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- Abedin, Raschid, et al.
(författare)
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Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]
- 2024
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Ingår i: Communications in Mathematical Physics. - 1432-0916 .- 0010-3616. ; 405:1
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Tidskriftsartikel (refereegranskat)abstract
- The main result of the paper is classification of topological Lie bialgebra structures on the Lie algebra g[[x]] , where g is a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0. We introduce the notion of a topological Manin pair (L,g[[x]]) and present their classification by relating them to trace extensions of F[[x]] . Then we recall the classification of topological doubles of Lie bialgebra structures on g[[x]] and view it as a special case of the classification of Manin pairs. The classification of topological doubles states that up to an appropriate equivalence there are only three non-trivial doubles. It is proven that topological Lie bialgebra structures on g[[x]] are in bijection with certain Lagrangian Lie subalgebras of the corresponding doubles. We then attach algebro-geometric data to such Lagrangian subalgebras and, in this way, obtain a classification of all topological Lie bialgebra structures with non-trivial doubles. For F= C the classification becomes explicit. Furthermore, this result enables us to classify formal solutions of the classical Yang–Baxter equation.
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| 2. |
- Abedin, R., et al.
(författare)
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Topological Manin pairs and (n, s)-type series
- 2023
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Ingår i: Letters in Mathematical Physics. - 0377-9017 .- 1573-0530. ; 113:3
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Tidskriftsartikel (refereegranskat)abstract
- Lie subalgebras of L = g((x)) x g[x]/x(n)g[x], complementary to the diagonal embedding Delta of g[x] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series g[x]. In this work we consider arbitrary subspaces of L complementary to Delta and associate them with so-called series of type (n, s). We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s)-type series and topological quasi-Lie bialgebra structures on g[x]. Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s), solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.
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| 3. |
- Karolinsky, Eugene, 1972, et al.
(författare)
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Classification of Quantum Groups via Galois Cohomology
- 2020
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 377:2, s. 1099-1129
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Tidskriftsartikel (refereegranskat)abstract
- The first example of a quantum group was introduced by P. Kulish and N. Reshetikhin. In the paper Kulish et al. (J Soviet Math 23:2435–2441, 1983), 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 Faddeev et al. (Leningr Math J 1:193–225, 1990). Recently, the so-called Belavin–Drinfeld cohomology (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 cohomology in terms of non-abelian Galois cohomology H1(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 cohomology and thus, to present classification of quantum groups in terms of Galois cohomology 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 Delone and Faddeev (The theory of irrationalities of the third degree. Translations of mathematical monographs, vol 10. AMS, Providence, 1964). Our results show that there exist yet unknown quantum groups for Lie algebras of the types An, D2n+1, E6, not mentioned in Etingof et al. (J Am Math Soc 13:595–609, 2000).
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