On the singularities of Mishchenko-Fomenko systems

TL;DR

This paper investigates the singularities of Mishchenko-Fomenko integrable systems associated with complex semisimple Lie algebras, establishing precise codimension results for their bifurcation diagrams depending on the nature of the regular element.

Contribution

It provides sharp bounds on the codimension of bifurcation diagrams for Mishchenko-Fomenko systems, distinguishing between nilpotent and non-nilpotent regular elements.

Findings

Codimension of bifurcation diagram is one for non-nilpotent regular elements.

Codimension is one or two for nilpotent regular elements.

Both codimension cases are shown to be attainable.

Abstract

To each complex semisimple Lie algebra $\mathfrak{g}$ and regular element $a\in\mathfrak{g}_{\text{reg}}$, one associates a Mishchenko-Fomenko subalgebra $\mathcal{F}_a\subseteq\mathbb{C}[\mathfrak{g}]$. This subalgebra amounts to a completely integrable system on the Poisson variety $\mathfrak{g}$, and as such has a bifurcation diagram $\Sigma_a\subseteq\mathrm{Spec}(\mathcal{F}_a)$. We prove that $\Sigma_a$ has codimension one in $\mathrm{Spec}(\mathcal{F}_a)$ if $a\in\mathfrak{g}_{\text{reg}}$ is not nilpotent, and that it has codimension one or two if $a\in\mathfrak{g}_{\text{reg}}$ is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of $\Sigma_a$.