Quantum Groups: From the Kulish–Reshetikhin Discovery to Classification

Kadets, Borys; Karolinsky, Eugene; Pop, Iulia; Stolin, Alexander

doi:10.1007/s10958-016-2735-z

Cited by 4 publications

(19 citation statements)

References 16 publications

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“…We now recall (with our notation) the Belavin-Drinfeld cohomology definitions and results developed in [9]. Let r BD ∈ g ⊗ K g be a Belavin-Drinfeld r -matrix.…”

Section: Untwisted Belavin-drinfeld Cohomologymentioning

confidence: 99%

“…The linearization problem is an extremely technical construction brought forward as a conjecture in the work of Drinfeld [5] (see also [3] and [4]), and proved in the seminal work of Etingof and Kazhdan (see [6,7]). An outline of this correspondence can be found in the Introductions of [9,11], wherein one can also find an explanation of why the description of which Lie bialgebras structures exists on the Lie algebra g ⊗ k k((t)), with g simple finite dimensional over an algebraically closed field k of characteristic 0, arise naturally in the classification of quantum groups. The approach to the classification of Lie bialgebra structures on g ⊗ k k((t)) developed in [9][10][11] and [14] is by the introduction of the so-called "Belavin-Drinfeld cohomologies".…”

Section: Introductionmentioning

confidence: 99%

“…An outline of this correspondence can be found in the Introductions of [9,11], wherein one can also find an explanation of why the description of which Lie bialgebras structures exists on the Lie algebra g ⊗ k k((t)), with g simple finite dimensional over an algebraically closed field k of characteristic 0, arise naturally in the classification of quantum groups. The approach to the classification of Lie bialgebra structures on g ⊗ k k((t)) developed in [9][10][11] and [14] is by the introduction of the so-called "Belavin-Drinfeld cohomologies". The calculation of these cohomologies is mostly done on a case-by-case basis in the classical types using realizations of the relevant objects as matrices.…”

Section: Introductionmentioning

confidence: 99%

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Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations

Pianzola

Stolin

2016

Bull. Math. Sci.

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We relate the Belavin-Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field K of characteristic 0 to the standard non-abelian Galois cohomology H 1 (K, H) for a suitable algebraic K-group H. The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.

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“…We now recall (with our notation) the Belavin-Drinfeld cohomology definitions and results developed in [9]. Let r BD ∈ g ⊗ K g be a Belavin-Drinfeld r -matrix.…”

Section: Untwisted Belavin-drinfeld Cohomologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations

Pianzola

Stolin

2016

Bull. Math. Sci.

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“…For a motivation of these two definitions see [13,14]. The twisted Belavin-Drinfeld cohomology provides a classification of quantum groups modulo the action of the gauge group G ad (K).…”

Section: Twisted Belavin-drinfeld Cohomologymentioning

confidence: 99%

Classification of Quantum Groups via Galois Cohomology

Karolinsky¹,

Pianzola²,

Stolin³

2019

Commun. Math. Phys.

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The first example of a quantum group was introduced by P. Kulish and N. Reshetikhin. In the paper [16], they found a new algebra which was later called Uq(sl(2)). Their example was developed independently by V. Drinfeld and M. Jimbo, which resulted in the general notion of quantum group. Later, a complimentary approach to quantum groups was developed by L. Faddeev, N. Reshetikhin, and L. Takhtajan in [11].Recently, the so-called Belavin-Drinfeld cohomologies (twisted and non-twisted) have been introduced in the literature to study and classify certain families of quantum groups and Lie bialgebras. Later, the last two authors interpreted non-twisted Belavin-Drinfeld cohomologies in terms of non-abelian Galois cohomology H 1 (F, H) for a suitable algebraic F-group H. Here F is an arbitrary field of zero characteristic. The non-twisted case is thus fully understood in terms of Galois cohomology.The twisted case has only been studied using Galois cohomology for the so-called ("standard") Drinfeld-Jimbo structure.The aim of the present paper is to extend these results to all twisted Belavin-Drinfeld cohomologies and thus, to present classification of quantum groups in terms of Galois cohomologies and the so-called orders. Low dimensional cases sl(2) and sl(3) are considered in more details using a theory of cubic rings developed by B. N. Delone and D. K. Faddeev in [5].Our results show that there exist yet unknown quantum groups for Lie algebras of the types An, D2n+1, E6.

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“…To determine the possible values of λ for which one also has a Lie bialgebra structure on g, the following result from [10] is useful. Lemma 2.5.…”

Section: 1mentioning

confidence: 99%

Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem

Alsaody

Stolin

2017

Journal of Algebra

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Abstract. We investigate Lie bialgebra structures on simple Lie algebras of non-split type A. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin-Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

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Quantum Groups: From the Kulish–Reshetikhin Discovery to Classification

Abstract: The aim of this paper is to provide an overview of the results about classification of quantum groups that were obtained in [10], [11].Mathematics Subject Classification (2010): 17B37, 17B62.

Cited by 4 publications

References 16 publications

Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations

Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations

Classification of Quantum Groups via Galois Cohomology

Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem

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