Classification of quantum groups and Belavin–Drinfeld cohomologies for orthogonal and symplectic Lie algebras

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

doi:10.1063/1.4950895

Cited by 4 publications

(19 citation statements)

References 9 publications

(7 reference statements)

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“…Consequently, we get a system of equations which might lead to non-connectedness of C(G, r BD ) as it happened for G = SO 2n , see [15]. Remark 6.6.…”

Section: Classification Of Lie Bialgebrasmentioning

confidence: 98%

“…For the first two cases (see again [14] for details and further references), the classification is given in terms of what the authors call nontwisted and twisted Belavin-Drinfeld cohomologies, and the corresponding Lie bialgebra structures are called of non-twisted and of twisted type respectively. It was also notices in [15] that certain non-twisted Belavin-Drinfeld cocycles are Galois cocycles.…”

Section: Introductionmentioning

confidence: 94%

“…It is not so easy to describe explicitly all discrete parameters in the case g = A n that satisfy the conclusions of Proposition 4.10. On the other hand, all possible discrete parameters for D 2n+1 were found in [15]. There, it was also noticed that if the discrete parameter satisfies the conclusions of Proposition 4.10, then the set of the corresponding continuous parameters is non-empty.…”

Section: Twisted Belavin-drinfeld Cohomologymentioning

confidence: 99%

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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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“…Consequently, we get a system of equations which might lead to non-connectedness of C(G, r BD ) as it happened for G = SO 2n , see [15]. Remark 6.6.…”

Section: Classification Of Lie Bialgebrasmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 94%

Section: Twisted Belavin-drinfeld Cohomologymentioning

confidence: 99%

See 1 more Smart Citation

Classification of Quantum Groups via Galois Cohomology

Karolinsky¹,

Pianzola²,

Stolin³

2019

Commun. Math. Phys.

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“…In the same way one can use Theorem 1.3 as an approach to the classification of quantum groups over semisimple Lie algebras. This was done in the works [10], [11]. The rest of the paper is dedicated to the exposition of the main results of that works.…”

Section: Conjecture 12 (Drinfeld's Quantization Conjecture)mentioning

confidence: 99%

“…

AbstractThe 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.

…”

mentioning

confidence: 99%

Quantum Groups: From the Kulish–Reshetikhin Discovery to Classification

et al. 2016

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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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Classification of quantum groups and Belavin–Drinfeld cohomologies for orthogonal and symplectic Lie algebras

Cited by 4 publications

References 9 publications

Classification of Quantum Groups via Galois Cohomology

Classification of Quantum Groups via Galois Cohomology

Quantum Groups: From the Kulish–Reshetikhin Discovery to Classification

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

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