Poisson–Lie Algebras and Singular Symplectic Forms Associated to Corank 1 Type Singularities

Abstract

We show that there exists a natural Poisson–Lie algebra associated to a singular symplectic structure $$\omega$$ . We construct Poisson–Lie algebras for the Martinet and Roussarie types of singularities. In the special case when the singular symplectic structure is given by the pullback from the Darboux form, $$\omega=F^*\omega_0$$ , this Poisson–Lie algebra is a basic symplectic invariant of the singularity of the smooth mapping $$F$$ into the symplectic space $$({\mathbb R}^{2n},\omega_0)$$ . The case of $$A_k$$ singularities of pullbacks is considered, and Poisson–Lie algebras for $$\Sigma_{2,0}$$ , $$\Sigma_{2,2,0}^{\textrm{e}}$$ and $$\Sigma_{2,2,0}^{\textrm{h}}$$ stable singularities of $$2$$ -forms are calculated.