标题： 与奇异泊松代数相关的辛形式 显示英文标题

标题： The symplectic form associated to a singular Poisson algebra

摘要： 给定一个奇异的仿射泊松代数，可以问是否存在相关的辛形式。在光滑情况下，答案是明显的：为了存在辛形式，泊松张量必须是可逆的。然而在奇异情况下，导子不形成一个投射模，非退化条件更为微妙。对于一个辛奇点，可以天真地问是否存在一个辛形式的类似物。我们研究了一个辛奇点的例子，即双重锥，并证明在这里确实存在这样的辛形式。我们使用了李-里纳尔特代数的朴素德拉姆复形。我们对双重锥的分析使用了格罗布纳基计算。我们还给出了一个辛形式的替代构造，该构造可以推广到余切提升表示的有限群的范畴商。我们使用相同的公式在简单锥上构造一个辛形式，将其视为泊松微分空间，并将构造推广到线性辛轨道空间。我们提供了有用的辅助结果，使得能够显式确定仿射簇的导子模的生成元。后者可以被理解为微分空间。 显示英文摘要

摘要： Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the singular case, however, derivations do not form a projective module and the nondegeneracy condition is more subtle. For a symplectic singularity one may naively ask if there is indeed an analogue of a symplectic form. We examine an example of a symplectic singularity, namely the double cone, and show that here such a symplectic form exists. We use the naive de Rham complex of a Lie-Rinehart algebra. Our analysis of the double cone uses Gr\"obner bases calculations. We also give an alternative construction of the symplectic form that generalizes to categorical quotients of cotangent lifted representations of finite groups. We use the same formulas to construct a symplectic form on the simple cone, seen as a Poisson differential space and generalize the construction to linear symplectic orbifolds. We present useful auxiliary results that enable to explicitly determine generators for the module of derivations an affine variety. The latter may be understood as a differential space.