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

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 $\Sigma _{2,0}$, $\Sigma _{2,2,0}^\mathrm{e}$ and $\Sigma _{2,2,0}^\mathrm{h}$ stable singularities of $2$-forms are calculated.