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.
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The authors are grateful to the referees for helpful suggestions.
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The work was partially supported by the NCN grant no. DEC-2013/11/B/ST1/03080.
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Dedicated to Professor Armen Sergeev on his 70th birthday
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Fukuda, T., Janeczko, S. Poisson–Lie Algebras and Singular Symplectic Forms Associated to Corank 1 Type Singularities. Proc. Steklov Inst. Math. 311, 129–151 (2020). https://doi.org/10.1134/S0081543820060085
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DOI: https://doi.org/10.1134/S0081543820060085