发表机构
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University; School of Mathematics and Statistics, Guangdong University of Technology; Department of Mathematics, Nanjing University（南京师范大学数学科学学院数学研究所; 广东工业大学数学学院; 南京大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究代数广义Ricci孤立子的调和性与存在性，刻画了调和挠率条件，构造了无经典nilsoliton的三步幂零李代数上的广义nilsoliton，并给出谱障碍，证明filiform李代数m_2(n)承认广义nilsoliton当且仅当5≤n≤8。
AI 中文摘要
本文研究代数广义Ricci孤立子的调和性与存在性。首先，我们通过一个涉及Killing形式的恒等式，刻画了任意度量李代数上代数广义Ricci孤立子的调和挠率。特别地，Killing形式的半正定性蕴含调和性，无需单模性假设。然后，我们在不可分解的三步幂零李代数上构造了具有非零挠率的广义nilsoliton，这些李代数不承认经典nilsoliton，维度为七以及$6m+d$（其中$m>d\geq1$）。此外，我们为具有阿贝尔导出的幂零李代数提供了一个谱障碍，并给出了一个基于导子在中心商上作用的障碍。结合这些障碍与显式构造，我们证明了filiform李代数$\mathfrak m_2(n)$（$n\geq5$）承认广义nilsoliton当且仅当$5\leq n\leq8$。
英文摘要
In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions $6m+d$ for $m>d\geq1$. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra $\mathfrak m_2(n)$, $n\geq5$, admits a generalized nilsoliton if and only if $5\leq n\leq8$.
