On Poisson pairs associated to modified \(R\)-matrices
Let \(g\) be a finite dimensional semi-simple complex Lie algebra with adjoint group \(G\). Let \({\mathcal O} \subset g^*\) be a coadjoint orbit of \(G\) and \(r \in g \wedge g\) satisfy \[ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = z \phi,\quad z \in \mathbb{C}^*,\quad \phi \in (\wedge^ 3 g)^ G. \] The authors classify those orbits \(\mathcal O\) for which the skew biderivation of \(C^ \bullet ({\mathcal O})\) \[ \{f,g\}_ r = m(df \otimes dg (\rho \otimes \rho(r))),\quad f,g \in C^ \bullet ({\mathcal O}) \] satisfies the Jacobi identity. Here \(m\) is multiplication in \(C^ \bullet ({\mathcal O})\) and \(\rho : g \to \text{Vect} ({\mathcal O})\) is the infinitesimal left \(g\)-action on \(\mathcal O\). The compatibility of \(\{ , \}_ r\) with the Kostant-Kirillov-Souriau symplectic structure of \(\mathcal O\) is used to recover a result of \textit{S. Khoroshkin, A. Radul} and \textit{V. Rubtsov} [Commun. Math. Phys. 152, No. 2, 299-315 (1993; Zbl 0785.58024)] according to which for \(\mathcal O\) hermitian symmetric, the reduced Sklyanin bracket is compatible with the KKS structure.
scientific article; zbMATH DE number 572060