Poisson structures due to Lie algebra representations
A method of constructing homogeneous quadratic Poisson structures (HQPS) is described. One considers a real, finite-dimensional Lie algebra \(G\). Then, the notion of unitary solution \(R\) of the classical Yang-Baxter equation (CYBE) on \(G\) is introduced. If \(\Phi\) is a finite-dimensional representation of \(G\) in \(V\), then \(\Phi (R)\) is a unitary solution of CYBE on \(\text{End} (V)\). To any \(\Phi (R)\) there corresponds an HQPS on \(V^*\), denoted by \(\widetilde {\Phi (R)}\). It is shown that if \(\Phi\) and \(\Psi\) are two equivalent representations of \(G\) in \(V\) and \(W\) respectively, then their corresponding Poisson structures \(\widetilde {\Phi (R)}\) and \(\widetilde {\Psi (R)}\) are isomorphic. The Poisson structures due to equivalent representations give rise to isomorphic symplectic foliations. Some examples where the symplectic leaves are explicitly calculated are given.
scientific article; zbMATH DE number 836141