| 1. |
- Abedin, Raschid, et al.
(författare)
-
Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]
- 2024
-
Ingår i: Communications in Mathematical Physics. - 1432-0916 .- 0010-3616. ; 405:1
-
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.
|
|
| 2. |
- Abedin, R., et al.
(författare)
-
Topological Manin pairs and (n, s)-type series
- 2023
-
Ingår i: Letters in Mathematical Physics. - 0377-9017 .- 1573-0530. ; 113:3
-
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.
|
|
| 3. |
-
Algebra, Geometry, and Mathematical Physics 2010
- 2012
-
Ingår i: Journal of Physics: Conference Series. - : IOP Publishing. - 1742-6596 .- 1742-6588.
-
Samlingsverk (redaktörskap) (övrigt vetenskapligt/konstnärligt)abstract
- This proceedings volume presents results obtained by the participants of the 6th Baltic–Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25–30, 2010. The Baltic–Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic–Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic–Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers.
|
|
| 4. |
- Alsaody, Seidon, 1986, et al.
(författare)
-
Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem
- 2017
-
Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 476, s. 368-394
-
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.
|
|
| 5. |
|
|
| 6. |
|
|
| 7. |
|
|
| 8. |
- Gekhtman, M., et al.
(författare)
-
Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians
- 2012
-
Ingår i: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 100:2, s. 139-150
-
Tidskriftsartikel (refereegranskat)abstract
- The present paper is a first step toward establishing connections between solutions of the classical Yang-Baxter equations and cluster algebras. We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian G(k)(n) and show that any such bracket endows G(k)(n) with a structure of a Poisson homogeneous space with respect to the natural action of SLn equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.
|
|
| 9. |
|
|
| 10. |
|
|
