Open Problems in Integrable Systems — Quantum integrability for polynomial in momenta integrals

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathfrak g$ be an arbitrary finite-dimensional real Lie algebra. Identify $S(\mathfrak g)$ with polynomial functions on $\mathfrak g^*$ and equip it with the Lie--Poisson bracket $\{f,g\}(x)=\langle x,[df_x,dg_x]\rangle$. Let $a\in\mathfrak g^*$ be regular: its coadjoint stabilizer has the minimum possible dimension. For every smooth local coadjoint-invariant function $f$ near $a$, let $f_j(x)$ be the degree-$j$ homogeneous Taylor coefficient in \[ f(a+\lambda x)\sim f(a)+\sum_{j\geq1}\lambda^j f_j(x). \] The Mischenko--Fomenko algebra $\mathcal F_a\subseteq S(\mathfrak g)$ is generated by all these $f_j$ for all $f$ and $j\geq1$; it is Poisson commutative.

The enveloping algebra $U(\mathfrak g)$ is the tensor algebra modulo $xy-yx-[x,y]$. With its standard degree filtration, the Poincaré--Birkhoff--Witt theorem identifies $\operatorname{gr}U(\mathfrak g)$ with $S(\mathfrak g)$; the leading homogeneous part of an element is its principal symbol.

Statement recorded in the supplied source.

For every $\mathfrak g$ and regular $a$, construct a commutative subalgebra $\widehat{\mathcal F}_a\subseteq U(\mathfrak g)$ whose associated graded algebra is exactly $\mathcal F_a$, or identify an obstruction to such a quantization. Equivalently, all its principal symbols must lie in $\mathcal F_a$ and those symbols must generate $\mathcal F_a$. [S2]

Short English statement

Lift the argument-shift algebra of any finite-dimensional Lie algebra to an exactly commuting enveloping-algebra subalgebra, or find an obstruction.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100711 (AMR-020-0711). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 7.12 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.