Relating Homogeneous Cones and Positive Definite Cones via T-Algebras

Abstract

T-algebras are nonassociative algebras defined by Vinberg in the early 1960s for the purpose of studying homogeneous cones. Vinberg defined a cone $K({\cal A})$ for each T-algebra ${\cal A}$ and proved that every homogeneous cone is isomorphic to one such $K({\cal A})$. We relate each T-algebra ${\cal A}$ with a space of linear operators in such a way that $K({\cal A})$ is isomorphic to the cone of positive definite self-adjoint operators. Together with Vinberg's result, we conclude that every homogeneous cone is isomorphic to a "slice" of a cone of positive definite matrices.