| 1. |
- Alsaody, Seidon, 1986, et al.
(författare)
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Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem
- 2017
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 476, s. 368-394
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Tidskriftsartikel (refereegranskat)abstract
- © 2016 Elsevier Inc.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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| 2. |
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| 3. |
- Kadets, B., et al.
(författare)
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Classification of Quantum Groups and Belavin-Drinfeld Cohomologies
- 2016
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 344:1, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. This problem is reduced to the classification of all Lie bialgebra structures on g(K) , where K= C((ħ)). The associated classical double is of the form g(K) ⊗ KA, where A is one of the following: K[ ε] , where ε2= 0 , K⊕ K or K[ j] , where j2= ħ. The first case is related to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin–Drinfeld cohomology associated to any non-skewsymmetric r-matrix on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984). We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on g(K) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.
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| 4. |
- Kadets, Boris, et al.
(författare)
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Classification of quantum groups and Belavin-Drinfeld cohomologies for orthogonal and symplectic Lie algebras
- 2016
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 57:5
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we continue to study Belavin-Drinfeld cohomology introduced in Kadets et al., Commun. Math. Phys. 344(1), 1-24 (2016) and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric r-matrices on the Belavin-Drinfeld list for simple Lie algebras of type B, C, and D.
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| 5. |
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| 6. |
- Kadets, B., et al.
(författare)
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Quantum Groups: From the Kulish–Reshetikhin Discovery to Classification
- 2016
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Ingår i: Journal of Mathematical Sciences (United States). - : Springer Science and Business Media LLC. - 1072-3374 .- 1573-8795. ; 213:5, s. 743-749
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this paper is to provide an overview of results about classification of quantum groups which were obtained by the authors. Bibliography: 17 titles.
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| 7. |
- Pianzola, Arturo, et al.
(författare)
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Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations
- 2018
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Ingår i: Bulletin of Mathematical Sciences. - : World Scientific Pub Co Pte Lt. - 1664-3607 .- 1664-3615. ; 8:1, s. 1-14
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Tidskriftsartikel (refereegranskat)abstract
- 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 H1(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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| 8. |
- Stolin, Alexander, 1953, et al.
(författare)
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Classification of quantum groups and Lie bialgebra structures on sl(n, F). Relations with Brauer group
- 2016
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 293, s. 324-342
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Tidskriftsartikel (refereegranskat)abstract
- Given an arbitrary field F of characteristic 0, we study Lie bialgebra structures on sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure.5, the classical double D(sl(n, F), delta) is isomorphic to sl(n,F) circle times(F) A, where A is either F[epsilon], with epsilon(2) = 0, or F circle plus F or a quadratic field extension of F. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n,F). In the second and third cases, a Belavin-Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The Belavin Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed. For the Cremmer-Gervais r-matrix in sl(3), we also construct a natural map of sets between the total Belavin-Drinfeld twisted cohomology set and the Brauer group of the field F.
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