Poisson structures transverse to coadjoint orbits.
The authors prove a conjecture by \textit{P. Damianou} [ibid. 120, No. 2, 195--214 (1996; Zbl 0853.58048)], that the transverse Poisson structure to a coadjoint orbit of a semisimple, real, Lie algebra is polynomial. The precise statement is as follows. Let \(\mathfrak{g}\) be a semisimple, real, Lie algebra. Let \(\langle\cdot,\cdot\rangle\) denote the Killing form, and fix \(\nu\in\mathfrak{g}^*\). Also, let \(\mathfrak{g}_\nu\) be the isotropy Lie algebra of \(\nu\) and denote by \(\mathfrak{n}\) its orthogonal complement, so that \( \mathfrak{g}=\mathfrak{g}_\nu\oplus \mathfrak{n}.\) This gives rise to a dual decomposition \( \mathfrak{g}^*=\mathfrak{g}_\nu^0\oplus \mathfrak{n}^0.\) The affine space \(N=\nu+\mathfrak{n}^0\) is transverse at \(\nu\) to the coadjoint orbit through \(\nu\), hence, carries a Poisson structure in a neighborhood of \(\nu\). The authors show that this Poisson structure is polynomial.
scientific article; zbMATH DE number 1886864