Four-dimensional Lorentzian Lie groups

We describe four-dimensional Lie groups equipped with a left-invariant Lorentzian metric, obtaining a complete classification of the Einstein and Ricci-parallel examples.     

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Singular Ricci Solitons and Their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. The main objective of this paper is to study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the “opening up” of the singularity persists for the perturbations as well.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

We determine the admissible forms for the Ricci operator of three-dimensional locally homogeneous Lorentzian manifolds.      

Canonical variation of a Lorentzian metric

Given a Lorentzian manifold (M,_gL)$(M,g_L)$ and a timelike unitary vector field E  , we can construct the Riemannian metric _gR=_gL+2ω⊗ω$g_R=g_L+2\omega \otimes \omega$, ω being the metrically equivalent one form to E. We relate the curvature of both metrics, especially in the case of E   being Killing or closed, and we use the relations obtained to give some results about (M,_gL)$(M,g_L)$.     

Isometry Groups of Unimodular Simply Connected 3-Dimensional Lie Groups

We characterize the left-invariant Riemannian metrics on each of the six unimodular simply connected 3-dimensional Lie groups which give rise to 3-, 4-, or 6-dimensional isometry groups. It turns out that this classification is independent of curvature properties.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations

In the canonical smooth fiber bundles : ^{n+1 n} endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups and construct -invariant generalizations of differentiable connections. In both regular and special cases of the representations of the relevant groups , we found all the affine nonholonomic ₁-, ₂-, and _{1, 2}-connections of the first order (see [4]) possessing the local Lie groups of transformations and also described the respective -invariant planar connections.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups (1, n) and construct (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups (1, n), we find all affine nonholonomic , and _1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations (1, n) and also describe the corresponding (1, n)invariant planar connections.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. II

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups and construct -invariant generalizations of differentiable connections. In both regular and special cases of the representations of the relevant groups , we found all the affine nonholonomic -, -, and -connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations and also described the respective -invariant planar connections.     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

Solvable extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left‐invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one‐dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.     

Some Weyl Modules for Simple Algebraic Groups of Type B4 and D4

Let G be a simply connected, semisimple algebraic group of type B₄ or D₄ over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.     

Lie Regular Generators of General Linear Groups

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ?_{2 ⁿ} ), GL(2, ? _{p ⁿ} ), and GL(2, ?_2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ?₄), GL(2, ?₆), GL(2, ?₈), and GL(2, ?₁₀) are also given.     

Automorphism Groups of Restricted Cartan-Type Lie Superalgebras

Let X denote the restricted Lie superalgebras of Cartan type W, S, H, or K over a field of characteristic p > 3, and 𝔄 the corresponding underlying superalgebra of X. Employing the invariance of the filtration of X we construct an isomorphism of Aut X to Aut(𝔄:X), the admissible automorphism group of the associative super-commutative superalgebra 𝔄. Moreover, it is proved that the group isomorphism above maps the standard normal series of Aut X to the one of Aut(𝔄:X), and also maps the homogeneous automorphism group of X to the admissible homogeneous automorphism group of 𝔄.     

Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

We introduce the notion of the -prolongation of Lie algebras of differential operators on homogeneous spaces. The -prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where V is a homogeneous space. We apply the obtained results to the spaces S ¹ (the Virasoro algebra) and .     

Nilpotient Lie群上Herz空间及它的某些应用

§1 .　IntroductionLetGbeaconnectedandsimplyconnectednilpotentLiegroupwith(bi invariant)HaarmeasuredgandLiealgebraG .Theexponetialmapissurjectiveby [12 ],Theorem 3.6 .1.Onecanassociatedasubellipticdistance (g ,h)d′( g ;h)witheachfixedalgeraicbasisa1 ,a2 ,…ad′ofG .Foralli∈ { 1,2 ,… ,d′} ,letAi =dL(ai)denotethegeneratorsoflefttranslationsactingontheclass{ai} .Thisdistancehasthecharacterizationd′( g ;h) =sup{ ｜ψ( g) - ψ(h) ｜ :ψ∈C∞0 (G) ,∑d′i=1｜ (Aiψ) ｜2 ≤ 1} ,(see [9],…     

Topology of steady and expanding gradient Ricci solitons via f-harmonic maps

In this paper we give some results on the topology of manifolds with ∞-Bakry–Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yau?s theory of harmonic maps.     

Lie Powers of Modules for Groups of Prime Order

Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Lⁿ(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Lⁿ(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J₁,...,J_p, where J_r has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J₁,...,J_p in the Lie powers Lⁿ(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J₁ or to an infinite sum of the form J_r J_{p-1} J_{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J₁,..., J_p in Lⁿ(J_p and Lⁿ(J_{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Lⁿ(J_r) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20.