International Geometry Symposium in Memory of Prof. Erdoğan ESİN, Ankara, Turkey, 09 February 2023, pp.37
A Riemannian metric is an important tool for the geometric structure of a given manifold. Particularly
on Lie groups, it is possible to define several different Riemannian metrics. Considering any
Riemannian metrics, Einstein metrics are the most preferable metric, as the Ricci tensor complies
the Einstein metric. But, it is not possible to define Einstein metrics on non-abelian nilpotent Lie
algebras. Therefore the notion of nilsoliton metrics became an alternative option with a weaker condition
on a left invariant metric on a nilpotent Lie group G.
A nilsoliton is a Nilpotent Lie algebra with a Riemannian metric with the aforementioned weaker
condition. Nilsolitons are an important topic in mathematics for several reasons. First, nilsoliton
metric Lie algebras are unique up to isometry and scaling. A nilpotent Lie algebra is an Einstein
nilradical if and only if it admits a nilsoliton metric. Therefore classification of nilsoliton metrics
on a nilpotent Lie algebra is equivalent to the same of Einstein nilradicals. On the other hand, an
Einstein solvmanifold can completely be determined by the Lie algebra with the Lie bracket of the
solvmanifold with itself .Therefore the study of solvmanifolds are actually the study of nilsolitons.
Recently symbolic computation methods have been used for Lie algebras. By the formulation of
algorithms, one can easily compute exact solutions of symbolic mathematical problems with the
help of computer algebra programming languages. By this way, the computations can be done more
productively and accurately than by hand. To use symbolic computation, we represent a Lie algebra
by using table of its structure constants with a fixed basis explicitly by given multiplication table,
consisting of structure constants.
In this study, we present a computational procedure for the classifications of all possible eigenvalues
of simple nilsoliton derivations in dimension 9. We particularly classify the derivations of nilsolitons
with non-singular Gram matrix. The computational procedure was implemented using Matlab
R2022b with symbolic computation package on Intel(R)Core(TM) i3-5015U CPU at 2.10 GHz processor
and 4 GB of RAM.