| 1. |
- Abedin, Raschid, et al.
(författare)
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Classification of classical twists of the standard Lie bialgebra structure on a loop algebra
- 2021
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Ingår i: Journal of Geometry and Physics. - : Elsevier BV. - 0393-0440. ; 164
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Tidskriftsartikel (refereegranskat)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.
(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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| 3. |
- Maximov, Stepan, 1994
(författare)
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Classification of classical twists of the standard Lie bialgebra structure on a loop algebra
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This licentiate thesis is based on the work "Classification of classical twists of the standard Lie bialgebra structure on a loop algebra" by R. Abedin and the author of this thesis. 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. We study classical twists of the induced Lie bialgebra structures and obtain their full classification in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced Lie bialgebra structures are determined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of the CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by A. Belavin and V. 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 the CYBE.
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| 4. |
- Maximov, Stepan, 1994
(författare)
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Infinite-dimensional Lie bialgebras and Manin pairs
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis is devoted to the theory of infinite-dimensional Lie bialgebra structures as well as their close relatives such as r-matrices and Manin pairs. The thesis is based on three papers. Paper I. The standard structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. We obtain a full classification of the induced twisted Lie bialgebra structures in terms of Belavin-Drinfeld quadruples. First, we prove that the induced structures are pseudo quasi-triangular. Then, using the algebro-geometric theory of the classical Yang-Baxter equation (CYBE), we reduce the problem of classification to the well-known Belavin-Drinfeld list of trigonometric solutions. Paper II. We classify topological Lie bialgebra structures on the Lie algebra of Taylor series g[[x]], where g is a simple Lie algebra over an algebraically closed field F of characteristic 0. We formalize the notion of a topological Lie bialgebra and introduce topological analogues of Manin pairs, Manin triples, Drinfeld doubles and twists. By relating topological Manin pairs with trace extension of F[[x]] we obtain their complete classification. The classification of topological doubles, which was known before, becomes a special case of the classification of Manin pairs. The classification of doubles tells us that there are only three non-trivial doubles over g[[x]], namely g((x))×(g[x]/x^n g[x]), n ∈ {0, 1, 2}. We prove that topological Lie bialgebra structures on g[[x]] are in one-to-one correspondence with Lagrangian Lie subalgebras of these doubles complementary to the diagonal embedding Δ of g[[x]]. The classification of topological Lie bialgebra structures is then obtained by associating the corresponding Lagrangian subalgebras with algebro-geometric datum. When the underlying field F is the field of complex numbers, the classification becomes explicit. Paper III. In this paper we associate arbitrary subspaces of g((x))×(g[x]/x^n g[x]) complementary to Δ with so-called series of type (n, s). We prove that skew-symmetric (n, s)-type series are in bijection with Lagrangian subspaces and topological quasi-Lie bialgebra structures on g[[x]]. We classify all quasi-Lie bialgebra structures using the classification of Manin pairs from Paper II. We show that series of type (n, s), solving the generalized CYBE, correspond to Lie subalgebras.
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