Yang-Mills homogeneous connections on compact simple Lie groups
type: Item
instance of: scholarly article
title: Yang-Mills homogeneous connections on compact simple Lie groups
zbMATH Open document ID: 0890.53035
author: Osamu Ikawa
published in: Kodai Mathematical Journal
publication date: 1998-06-25T00:00:00Z
review text: Let \(M\) be a compact Riemannian manifold and \(P\) a principal \(G\)-bundle, where \(G\) is a compact Lie group with a fixed bi-invariant Riemannian metric. A Yang-Mills connection is a critical point of the Yang-Mills functional; if the second variation is non-negative, then the connection is said to be stable. The compact Riemannian manifold \(M\) is Yang-Mills unstable if for every choice of \(G\) and \(P\), a stable Yang-Mills connection is flat. In an interesting paper [Math. Z. 193, 165-189 (1986; Zbl 0634.53022)], \textit{S. Kobayashi, Y. Ohnita} and \textit{M. Takeuchi} posed the question: Is every simply connected compact simple Lie group Yang-Mills unstable? The author considers an equivariant \(G\)-bundle \(P\) over a compact connected simple Lie group \(L\), obtained by a Lie homomorphism \(R\). (1), (2) and (3) are the following properties with respect to homogeneous connections on \(P\): (1) \(R\) is indecomposable; (2) flat bihomogeneous connections are only (+)-connections and \((-)\)-connections; (3) the (0)-connection is a unique non-flat Yang-Mills homogeneous connection. Then (1) and (2) are equivalent; (3) implies (1). Moreover, if the image of \(R\) contains a regular element, then any non-flat Yang-Mills homogeneous connection is unstable.
reviewed by: Valentin Boju
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 949859
zbMATH Keywords: unstable Yang-Mills connection
zbMATH Keywords: Whitehead's vanishing theorem
zbMATH Keywords: cohomology group
zbMATH Keywords: Casimir operator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043596
OpenAlex ID: W2062609987
cites work: Q3897665
cites work: Q5731354
cites work: On instability of Yang-Mills connections
cites work: Stability properties of the Yang-Mills functional near the canonical connection
cites work: Invariant Affine Connections on Lie Groups
cites work: Q5732712
Wikidata QID: Q115224574
DOI: 10.2996/KMJ/1138043596
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description: scientific article; zbMATH DE number 949859