Symmetric cone

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1933). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity. A convex cone C in a finite-dimensional real inner product space V is a convex set invariant under multiplication by positive scalars. It spans the subspace C – C and the largest subspace it contains is C ∩ (−C). It spans the whole space if and only if it contains a basis. Since the convex hull of the basis is a polytope with non-empty interior, this happens if and only if C has non-empty interior. The interior in this case is also a convex cone. Moreover, an open convex cone coincides with the interior of its closure, since any interior point in the closure must lie in the interior of some polytope in the original cone. A convex cone is said to be proper if its closure, also a cone, contains no subspaces. Let C be an open convex cone. Its dual is defined as It is also an open convex cone and C** = C. An open convex cone C is said to be self-dual if C* = C. It is necessarily proper, sinceit does not contain 0, so cannot contain both X and −X. The automorphism group of an open convex cone is defined by Clearly g lies in Aut C if and only if g takes the closure of C onto itself. So Aut C is a closed subgroup of GL(V) and hence a Lie group. Moreover, Aut C* = (Aut C)*, where g* is the adjoint of g. C is said to be homogeneous if Aut C acts transitively on C. The open convex cone C is called a symmetric cone if it is self-dual and homogeneous. In their classic paper, Jordan, von Neumann & Wigner (1934) studied and completely classified a class of finite-dimensional Jordan algebras, that are now called either Euclidean Jordan algebras or formally real Jordan algebras.