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.

中文翻译：

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与Corank 1型奇异性相关的Poisson-Lie代数和奇异辛形式

摘要

我们证明了存在一个与奇异辛结构\（\ omega \）相关的自然泊松-李代数。我们为奇异的Martinet和Roussarie类型构造Poisson-Lie代数。在当奇异辛结构由拉回从达布形式给出的特殊情况下，\（\欧米加= F ^ * \ omega_0 \） ，这泊松李代数是光滑映射的奇异性的基本辛不变\ （F \）进入辛空间\（（{{\ mathbb R} ^ {2n}，\ omega_0）\）。考虑了回调的\（A_k \）奇异的情况，以及\（\ Sigma_ {2,0} \），\（\ Sigma_ {2,2,0} ^ {\ textrm {e}的Poisson-Lie代数} \）和\（\ Sigma_ {2,2,0} ^ {\ textrm {h}} \）计算出\（2 \）-形式的稳定奇点。