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Search: WFRF:(Abedin Raschid)

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1.
  • Abedin, Raschid, et al. (author)
  • Classification of classical twists of the standard Lie bialgebra structure on a loop algebra
  • 2021
  • In: Journal of Geometry and Physics. - : Elsevier BV. - 0393-0440. ; 164
  • Journal article (peer-reviewed)abstract
    • The standard Lie bialgebra structure on an affine Kac–Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin–Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang–Baxter equation (CYBE) with two parameters. Then, using the algebro–geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.
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2.
  • Abedin, Raschid, et al. (author)
  • Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]
  • 2024
  • In: Communications in Mathematical Physics. - 1432-0916 .- 0010-3616. ; 405:1
  • Journal article (peer-reviewed)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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  • Result 1-2 of 2
Type of publication
journal article (2)
Type of content
peer-reviewed (2)
Author/Editor
Abedin, Raschid (2)
Maximov, Stepan, 199 ... (1)
Maximov, Stepan (1)
Stolin, Alexander, 1 ... (1)
Zelmanov, Efim (1)
University
Chalmers University of Technology (2)
University of Gothenburg (1)
Language
English (2)
Research subject (UKÄ/SCB)
Natural sciences (2)

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