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The existence of soliton metrics for nilpotent Lie groups. (English) Zbl 1189.53048
The primary aim of this work is to investigate nilsoliton metrics using algebraic and combinatorial methods. Each nonabelian metric algebra $(\mathfrak n_{\mu},Q)$ is associated to a vector $[\alpha^2]$ listing the squares of the non trivial structure constants. It is shown that the nil Ricci endomorphism satisfies the nilsoliton condition if and only if the vector $[\alpha^2]$ is a solution to the matrix equation $U v=[1]$ being $[1]$ a vector with every entry a one. A generalized Cartan matrix $A$ is defined in terms of $U$. So the authors characterizes the solution spaces of $(cA+d[1])v=-2\beta [1]$, $c,d$ positive, by the properties of $A$. Several examples of nilsoliton metrics are presented, and also a metric nilpotent Lie algebra admitting a semisimple derivation with rational eigenvalues but no nilsoliton metric.

53C30Homogeneous manifolds (differential geometry)
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
22E25Nilpotent and solvable Lie groups
22F30Homogeneous spaces
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