Classification of three-dimensional Lie bialgebras (Q2738090)

A classification of the complex 3-dimensional Lie bialgebras was obtained by \textit{J. M. Figueroa-O'Farill} [Commun. Math. Phys. 177, 129-156 (1996; Zbl 0864.17007))]. That work was related to conformal field theory and the \(N=2\) Sugawara construction over solvable Lie algebras. The author of the paper under review says that the calculations involved seemed to be heavy and rather intricate, and that his point of view is closer to the basic definition of Lie bialgebra. His classification uses extensively the idea of twisting, i.e., adding an appropriate coboundary to a Lie cobracket. His methods lead to a classification of both the complex and real 3-dimensional Lie bialgebras. He also gives a description of the corresponding Drinfeld doubles. The classification was known when the 3-dimensional Lie algebra \(L\) is simple. Otherwise, the author deals separately with the two cases where \(L\) is solvable but non-nilpotent, and where \(L\) is nilpotent, with most of the work dealing with the former case.