Detection and Uniqueness of Lie Quasi States

Definitions and mathematical statement

Let $G$ be a Lie group with real Lie algebra $\mathfrak g$, and let $K\subseteq\mathfrak g$ be the adjoint-invariant cone in the source. A Lie quasi-state in the stated algebraic sense is a function $\zeta:\mathfrak g\to{\mathbb{R}}$ such that \[ [X,Y]=0\quad\Longrightarrow\quad \zeta(aX+bY)=a\zeta(X)+b\zeta(Y) \quad(a,b\in{\mathbb{R}}), \] with $\zeta(X)\geq0$ for $X\in K$. Equivalently, its restriction to every commutative linear subspace is linear. The general request is to detect or characterize all such functions for the given $(\mathfrak g,K)$. Adjoint invariance of $K$ is a hypothesis on the cone; it does not itself impose $\zeta(\operatorname{Ad}(g)X)=\zeta(X)$. Continuity, measurability, local boundedness, and invariance of the quasi-state are not stated.

The source then asks three groups of special questions. First, is the Maslov Lie quasi-state for $\operatorname{Sp}(2n)$ unique up to the intended isomorphism? The usual setting is the real symplectic group and its Lie algebra, but the cone, normalization, range of $n$, regularity class, and meaning of isomorphism are absent. Positive scaling and pullback by automorphisms are different equivalences; none is selected.

Second, does $(T^4,\omega_{\mathrm{std}})$ admit a nonlinear Lie quasi-state? Third, how many do $(\mathbb{CP}^2,\omega_{\mathrm{std}})$ and $(S^2\times S^2,\omega_{\mathrm{std}}\oplus\omega_{\mathrm{std}})$ admit? A manifold itself is not a Lie algebra. The standard symplectic interpretation uses $C^\infty(M,{\mathbb{R}})$ with Poisson bracket and the cone of pointwise nonnegative functions, giving \[ \{f,g\}=0\quad\Longrightarrow\quad \zeta(af+bg)=a\zeta(f)+b\zeta(g). \] Nonlinear means failing to be linear on all functions, rather than on just the Poisson-commuting subspaces. The source does not specify whether this full Poisson algebra, functions modulo constants, or another Hamiltonian Lie algebra is intended, nor whether normalization $\zeta(1)=1$, monotonicity, continuity, or symplectic invariance is additionally required. Definitions of symplectic quasi-states used in the cited work can include more axioms and must not be equated automatically with the minimal Lie quasi-state definition above.

The supplied assessment dated 2026-08-21 reports uniqueness and multiplicity results in important cases, including rank-at-least-three cases and $S^2\times S^2$, but says full detection/classification and the $T^4$ question remain unresolved. It supplies neither the regularity hypotheses nor exact counts or equivalence classes of those results.

Short English statement

Classify functions linear on commuting Lie-algebra elements and positive on a given invariant cone, then settle the stated Maslov uniqueness and symplectic-manifold existence/counting questions. Regularity, normalization, equivalence, and the precise Poisson-algebra interpretation are essential missing choices.

Sources

• Local UnsolvedMath registry, permanent ID $30001039$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406. https://doi.org/10.4171/owr/2008/42.
• Entov--Polterovich, "Lie quasi-states," https://arxiv.org/abs/0906.4901.
• Eliashberg--Polterovich, "symplectic quasi-states," https://arxiv.org/abs/1006.2501.
• The supplied corrected_verified note reports preservation of the full question with notation cleanup. The partially solved label and assessment checked 2026-08-21 are supplied dated curation, not independently verified; exact rank, regularity, normalization, and manifold hypotheses require review.