Chapter II: Prehomogeneous symmetric spaces and Jordan algebras

Abstract

Prehomogeneous symmetric spaces are symmetric spaces which are open orbits under the action of a linear group in a vector space. The interaction of the symmetric space structure with the flat structure of the ambient vector space gives rise to new features: a new, commutative algebra, the so-called Lie triple algebra emerges (Section 1). The most important examples of these algebras are Jordan algebras. They are characterized by the fact that the quadratic map of the corresponding symmetric space extends to a quadratic polynomial defined on the ambient vector space. We call the corresponding prehomogeneous symmetric spaces quadratic (Section 2). Important examples of such spaces are the general linear groups and the cones of positive definite symmetric matrices (Section 3). In the final section we introduce briefly two other classes of symmetric orbits in vector spaces; a typical example is given by the orthogonal groups which are certain closed symmetric orbits in the vector space of real or complex matrices.