The existence of soliton metrics for nilpotent Lie groups

By Tracy L. Payne. 2008

72 citations recorded by the source corpus.

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/15ace37a95c4183e9e4dc819b145ff2fdc93a240

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Einstein solvmanifolds and nilsolitons (2008, methodology): “• [Payne 05] Any filiform (i.e. n-dimensional and (n − 1)-step nilpotent) Lie algebra with at least two linearly independent semisimple derivations.”
Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces (2012, background): “1) holds for a three-dimensional Lie group [12] (see also [18, 24]).”
Einstein solvmanifolds: existence and non-existence questions (2006, background): “Such distinguished metrics are called minimal (see [11] for further information) or sometimes nilsoliton metrics (see [7,19]).”