Quasi-state Rigidity for Finite-dimensional Lie Algebras

We say that a Lie algebra $\text{gfr}$ is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras ${\text{C}}^n⋊\text{L}u(n)$, $n \geq 1$, are rigid. On the other hand, a Lie algebra which surjects onto the three-dimensional Heisenberg algebra is not rigid. For Lie algebras of dimension $\le 3$ and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity.