Solvable Lie algebras obtained by quivers and Einstein metrics

In geometry, it is important to study whether a given Lie group admits a special left-invariant geometric structure.

In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing paths within these quivers.

Also, we proved that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons.

In this paper, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero.

We demonstrate that the simply-connected Lie groups corresponding to these solvable Lie algebras always admit left-invariant Ricci solitons.

Moreover, we prove that when the quivers are oriented multi-trees, these Ricci soliton Lie groups are rigid, that is the direct product manifold of a flat manifold and an Einstein manifold.