Toric geometry of Schouten squares on rotational double extensions

Abstract

Let $\g=\fb\oplus V\oplus\fb^{*}$ be a Medina-Revoy rotational double extension in which an abelian Lie algebra $\fb$ acts by scalar rotations on an orthogonal sum of oriented Euclidean two-planes. We analyse the restriction of the Schouten square $r\mapsto[r,r]$ to the block-rank-one locus $\Rone\subset\fb\wedge V$ and to its nondegenerate open part $\Rplus$. On every marked slice $S_{H}\subset\Rplus$ the Schouten square splits orthogonally into a radial and an angular component. The radial component factors through the standard toric moment map: its image is a simplicial cone, it satisfies a sharp quadratic coercive estimate with an intrinsic optimal constant, and the restricted Schouten map is proper with compact semialgebraic affine fibres. The angular component becomes Laurent monomial in joint eigencoordinates; a marker-cancellation criterion determines its signed exponent configuration, whose integral lattice controls the differential rank, the compact isotropy, its component group, and the Laurent binomial ideal defining the Zariski closure of the complexified image. Smith normal form makes each invariant algorithmic. All statements hold for arbitrary $\dim\fb$ and impose no nonresonance assumption. A corollary of the underlying disjoint-support decomposition classifies the classical Yang-Baxter equation on $\Rone$: for any bivector whose $\fb\wedge V$ component lies in $\Rone$, triangularity forces every block detected by its rotation weight to vanish, regardless of the $\Lambda^{2}\fb$, $\Lambda^{2}V$, or central-wedge terms.