Abstract
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.
Highlights
Lie bialgebras [1,2,3,4] appeared as a tool to study integrable systems [5,6]
We prove that the orbits of Aut(g) on Yg that project onto the same orbit of the natural action of Aut(g) on Λ2g/(Λ2g)g are exactly the r-matrices that lead to equivalent coboundary Lie bialgebras on g
As we are interested in determining the strata of Eg within Yg, we look for an 1 containing some solution to the modified classical Yang–Baxter equation (mCYBE), namely 1 ∩ Yg = ∅
Summary
Lie bialgebras [1,2,3,4] appeared as a tool to study integrable systems [5,6]. A Lie bialgebra is a Lie algebra g, along with a Lie bracket on its dual space g∗, which amounts to a cocycle in a Chevalley–Eilenberg cohomology of g. In gαβγ this fγ for work, Darboux families are studied and employed to classify, up to Lie algebra automorphisms, the r-matrices and coboundary Lie bialgebras on real four-dimensional Lie algebras that are indecomposable [24], i.e., they cannot be written as a direct sum of two proper ideals, namely ideals that are different for the zero and the total Lie algebra. As a byproduct of our research, a method for the matrix representation of a class of finite-dimensional Lie algebras with a non-trivial center, which cannot be represented through the matrices of the adjoint representation, is given This is interesting for our purposes, and in many other works where such a representation is employed in practical calculations, e.g., [37].
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