Quasi-state rigidity for finite-dimensional Lie algebras

Abstract

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 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. MICHAEL BJÖRKLUND AND TOBIAS HARTNICKis primarily interested in the infinite-dimensional case, such an analysis is relevant in order to understand the behaviour of Lie quasi-states along finite-dimensional subalgebras. However, to the best of our knowledge [15] remains the only paper so far which concerns Lie quasi-states on finite-dimensional Lie algebras.The purpose of the present paper is to extend some of the results from [15] to larger classes of finite-dimensional Lie algebras and to obtain a clearer picture about Lie quasi-states on general finite-dimensional Lie algebras through some key examples. Our main focus will be on Adinvariant Lie quasi-states, since these are comparatively easy to handle and at the same time the most relevant ones in applications. One particular goal of this article is to understand their connection with homogeneous quasimorphisms on finite-dimensional Lie groups. Integrable Lie quasi-states and homogeneous quasimorphisms.From now on we assume that all Lie algebras are real and finite-dimensional. We denote by Q(g) the space of Lie quasi-states on g and by Q(g) ⊂ Q(g) the subspace of continuous Lie quasi-states on g respectively. Note that the adjoint group associated to the Lie algebra g acts on these spaces by g.ζ(X ) = ζ(Ad(g) −1 (X )).