Quasi-state Rigidity for Finite-dimensional Lie Algebras

TL;DR

This paper introduces the concept of quasi-state rigidity in finite-dimensional Lie algebras, characterizes which algebras are rigid, and extends previous work to include new classes of Lie algebras.

Contribution

It extends the theory of quasi-state rigidity to a broader class of Lie algebras, including reductive and certain semi-direct product algebras, and identifies obstructions to rigidity.

Findings

Reductive Lie algebras are rigid.

Algebras surjecting onto the Heisenberg algebra are not rigid.

Rigidity is characterized for low-dimensional and certain solvable Lie algebras.

Abstract

We say that a Lie algebra $\gfr$ 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 $\C^n \rtimes \L{u}(n)$, $n \geq 1$, 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 $\leq 3$ and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity.