乘法有向图品种上的兼容泊松结构
Compatible Poisson structures on multiplicative quiver varieties
AI总结：
本文研究了乘法有向图品种上的兼容泊松结构，通过构造与准哈密顿泊松结构的约简，探讨了其与辛结构和准哈密顿结构的关系，并应用于旋转向量Ruijsenaars-Schneider相空间。
摘要：
任何乘法有向图品种都配备了通过Van den Bergh构造的泊松结构，该结构通过从哈密顿准泊松结构的约简得到。光滑区域携带由Yamakawa通过准哈密顿约简定义的相应辛形式。本文将泊松结构作为乘法有向图品种上兼容泊松结构的圆柱的一部分。该圆柱由从具有维度ℓ(ℓ-1)/2的准哈密顿准泊松结构圆柱的约简定义，其中ℓ是有向图底层的箭头数量。对于圆柱中的每个元素，我们展示了相应的兼容辛或准哈密顿结构。我们还讨论了特征品种和有向图品种的类似构造。该形式化方法应用于旋转向量Ruijsenaars-Schneider相空间，以解释两种近期文献中出现的兼容泊松结构的兼容性。
英文摘要：
Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa through quasi-Hamiltonian reduction. In this note, we include the Poisson structure as part of a pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of Hamiltonian quasi-Poisson structures which has dimension $\ell(\ell-1)/2$, where $\ell$ is the number of arrows in the underlying quiver. For each element of the pencil, we exhibit the corresponding compatible symplectic or quasi-Hamiltonian structure. We comment on analogous constructions for character varieties and quiver varieties. This formalism is applied to the spin Ruijsenaars-Schneider phase space in order to explain the compatibility of two Poisson structures that have recently appeared in the literature.
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