An LMI description for the cone of Lorentz-positive maps

Let Ln be the n-dimensional second order cone. A linear map from Rm to Rn is called positive if the image of Lm under this map is contained in Ln. 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 (R), Spin 1,m−1 (R). We also show that the positive cone is not hyperbolic for min(n,m) ≥ 3.