QuasiI-State Rigidity for Finite-Dimensional Lie Algebras

Journal article, 2017

We say that a Lie algebra g 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 C-n x u( n), n = 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 <= 3 and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity.