Haantjes挠率与可积性：Bolsinov-Konyaev-Matveev猜想的证明
Haantjes torsion and integrability: a proof of Bolsinov-Konyaev-Matveev's conjecture
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中文总结 AI 辅助
该研究证明了Bolsinov等人的猜想，即$\boldsymbol{\frak{gl}}$-正则流体动力学型系统在某点可积则其Haantjes张量为零，还推导得出这类系统在代数通有点邻域内的特定表达式。
中文摘要 AI 辅助
我们证明了Bolsinov、Konyaev和Matveev在文献[7]中提出的猜想：对于具有$\boldsymbol{\frak{gl}}$-正则算子场$A$的流体动力学型系统$\boldsymbol{u}_t=A(\boldsymbol{u})\boldsymbol{u}_x$，其在点$p$处的可积性，意味着在$p$的邻域内$A$及其所有对称性的Haantjes张量均为零。基于此，结合文献[8]的结果，在代数通有点的邻域内，任意由$\boldsymbol{\frak{gl}}$-正则算子场定义的可积流体动力学型系统，均可表示为$\boldsymbol{u}_t=X(\boldsymbol{u})\boldsymbol{\frak{o}}\boldsymbol{u}_x$，其中$X$为向量场，$\boldsymbol{\frak{o}}$为满足Hertling-Manin条件的交换结合乘积。
英文摘要
We prove a conjecture formulated by Bolsinov, Konyaev and Matveev in [7] stating that, integrability of a system of hydrodynamic type ${\bf u}_t=A({\bf u}) {\bf u}_x$ with $\mathfrak{gl}$-regular $A$ at a point $p$ implies the vanishing of the Haantjes tensor of $A$ and of all its symmetries in a neighborhood of $p$. As a consequence, leveraging on the result of [8], in a neighbourhood of an algebraically generic point, any integrable system of hydrodynamic type defined by a $\mathfrak{gl}$-regular operator field can be written as ${\bf u}_t=X({\bf u})\circ {\bf u}_x$ where $X$ is a vector field and $\circ$ is a commutative associative product satisfying Hertling-Manin conditions.
