darboux polynomial
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Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 465
Author(s):  
Javier de Lucas ◽  
Daniel Wysocki

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.


Author(s):  
Yohann De Castro ◽  
Fabrice Gamboa ◽  
Didier Henrion ◽  
Jean Bernard Lasserre

2020 ◽  
Vol 118 ◽  
pp. 103284
Author(s):  
M. Manoranjani ◽  
R. Mohanasubha ◽  
V.K. Chandrasekar ◽  
M. Senthilvelan

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