Existence of triangular Lie bialgebra structures. II
The author considers the question of which finite-dimensional Lie algebras \(L\) over a field \(F\) of characteristic zero admit a triangular Lie bialgebra structure, i.e., the cobracket is a coboundary given by a solution of the classical Yang-Baxter equation. In earlier papers, he studied the case of \(F\) algebraically closed, the answer being that \(L\) should be non-abelian and not a 3-dimensional Heisenberg algebra [J. Pure Appl. Algebra 134, No. 1, 1--14 (1999; Zbl 0933.17008)] and the case of \(F\) the real numbers [Abh. Math. Sem. Univ. Hamb. 71, 297--304 (2001; Zbl 0994.17009)]. These results are extended to arbitrary fields \(F\) of characteristic zero in the paper under review. The main theorem is that \(L\) will have a triangular Lie bialgebra structure if and only if \(L\) is non-abelian and not one of three types of Lie algebras of dimension 3. The exceptions are a non-split 3-dimensional simple Lie algebra, the Heisenberg algebra over \(F\), and a solvable Lie algebra determined by a 2 by 2 matrix over \(F\) whose trace is zero and whose determinant is not the negative of a square in \(F\). However, in each of these three exceptional cases, there is a coboundary Lie bialgebra structure, i.e., one determined by a skew-symmetric element \(r\) of \(L\) tensor \(L\) which is not a solution of the classical Yang-Baxter equation, but rather the classical Yang-Baxter operator on \(r\) is an L-invariant of \(L\) tensor \(L\) tensor \(L\). Thus every finite-dimensional non-abelian Lie algebra over a field of characteristic zero admits a non-trivial coboundary Lie bialgebra structure.
scientific article; zbMATH DE number 2173217
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