Lifting smooth curves over invariants for representations of compact Lie groups

将微分同胚提升至向量丛

Lifting diffeomorphisms to vector bundles

本文提出了光滑流形$M$的微分同胚提升为实向量丛$p\colon V\rightarrow M$的线性自同构的准则，并通过示例探讨了度量与复向量丛的情形。

Linear natural liftings of forms to Weil bundles with Weil algebras ${\mathbb D}^r_k$

We give an explicit description and calculate the dimension of the vector space of linear natural liftings of $p$-forms on $n$-dimensional manifolds $M$ to $q$-forms on $T^{{\mathbb D}^r_k}M$, where ${\mathbb D}^r_k$ is the Weil algebra of $r$-jets at 0 of smooth functions ${\mathbb R}^k\longrightarrow{\mathbb R}$, for all non-negative integers $n$, $p$, $q$, $r$, $k$ except the case $p=n$ and $q=0$.

提升统计结构

Lifting statistical structures

该研究提出统计流形几何对象与势函数到高阶切丛的自然函子性提升方法，证明提升后仍构成伪黎曼度量的统计流形结构，并拓展至李代数胚与李群胚场景，辅以显式示例验证。

Lifting spectral triples to noncommutative principal bundles

Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.

曲线在指数型映射下的提升原理

A Lifting principle of curves under exponential-type maps

本文提出曲线在指数型映射下的提升原理，克服经典障碍，通过局部提升得到全局提升，为半黎曼几何中的基础定理提供新证明。

Rothschild-Stein 提升方法向非幂零群的一个推广

An extension to non-nilpotent groups of Rothschild-Stein lifting method

本文提出一种与有限维非幂零李代数相关的李群全局构造，避免空间倾斜问题，并通过饱和方法从提升算子的基本解获得原微分算子的基本解，推广了已有结果。

Differential Operator Endomorphisms of an Euler-Lagrange Complex

The main results of our paper deal with the lifting problem for multilinear differential operators between complexes of horizontal de Rham forms on the infinite jet bundle. We answer the question when does an n-multilinear differential operator from the space of (N,0)-forms (where N is the dimension of the base) to the space of (N-s,0)-forms allow an n-multilinear extension of degree (-s,0) defined on the whole horizontal de Rham complex. To study this problem we define a differential graded operad DEnd of multilinear differential endomorphisms, which we prove to be acyclic in positive degrees…

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

We study Lie group structures on groups of the form C^\infty(M,K)}, where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra C^\infty(M,k) for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in Ω^1(M,k) is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these …

Regular infinite dimensional Lie groups

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply con…

A Dixmier-Malliavin theorem for Lie groupoids

A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. Most of the heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from…

Lifting differentiable curves from orbit spaces

Let $ρ: G \rightarrow \operatorname{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $σ= (σ_1,\ldots,σ_n) : V \to \mathbb R^n$, where $σ_1,\ldots,σ_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} σ_i$. We prove that for each $C^{d-1,1}$-curve $c$ in $σ(V) \subseteq \mathbb R^n$ there exits a locally Lipschitz lift over $σ$, i.e., a locally Lipschitz curve $\overline c$ in $V$ so that $c = σ\circ \overline c$, and we obtain explicit bounds for the Lipschitz constant of $\o…

提升四元数环面作用的障碍

Obstructions to lifting quaternionic torus actions

该研究构建四元数环面作用提升的障碍理论，给出全局与局部四元数环面作用可提升的充要条件，阿贝尔情形退化为经典局部环面作用提升理论。

The Ring of Invariants for Smooth Completions of Kac-Moody Lie Algebras

It is proved that the ring of invariants of the standard smooth completion of a Kac-Moody Lie algebra is functionally generated by two elements: the coefficient of the center and the Killing form.

通过黎曼流形上基于测度的提升来正则化 Cryo-EM 三维图谱精修中的方向估计

Regularising orientation estimation in Cryo-EM 3D map refinement through measure-based lifting over Riemannian manifolds

本文提出椭球支撑提升（ESL）以正则化黎曼流形上的旋转估计，并将其融入 Cryo-EM 三维图谱与姿态联合重建的交替更新框架，证明适定性与误差界且实验显示旋转更接近真值。

Lie algebras of smooth sections

Lie algebras of smooth sections are Lie algebras obtained from bundles of Lie algebras, where the latter are vector bundles of which the fibers are Lie algebras. We also consider the $\operatorname{C}^k$-sections for $k \in \mathbb{N}$. This paper studies the derivations, the centroid and the isomorphisms of such Lie algebras and generalizes some facts from Pierre Lecomte's publications in 1979 and 1980 to the case where the fiber is perfect or centerfree and it gives some more explicit proofs.

Poisson 2-groups

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a "universal lifting theorem" for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.

Smooth Lie group actions are parametrized diffeological subgroups

We show that every effective smooth action of a Lie group G on a manifold M is a diffeomorphism from G onto its image in Diff(M), where the image is equipped with the subset diffeology of the functional diffeology.

Totally Geodesic Foliations and Doubly Ruled Surfaces in a Compact Lie Group

For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is "doubly ruled" (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define "doubly ruled parameterized surfaces" in any Riemannian manifold, independent of Riemannian su…

Lifting a Weak Poisson Bracket to the Algebra of Forms

We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian fu…

光滑Cartan三元组与Hausdorff étale Lie群胚上的Lie扭转

Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids

本文基于Connes重建定理思想，提出从泛函分析数据恢复Hausdorff étale群胚上扭转的Lie结构，引入Lie扭转概念，并给出使扭转成为Lie扭转的截面条件，进而应用于Cartan对与正规化子，使Renault Weyl扭转成为Lie扭转。

李群胚上主2-丛中的规范理论与平行传输研究

On gauge theory and parallel transport in principal 2-bundles over Lie groupoids

本研究引入李群胚上的主李2-群丛概念，构建其Atiyah序列并定义严格与半严格联络，揭示半严格联络的扩展对称性，建立Haefliger路径上的平行传输理论，沟通了规范理论与纤维化范畴的核心概念。

nilpotent Lie algebras的gradings与smooth complex algebraic varieties的nilpotent fundamental groups相关

Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties

本文研究了nilpotent Lie代数的gradings与smooth complex algebraic varieties的nilpotent fundamental groups之间的关系，并给出了晶格同构于基本群的必要条件。