On the diagonalization of the Ricci flow on Lie groups

TL;DR

This paper proves that for nilpotent Lie algebras, bases with diagonal Ricci tensors impose specific simple structural constants, and explores implications for diagonalizing Ricci flow on Lie groups.

Contribution

It characterizes the structural constants of nilpotent Lie algebras with diagonal Ricci tensors and applies these results to the Ricci flow on Lie groups.

Findings

Structural constants are restricted to multiples of basis elements

Only brackets of disjoint pairs can be nonzero

Applications to Ricci flow diagonalization

Abstract

The main purpose of this note is to prove that any basis of a nilpotent Lie algebra for which all diagonal left-invariant metrics have diagonal Ricci tensor necessarily produce quite a simple set of structural constants; namely, the bracket of any pair of elements of the basis must be a multiple of some of them and only the bracket of disjoint pairs can be nonzero multiples of the same element. Some applications to the Ricci flow of left-invariant metrics on Lie groups concerning diagonalization are also given.