[Paper Review] Ricci Yang-Mills solitons on nilpotent Lie groups

Michael Jablonski, Andrea Young|arXiv (Cornell University)|Jul 7, 2009

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper introduces Ricci Yang-Mills solitons on 2-step nilpotent Lie groups, showing they are strictly weaker than Ricci solitons. Using Geometric Invariant Theory and moment map techniques, the authors construct explicit examples of Lie groups that admit Ricci Yang-Mills solitons but not Ricci solitons, demonstrating the existence of non-trivial soliton structures in previously excluded cases.

ABSTRACT

The purpose of this paper is to introduce the Ricci Yang-Mills soliton equations on nilpotent Lie groups. In the 2-step nilpotent setting, we show that these equations are strictly weaker than the Ricci soliton equations. Using techniques from Geometric Invariant Theory, we develop a procedure to build many different kinds of Ricci Yang-Mills solitons. We finish this note by producing examples of Lie groups that do not admit Ricci soliton metrics but that do admit Ricci Yang-Mills soliton metrics.

Motivation & Objective

• To define and study Ricci Yang-Mills solitons on nilpotent Lie groups, particularly in the 2-step nilpotent setting.
• To demonstrate that Ricci Yang-Mills solitons are strictly weaker than Ricci solitons in this context.
• To develop a method using Geometric Invariant Theory and moment maps to construct explicit examples of such solitons.
• To identify Lie groups that do not admit Ricci soliton metrics but do admit Ricci Yang-Mills soliton metrics.
• To resolve a technical question about moment maps in the 2-step nilpotent setting, as posed by Eberlein.

Proposed method

• The authors reframe the Ricci Yang-Mills soliton equations using moment maps in the context of a representation of $GL_n\mathbb{R}$, leveraging tools from Geometric Invariant Theory.
• They analyze the moment map $m_1$ for left-invariant metrics on 2-step nilpotent Lie groups, using its value to determine soliton conditions.
• The construction relies on concatenating specific matrix blocks representing the Lie algebra structure, parameterized by real coefficients $a_1, b_i, c_i, d_i$.
• A Ricci Yang-Mills soliton exists if the moment map value satisfies a specific algebraic condition: $a_1^2 = b_1^2 + c_1^2 = \cdots = d_1^2 + \cdots + d_j^2$.
• For non-trivial solitons, the moment map is not a multiple of the identity, and the stabilizer condition $B \in \text{Stab}(C)$ is used to verify soliton existence.
• The Lie derivative of left-invariant tensors under automorphisms is computed via the associated derivation $D \in \text{Der}(\mathfrak{g})$, linking geometric flows to algebraic structures.

Experimental results

Research questions

• RQ1 Can Ricci Yang-Mills solitons be defined and constructed on 2-step nilpotent Lie groups where Ricci solitons do not exist?
• RQ2 How do Ricci Yang-Mills solitons relate to Ricci solitons in the 2-step nilpotent setting—specifically, is the former strictly weaker?
• RQ3 Can Geometric Invariant Theory be used to systematically construct Ricci Yang-Mills solitons on nilpotent Lie groups?
• RQ4 What algebraic conditions on the Lie algebra structure ensure the existence of a Ricci Yang-Mills soliton?
• RQ5 Do there exist explicit examples of Lie groups that admit Ricci Yang-Mills solitons but not Ricci solitons?

Key findings

• The paper constructs a $n-1$-parameter family of non-isomorphic 2-step nilpotent Lie algebras that admit Ricci Yang-Mills solitons but do not admit Ricci soliton metrics.
• For algebras where the moment map $m_1(C)$ is a multiple of the identity, Ricci Yang-Mills solitons exist if and only if $a_1^2 = b_1^2 + c_1^2 = \cdots = d_1^2 + \cdots + d_j^2$.
• In non-trivial cases where $m_1(C) \neq r\text{Id}$, a $j-1$-parameter family of non-isometric Ricci Yang-Mills solitons exists, satisfying $4a_1^2 = 6\lambda^2 = 4(b_1^2 + \cdots + b_j^2)$.
• The existence of such solitons is confirmed by verifying that the matrix $B$ is a stabilizer of the algebraic data $C$, satisfying the conditions of Theorem 3.21.
• The authors resolve a question posed by Eberlein regarding moment maps in the 2-step nilpotent setting by providing a constructive framework.
• The results show that Ricci Yang-Mills solitons provide a strictly weaker notion of 'best' metric than Ricci solitons, extending the class of solvable Lie groups with special geometric structures.