Geodesic
Poisson--Lie Algebras and Singular Symplectic Forms Associated to Corank 1 Type Singularities
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 140-163

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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.
Keywords: implicit Hamiltonian system, solvability, singularities, singular symplectic structures.
Mots-clés : Poisson–Lie algebra 
@article{TM_2020_311_a7,
     author = {T. Fukuda and S. Janeczko},
     title = {Poisson--Lie {Algebras} and {Singular} {Symplectic} {Forms} {Associated} to {Corank} 1 {Type} {Singularities}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {140--163},
     publisher = {mathdoc},
     volume = {311},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2020_311_a7/}
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T. Fukuda; S. Janeczko. Poisson--Lie Algebras and Singular Symplectic Forms Associated to Corank 1 Type Singularities. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 140-163. http://geodesic.mathdoc.fr/item/TM_2020_311_a7/