Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]

Abedin，Raschid1; Maximov，Stepan2; Stolin，Alexander3; Zelmanov，Efim4

2024

10.1007/s00220-023-04911-6

Communications in Mathematical Physics   影响因子和分区

0010-3616

1432-0916

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.

SCI

英语

其他

Deutsche Forschungsgemeinschaft[SFB - TRR 358/1 2023 - 491392403] ; DFG[205607]

Physics

Physics, Mathematical

WOS:001147580100004

SPRINGER

PHYSICS

2-s2.0-85182644991

Scopus

1.Department of Mathematics,ETH Zürich,Zurich,8092,Switzerland
2.Institut für Mathematik,Universität Paderborn,Paderborn,33098,Germany
3.Department of Mathematical Sciences,Chalmers University of Technology and the University of Gothenburg,Gothenburg,412 96,Sweden
4.SICM,Southern University of Science and Technology,Shenzhen,518055,China

Abedin，Raschid,Maximov，Stepan,Stolin，Alexander,et al. Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]][J]. Communications in Mathematical Physics,2024,405(1).

Abedin，Raschid,Maximov，Stepan,Stolin，Alexander,&Zelmanov，Efim.(2024).Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]].Communications in Mathematical Physics,405(1).

Abedin，Raschid,et al."Topological Lie Bialgebras, Manin Triples and Their Classification Over g[[x]]".Communications in Mathematical Physics 405.1(2024).