Description of infinite dimensional abelian regular Lie groups

The Laplacian coflow on almost-abelian Lie groups

We find explicit solutions of the Laplacian coflow of $G_2-$structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existence.

Integrability Criterion for Abelian Extensions of Lie Groups

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension $\hat{G}$ of $G$ by $A$, where $G$ is a connected, simply connected Lie group and $A$ is a quotient of its Lie algebra by some discrete subgroup. When $G$ is non-simply connected, the kernel $A$ is replaced by a central extension $\hat{A}$ of $π_1(G)$ by $A$.

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedull…

几乎交换李群上的不变共形Killing形式

Invariant conformal Killing forms on almost abelian Lie groups

本文完全描述了几乎交换度量李代数上的共形Killing-Yano $p$-形式，证明了非平行形式仅存在于$p=1$或$p=n-1$时，并完成了至多5维的分类及格存在性区分。

Fino-Vezzoni猜想在具有余维二阿贝尔理想的李代数上

Fino-Vezzoni conjecture on Lie algebras with abelian ideals of codimension two

本文证实了Fino-Vezzoni猜想对含余维二阿贝尔理想的幺模李代数成立，推广了几乎阿贝尔情形，为3步可解流形上的猜想提供了新证据。

Locally conformal SKT almost abelian Lie algebras

A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure $(J, g)$ whose Bismut torsion 3-form $H$ satisfies the condition $dH = α\wedge H$, for some closed non-zero 1-form $α$. This condition was introduced in [6] as a generalization of the SKT (or pluriclosed) condition $dH= 0$. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure $(J, g)$ such that $dH = α\wedge H$, for some closed 1-form $α$. As an application we classifiy LCSKT almost abelian Lie algebras in dimension $6$. Finally, we also study on almost abelian Lie algebras the …

余维数为 $2$ 的交换理想的李代数的 Hermitian 几何

Hermitian geometry of Lie algebras with abelian ideals of codimension $2$

本文研究含余维数二交换理想的幺模李代数上的 Hermitian 度量，并分类了 Bismut Kähler-like 与挠率平行度量。

Abelian complex structures on solvable Lie algebras

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie groups are complex affine manifolds. It turns out that all 4-dimensional Lie algebras carrying abelian complex structures are central extensions of such affine Lie algebras.

Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebras with a codimension one Abelian ideal admit such structures. Moreover, we classify the seven-dimensional complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated (G_2)_C-structures.

Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to five or of dimension greater than or equal to three with rank one.

Odd dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triple…

Abelian Balanced Hermitian structures on unimodular Lie algebras

Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian and the fundamental form $F$ is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$ the dimension of the center of $\mathfrak{g}$. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent…

Affine Structures on abelian Lie Groups

The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.

Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension $4$, while in the latter case we provide examples of such Lie groups admitting lattices in any even dimension.

几乎阿贝尔李群上的对称 Killing 张量

Symmetric Killing tensors on almost abelian Lie groups

本文研究带左不变黎曼度量的几乎阿贝尔李群，完整刻画其左不变对称 Killing 张量，并证明它们均可表示为 Killing 向量场与黎曼度量的多项式。

Calibrated and parallel structures on almost Abelian Lie algebras

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the first example of a Ricci-flat calibrated G_2^*-structure on a compact manifold whose holonomy is not contained in G_2^*. Moreover, we get examples of non-flat parallel G_2^*-structures on almost Abelian Lie algebras g. We give a full classification of these G_2^*-structures if g…

Pseudo-Kähler Lie algebras with abelian complex structures

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which verifies certain compatibility conditions with respect to the associative product on U. The Riemannian and Ricci curvatures of the associated pseudo-Kähler metric are studied and a characterization of those Lie algebras which are Einstein but not Ricci flat is given. It is seen tha…

阿贝尔李群的局部自由等距作用的谱序列研究

A Study of the Spectral Sequence for Locally Free Isometric Actions of Abelian Lie Groups

该研究针对阿贝尔李群局部自由等距作用的谱序列，给出其退化页号的上界并证明界最优，还指出其在调和形式研究中的潜在应用，为相关谱序列研究提供了关键结果。

关于幂零李代数上的阿贝尔复结构

On abelian complex structures on Nilpotent Lie Algebras

本文提出幂零李代数上阿贝尔复结构等价分类的新方法，应用该方法得到4、6维已知分类，给出8维1-阿贝尔复结构分类及一般情况部分结果。

几乎阿贝尔伪Kähler李代数

Almost abelian pseudo-Kähler Lie algebras

研究可解流形上几乎阿贝尔李代数的不变伪Kähler结构，分类所有维数下的酉同构类，并刻画存在复结构和辛结构但无相容伪Kähler度量的李代数。

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.

具有大阿贝尔子代数的复辛李代数

Complex Symplectic Lie Algebras with Large Abelian Subalgebras

该研究提出两种构造大阿贝尔理想李代数上复辛结构的方法，完成几乎阿贝尔李代数复辛结构的分类，得到多类非（超）凯勒的紧复辛流形及带拉格朗日环面纤维的实例。

近阿贝尔李群上的不变几何结构

Invariant Geometric Structures on Almost Abelian Lie Groups

本文研究连通复近阿贝尔群上的不变 Hermite 与 Kähler 结构，利用不变标架分类 Hermite 形式并证明不存在不变 Kähler 形式，且将结论推广至一般连通情形。

几乎阿贝尔李代数上预辛形式的模空间

The moduli spaces of presymplectic forms on almost abelian Lie algebras

本文研究了几乎阿贝尔李代数上预辛形式的模空间，证明其辛形式的模空间有限且所有辛形式可由标准2-形式排列得到。

6维李代数上阿贝尔复结构的分类

Classification of abelian complex structures on 6-dimensional Lie algebras

本文分类了可赋予阿贝尔复结构的6维李代数，并在相差全纯同构的意义下对每个代数上这类结构的空间进行了参数化。

关于具有余维二交换理想的Hermitian李代数的曲率性质

Curvature property on Hermitian Lie algebras with abelian ideals of codimension two

本文研究含余维二交换理想的单模Hermitian李代数，证明常Chern全纯截面曲率蕴含Chern平坦，且其他典范联络的常全纯截面曲率导致Kähler平坦及商代数的交换性。

Non-Abelian Lie algebroids over jet spaces

We associate Hamiltonian homological evolutionary vector fields --which are the non-Abelian variational Lie algebroids' differentials-- with Lie algebra-valued zero-curvature representations for partial differential equations.

Complex and CR-structures on compact Lie groups associated to Abelian actions

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description of these structures. In this article we present an alternative and more geometric construction of this type of invariant structures on a compact Lie group K when it is semisimple. We prove that each left-invariant complex structure, or each CR-structure of maximal dimension with …

Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3 \times {\Bbb R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3 \times {\Bbb R}$, ~ $g_λ^{+}$ is a shrinking while $g_λ^{+}$ and $g_μ$ are expanding and also $g_0^1, g_0^2, g_0^3$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_4$ only $g_1^λ, g_2^λ$ have Lorentz Ricci solitons and $g_2^λ$ is a shrinking.

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.

Hyperkähler torsion structures invariant by nilpotent Lie groups

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying no…

几乎交换李群上的调和$G_2$-结构

Harmonic $G_2$-structures on almost Abelian Lie groups

本文研究了7维几乎交换李群上的左不变$G_2$-结构，通过对应李代数的李括号$A$刻画挠率张量并建立各挠率类的代数条件，最终给出满足调和条件$div T=0$的代数判据以识别调和挠率类。

关于弱爱因斯坦李群

On weakly Einstein Lie groups

本文研究带有左不变度量的弱爱因斯坦李群，证明了非阿贝尔2步幂零李群及维数不超过5的非阿贝尔幂零李群均不存在弱爱因斯坦结构，并给出几乎阿贝尔李群为弱爱因斯坦的算子刻画。

洛伦兹度量在近阿贝尔李群上的研究

Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups

本文研究四维及以上近阿贝尔李群上的洛伦兹左不变度量，构造了推广的皮托夫解，该解为 Ricci-flat 但非平坦，并具有封闭类时曲线。

Geodesic Bases for Lie Algebras

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algebras having a codimension one ideal of a particular kind, nonunimodular algebras of dimension \$\leq 4\$, and unimodular algebras of dimension 5.

一类秩一、可裂可解Abel型Lie群的体积熵

Volume entropy of a family of rank one, split-solvable Lie groups of Abelian type

本文研究推广SOL群与双曲空间对数模型的Euclid空间度量族，推导了Abel型Heintze群的体积熵公式，并解决了关于SOL群与双曲空间间插值3-流形族的猜想。

Pluriclosed and Strominger Kähler-like metrics compatible with abelian complex structures

We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be $2$-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step nilpotent Lie group preserves the Strominger Kähler-like condition.

The path group construction of Lie group extensions

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebras and to Lichnerowicz cocycles on the Lie algebra of divergence free vector fields.