Lie quasi-states

TL;DR

This paper investigates Lie quasi-states on symplectic Lie algebras, revealing a unique continuous non-linear quasi-state linked to the asymptotic Maslov index, highlighting its special structure in high-rank cases.

Contribution

It proves the uniqueness of the continuous non-linear Lie quasi-state on symplectic Lie algebras of rank at least 3, connecting it to the asymptotic Maslov index.

Findings

Only one continuous non-linear Lie quasi-state exists on the specified algebra

The unique quasi-state is related to the asymptotic Maslov index

The result applies to symplectic Lie algebras of rank ≥ 3

Abstract

Lie quasi-states on a real Lie algebra are functionals which are linear on any abelian subalgebra. We show that on the symplectic Lie algebra of rank at least 3 there is only one continuous non-linear Lie quasi-state (up to a scalar factor, modulo linear functionals). It is related to the asymptotic Maslov index of paths of symplectic matrices.