AI 中文总结
该注记源于鲁汶大学迷你课程，证明李群胚的乘法权重、李代数胚的李滤过、NQ流形的dg权重三者等价，其权重形变空间可用于次椭圆算子研究。
AI 中文摘要
本注记源于作者2026年6月在鲁汶大学（KU Leuven）讲授的迷你课程。我们回顾权重（或称拟齐次结构）、权重法丛及权重形变空间的定义，强调采用理想层与代数层特征谱的代数方法及函子性质。我们回顾李群胚$G\rightrightarrows M$上乘法权重的定义，并给出自包含证明，证明三者等价：(i) $G$沿$M$的乘法权重；(ii) $G$的李代数胚$A$的李滤过；(iii) NQ流形$A[1]$沿$M$的dg权重。对应的权重形变空间是切/绝热群胚的权重变体，用于次椭圆算子的研究。
英文摘要
This note originated with a mini-course taught at KU Leuven by the author in June 2026. We review the definition of weightings (or quasi-homogeneous structures), the weighted normal bundle, and the weighted deformation space. We emphasize the algebraic approach and functorial properties, using sheaves of ideals and the character spectrum for sheaves of algebras. We review the definition of a multiplicative weighting on a Lie groupoid $G\rightrightarrows M$, and give a self-contained proof of the three-way equivalence between (i) multiplicative weightings of $G$ along $M$, (ii) Lie filtrations of the Lie algebroid $A$ of $G$, (iii) dg weightings of the NQ manifold $A[1]$ along $M$. The corresponding weighted deformation space is a weighted variant of the tangent/adiabatic groupoid and is used in the study of hypoelliptic operators.
Comments28 pages
