An LMI description for the cone of Lorentz-positive maps
Résumé
Let $L_n$ be the $n$-dimensional second order cone. A linear map from $\mathbb R^m$ to $\mathbb R^n$ is called positive if the image of $L_m$ under this map is contained in $L_n$. For any pair $(n,m)$ of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality (LMI) that describes this cone. Namely, we show that its dual cone, the cone of Lorentz-Lorentz separable elements, is a section of some cone of positive semidefinite complex hermitian matrices. Therefore the cone of positive maps is a projection of a positive semidefinite matrix cone. The construction of the LMI is based on the spinor representations of the groups $\Spin_{1,n-1}(\mathbb R)$, $\Spin_{1,m-1}(\mathbb R)$. We also show that the positive cone is not hyperbolic for $\min(n,m) \geq 3$.