Left invariant Poisson structures on classical non-compact simple Lie groups
Let \(\{\;,\;\}\) be a Poisson structure on a differentiable manifold \(M\) and let \(\pi \in{\Gamma}({\wedge}^2TM)\) be the contravariant 2-tensor field on \(M\) defined by \(\langle \pi,df{\wedge}dg\rangle=\{f,g\}\), \(f,g{\in}C^{\infty}(M)\), where \(\langle\;,\;\rangle\) is the natural pairing [see \textit{I. Vaisman}, Lectures on the geometry of Poisson manifolds, Prog. Math. 118 ( Basel, 1994; Zbl 0810.53019)]. In the case when \(M\) is a Lie group \(G\), a Poisson structure on \(G\) is called left invariant if \(\pi\) is a left invariant 2-tensor field, i.e., \(\pi\) belongs to the space \({\wedge}^2g\), where \(g\) is the Lie algebra of \(G\), consisting of left invariant vector fields. The main purpose of the present paper is to give new examples of high rank left invariant Poisson structures for all non-compact classical real simple Lie groups. The result is equivalent to giving constant solutions of the classical Yang-Baxter equation \([r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13}, r^{23}]=0\), where \(r\) takes values in the space \({\wedge}^2g\) for these Lie groups.
scientific article; zbMATH DE number 1445780
scientific article