Maximal Poisson-Commutative Subalgebras and Flatness

Definitions and mathematical statement

Let $k$ be an algebraically closed field of characteristic zero and $\mathfrak q$ a finite-dimensional algebraic Lie algebra over $k$. Identify $S(\mathfrak q)=k[\mathfrak q^*]$, with its Lie--Poisson bracket determined on linear functions by $\{x,y\}(\lambda)=\lambda([x,y])$. Define \[ \mathfrak q_\lambda=\{x\in\mathfrak q:\lambda([x,y])=0\ \forall y\in\mathfrak q\},\qquad \ell=\operatorname{ind}\mathfrak q=\min_{\lambda\in\mathfrak q^*}\dim\mathfrak q_\lambda, \qquad b(\mathfrak q)=\frac{\dim\mathfrak q+\ell}{2}. \] A unital $k$-subalgebra $A\subseteq S(\mathfrak q)$ is Poisson-commutative when $\{A,A\}=0$, and maximal means maximal under inclusion among such subalgebras. The first question is \[ \forall A\subseteq S(\mathfrak q)\text{ maximal Poisson-commutative},\qquad \dim_{\mathrm{Krull}}A=b(\mathfrak q)\ ? \] Maximal under inclusion is the hypothesis; maximal possible dimension is the proposed conclusion. Ordinary associative commutativity would be automatic in $S(\mathfrak q)$ and is not the intended condition.

For the second question set \[ \mathfrak q^*_{\mathrm{reg}}=\{\lambda:\dim\mathfrak q_\lambda=\ell\},\qquad Z(\mathfrak q)=\{f\in S(\mathfrak q):\{f,g\}=0\ \forall g\in S(\mathfrak q)\}. \] Assume $\operatorname{codim}_{\mathfrak q^*}(\mathfrak q^*\setminus\mathfrak q^*_{\mathrm{reg}})\geq2$, and that the Poisson center $Z(\mathfrak q)$ contains algebraically independent homogeneous elements $f_1,\ldots,f_\ell$ satisfying $\sum_i\deg f_i=b(\mathfrak q)$. For every regular $\xi\in\mathfrak q^*_{\mathrm{reg}}$, write \[ f(\lambda+t\xi)=\sum_{j=0}^{\deg f}f_{\xi,j}(\lambda)t^j, \qquad B_\xi=F_\xi(Z(\mathfrak q)) =k[f_{\xi,j}:f\in Z(\mathfrak q),\ 0\leq j\leq\deg f]. \] Is the morphism induced by the inclusion $B_\xi\hookrightarrow S(\mathfrak q)$, \[ \pi_\xi:\mathfrak q^*=\operatorname{Spec}S(\mathfrak q) \longrightarrow\operatorname{Spec}B_\xi, \] flat for every such $\mathfrak q$ and $\xi$? Equivalently, is $S(\mathfrak q)$ a flat $B_\xi$-module? The second question has the additional hypotheses displayed above; they are not imposed on the first question.

Short English statement

The first question asks whether every inclusion-maximal Poisson-commutative subalgebra reaches the stated dimension bound. The second asks whether the map collecting all central argument-shift coefficients is flat under the codimension, degree-sum, and regularity hypotheses.

Sources

• Local UnsolvedMath registry, permanent ID $30000758$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Algebraic Groups (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/22, pp. 1191--1242. https://doi.org/10.4171/owr/2007/22.
• Supplied literature reference: Birtea--Ortega--Ratiu, "convexity methods," https://doi.org/10.1090/S0002-9947-08-04689-8.
• The corrected_verified note states that the official report supplies the index, dimension bound, codimension-two condition, homogeneous central elements, and regularity hypothesis. The assessment checked 2026-08-21 labels the questions open and says the cited results address special cases, not a general resolution. This is dated supplied triage requiring independent verification.