arXiv subjects

Zaili Yan

Publications and source records attributed to Zaili Yan.

6 recordsLinked to original sources

arXiv2025

Invariant Einstein metrics on basic classical Lie supergroups

This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$. We consider a natural family of left invariant metrics parameterized by scaling factors on the simple and Abelian components of the reductive even part, using the canonical bi-invariant bilinear form. Explicit expressions for the Levi-Civita connection and Ricci tensor are derived, and the Einstein condition is reduced to a solvable algebraic system. Our main result shows that, except for the cases of $\mathbf{A}(m,n)$ with $m\neq n$, $\mathbf{F}(4)$, and their real forms, every real basic classical Lie superalgebra admits at least two distinct Einstein metrics. Notably, for $\mathbf{D}(n+1,n)$ and $\mathbf{D}(2,1;\alpha)$, we obtain both Ricci flat and non Ricci flat Einstein metrics, a phenomenon not observed in the non-super setting.

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arXiv2024

On pseudo-Riemannian Ricci-parallel Lie groups which are not Einstein

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and B\'{e}rard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either $(X-\alpha)(X-\bar{\alpha})$ (type I), where $\alpha\in \mathbb{C}\setminus \mathbb{R}$, or $X^{2}$ (type II). Firstly, we obtain a complete description of Ricci-parallel metrics of type I. In particular, such a Ricci-parallel metric is uniquely determined by an Einstein metric and an invariant symmetric parallel complex structure up to isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type II by using double extension process. Surprisingly, we find that every double extension of a metric Abelian Lie algebra is Ricci-parallel and the converse holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II. Moreover, we construct infinitely many new explicit examples of Ricci-parallel metric Lie algebras which are not Einstein.

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arXiv2023

Curvatures of metric Jordan algebras

In this paper, we investigate metric Jordan algebras, and follow the lines of the paper (J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. (1976)). Firstly, we define the Jordan-Levi-Civita connection, then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection. Secondly, using the Jordan-Levi-Civita connection, we introduce three natural curvature tensors on metric Jordan algebras, and obtain the corresponding curvature formulas. Thirdly, based on these curvature formulas, we prove that every formally real Jordan algebra admits both a metric of non-positive Jordan curvature, and a Jordan-Einstein metric of negative Jordan scalar curvature. Besides, for nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics.

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arXiv2023

The moment map for the variety of associative algebras

We consider the moment map $m:\mathbb{P}V_n\rightarrow \text{i}\mathfrak{u}(n)$ for the action of $\text{GL}(n)$ on $V_n=\otimes^{2}(\mathbb{C}^{n})^{*}\otimes\mathbb{C}^{n}$, and study the critical points of the functional $F_n=\|m\|^{2}: \mathbb{P} V_n \rightarrow \mathbb{R}$. Firstly, we prove that $[μ]\in \mathbb{P}V_n$ is a critical point if and only if $\text{M}_μ=c_μ I+D_μ$ for some $c_μ \in \mathbb{R}$ and $D_μ \in \text{Der}(μ),$ where $m([μ])=\frac{\text{M}_μ}{\|μ\|^{2}}$. Then we show that any algebra $μ$ admits a Nikolayevsky derivation $ϕ_μ$ which is unique up to automorphism, and if moreover, $[μ]$ is a critical point of $F_n$, then $ϕ_μ=-\frac{1}{c_μ}D_μ.$ Secondly, we characterize the maxima and minima of the functional $F_n: \mathcal{A}_n \rightarrow \mathbb{R}$, where $\mathcal{A}_n$ denotes the projectivization of the algebraic varieties of all $n$-dimensional associative algebras. Furthermore, for an arbitrary critical point $[μ]$ of $F_n: \mathcal{A}_n \rightarrow \mathbb{R}$, we also obtain a description of the algebraic structure of $[μ]$. Finally, we classify the critical points of $F_n: \mathcal{A}_n \rightarrow \mathbb{R}$ for $n=2$, $3$, respectively.

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arXiv2018

Geodesic orbit Finsler metrics on Euclidean spaces

A Finsler space $(M,F)$ is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of $(M, F)$. In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case $(M, F)$ is a fiber bundle over a symmetric Finsler space $M_1$ of non-compact type such that each fiber $M_2$ is a totally geodesic nilmanifold with a step-size at most 2, and the projection $π:M\rightarrow M_1$ is a Finslerian submersion. Furthermore, when $M_1$ has no Hermitian symmetric factors, the fiber bundle description for $M$ can be strengthened to $M=M_1\times M_2$ as coset spaces, such that each product factor is totally geodesic in $(M,F)$ and is a geodesic orbit Finsler space itself. Finally, we use the techniques in this paper to discuss the interaction between the geodesic orbit spaces and the negative (non-positive) curved conditions, and provide new proofs for some of our previous results.

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arXiv2017

Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds is presented. In particular, we show that on the standard homogeneous Einstein manifolds, except for some special cases, there exist plenty of such metrics. A further interesting result of this paper is that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics.

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