Homogeneity rank of real representations of compact Lie groups

Toward a quasi-Möbius characterization of Invertible Homogeneous Metric Spaces

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Möbius self-homeomorphisms, connecting such homogeneity with the combinatio…

Supersymmetry and homogeneity of M-theory backgrounds

We describe the construction of a Lie superalgebra associated to an arbitrary supersymmetric M-theory background, and discuss some examples. We prove that for backgrounds with more than 24 supercharges, the bosonic subalgebra acts locally transitively. In particular, we prove that backgrounds with more than 24 supersymmetries are necessarily (locally) homogeneous.

The homogeneity conjecture for supergravity backgrounds

These notes record three lectures given at the workshop "Higher symmetries in Physics", held at the Universidad Complutense de Madrid in November 2008. In them we explain how to construct a Lie (super)algebra associated to a spin manifold, perhaps with extra geometric data, and a notion of privileged spinors. The typical examples are supersymmetric supergravity backgrounds; although there are more classical instances of this construction. We focus on two results: the geometric constructions of compact real forms of the simple Lie algebras of type B_4, F_4 and E_8 from S^7, S^8 and S^15, respec…

Explicit p-harmonic functions on rank-one Lie groups of Iwasawa type

We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Riemannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek-Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.

Functionally Graded Media

The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case Functionally Graded Media, which allows us to obtain some homogeneity conditions.

The X-ray transform on 2-step nilpotent Lie groups of higher rank

We prove injectivity and a support theorem for the X-ray transform on $2$-step nilpotent Lie groups with many totally geodesic $2$-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.

The spectrum of the Laplace operator on connected compact simple rank three Lie groups

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of obtained formulas with the number theory and integer ternary and binary quadratic forms.

分级超流形与齐性

Graded supermanifolds and homogeneity

本文提出齐性超流形的概念，通过赋予坐标齐性次数来定义额外几何结构，并证明了齐性Poincaré、Frobenius和Darboux定理。

The spectrum of the Laplace operator on connected compact simple rank four Lie groups. I

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank four Lie groups with biinvariant Riemannian metric, corresponding to root systems $B_4$, $C_4$, $D_4$ and established a connection of obtained formulas with the number theory and integer quadratic forms from two, three and four variables.

Geodesic orbit spaces of compact Lie groups of rank two

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H,g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.

Global stability of the Pluriclosed flow on compact simply-connected simple Lie groups of rank two

We compute the (1,1)-Aeppli cohomology of compact simply-connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply-connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott-Chern and the Aeppli cohomologies, arise from just the left-invariant forms and we computed the whole Bott-Chern diamonds of SU(3) and Spin(5) when th…

Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds

Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes anterior investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank. Combined with these results, we …

Clifford-Wolf homogeneous left invariant $(α,β)$-metrics on compact semi-simple Lie groups

Let $(M,F)$ be a connected Finsler space. An isometry of $(M,F)$ is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space $(M,F)$ is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close points $x_1,x_2\in M$, there exists a CW-translation $σ$ such that $σ(x_1)=x_2$. In this paper, we define the good normalized datum for a homogeneous non-Riemannian $(α,β)$-space, and use it to study the restrictive CW-homogeneity of left invariant $(α,β)$-metrics on a compact c…

The 2-ranks of connected compact Lie groups

The 2-rank of a compact Lie group $G$ is the maximal possible rank of the elementary 2-subgroup ${\mathbb Z}_{2}\times... {\mathbb Z}_{2}$ of $G$. The study of 2-ranks (and $p$-rank for any prime $p$) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups have been investigated by many mathematician. The 2-ranks of compact Lie groups relate closely with several important areas in mathematics. In this article, we survey important results concerning 2-ranks of compact Lie groups. In particular, we present the complete determination o…

关于具有常实秩的复李代数胚

On Complex Lie Algebroids with Constant Real Rank

该研究为复李代数胚定义实元素分布与实秩不变量，推导常实秩下的实李代数胚、极小子复李代数胚及局部分解，还引入斜代数胚复匹配对刻画实向量丛复化上的所有复李代数胚结构。

具有指定消失阶的奇异叶状结构的有限秩李代数胚

Finite-Rank Lie Algebroids for Singular Foliations of Prescribed Vanishing Order

该研究构造了一族有限秩李代数胚，其锚映射可实现原点处指定消失阶向量场的奇异叶状结构，该构造源自一类左对称代数胚的次相邻李代数胚。

Lie theorem via rank 2 distributions (integration of PDE of class ω=1)

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed form, as in the method of Darboux, and discuss nonlinear Laplace transformations and symmetric PDE models.

Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let Γ\subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of Γand G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserving, multiplicity free, trellised, Anosov actions on compact manifolds.

A natural Lie superalgebra bundle on rank three WSD manifolds

We determine the structure of the $*$-Lie superalgebra generated by a set of carefully chosen natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of $\mathbf{sl}(4,\C)$ with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of $\mathbf{so}(3,\R)$. We provide an explicit description of one of the real forms of this superalgebra, which is geometrically natural being made of $\mathbf{so}(3,\R)$-invariant oper…

紧李群上最小秩椭圆结构的一个分解定理及其应用

A decomposition theorem for elliptic structures of minimal rank on compact Lie groups and applications

本文证明奇数维紧李群上最小秩左不变椭圆结构具有代数分解，并据此将相关上同调归结为极大环面上的计算。

An embedding theorem for automorphism groups of Cartan geometries

We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an automorphism group; a result asserting local homogeneity and completeness of parabolic geometries admitting a maximal-rank group of automorphisms; and a local freeness theorem for actions additionally preserving a continuous volume form.

Natural Lie Algebra bundles on rank two s-Kähler manifolds, abelian varieties and moduli of curves

We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-Kähler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves.

亚交换李群中秩2极化的萨德性质

Sard property for rank 2 polarizations in metabelian Lie groups

研究亚交换李群上秩2极化的异常集维数界及相关萨德性质，通过给出维数界确立终点映射萨德性质，还得到特定亚交换李群戈赫异常集维数界，推断相关极化群满足极小化萨德性质。

Representations of Surface Groups and Right-Angled Artin Groups in Higher Rank

We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and show that the criterion is satisfied in particular cases. There are direct applications towards constructing representations of surface groups into higher-rank Lie groups, and, in particular, into lattices in higher-rank Lie groups.