Abstract
To each complex semisimple Lie algebra \( \mathfrak{g} \) and regular element a ∈ \( \mathfrak{g} \)reg, one associates a Mishchenko–Fomenko subalgebra \( \mathcal{F} \)a ⊆ ℂ[\( \mathfrak{g} \)]. This subalgebra amounts to a completely integrable system on the Poisson variety \( \mathfrak{g} \), and as such has a bifurcation diagram Σa ⊆ Spec(\( \mathcal{F} \)a). We prove that Σa has codimension one in Spec(\( \mathcal{F} \)a) if a ∈ \( \mathfrak{g} \)reg is not nilpotent, and that it has codimension one or two if a ∈ \( \mathfrak{g} \)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 Σa.
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