi:10.1090/tran/7438

Cited by 6 publications

(9 citation statements)

References 27 publications

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“…The primary result of this section is Theorem 7.5 below. Theorem 7.5, along with some information from [26], will imply Theorem A from the introduction. Recall our notation u q = u q (b), U q = U q (b).…”

Section: 2mentioning

confidence: 95%

“…(See Sections 3 and 8.1.) For the small quantum Borel, it is known that the unipotent algebraic group U, corresponding to the nilpotent subalgebra n = [b, b] ⊂ b, acts on the collection of twists for u q (b) by way of twisted automorphisms [6,26]. Basic considerations also establish an embedding Alt(G ∨ ) → Tw(u q (b)), where Alt(G ∨ ) denotes the set of alternating bilinear forms on the character group G ∨ of the Cartan subgroup G = G(u q (b)).…”

Section: Introductionmentioning

confidence: 99%

“…Recall that u q (g) is generated by the positive and negative Borel subalgebras u q (b ± ) ⊂ u q (g). We ask in [10,26] if there is an equality Tw(u q (g)) = G · BD(u q (g)), (1) where BD(u q (g)) denotes the collection of Belavin-Drinfeld twists for u q (g), and G is the simply-connected semisimple algebraic group with Lie algebra g. As with U above, G acts by twisted automorphisms on Tw(u q (g)).…”

Section: Introductionmentioning

confidence: 99%

“…We use the prescribed interaction between the positive and negative Borels, along with the R-matrix for u q (g), to produce an especially non-trivial twist for u q (g). For example, one can drastically change the group of grouplikes after applying such a twist to u q (g) (see [26]), and in an extreme dynamical case one can make u q (g) self-dual after twisting (see [13]).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 7 we prove a primordial version of Theorems A and B. In Section 8 we recall some information from [6,26], and prove Theorems A and B. In Section 9 we provide proofs of some technical results from Section 6.…”

Section: Introductionmentioning

confidence: 99%

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Twists of quantum Borel algebras

Negron

2018

Journal of Algebra

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We classify Drinfeld twists for the quantum Borel subalgebra uq(b) in the Frobenius-Lusztig kernel uq(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for uq(b) generate all twists for uq(b), under a certain algebraic group action. This implies a simple classification of Hopf algebras whose categories of representations are tensor equivalent to that of uq(b). We also show that cocycle twists for the corresponding De Concini-Kac algebra are in bijection with alternating forms on the aforementioned character group.

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Section: 2mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Twists of quantum Borel algebras

Negron

2018

Journal of Algebra

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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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Renormalized Hennings Invariants and 2 + 1-TQFTs

Renzi

Geer

Patureau-Mirand

2018

Commun. Math. Phys.

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We construct non-semisimple 2 + 1-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum sl 2 to the setting of finite-dimensional nondegenerate unimodular ribbon Hopf algebras. The tools used for this construction are a Hennings-augmented Reshetikhin-Turaev functor and modified traces. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a 2+1-TQFT on a not completely rigid monoidal subcategory of cobordisms.

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Small quantum groups associated to Belavin-Drinfeld triples

Cited by 6 publications

References 27 publications

Twists of quantum Borel algebras

Twists of quantum Borel algebras

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

Renormalized Hennings Invariants and 2 + 1-TQFTs

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