Relating homogeneous cones and positive definite cones via T-algebras

• Main Author: Chua, Chek Beng.
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2009
• Subjects: DRNTU::Science::Mathematics::Applied mathematics::Optimization
• Online Access: https://hdl.handle.net/10356/90513; http://hdl.handle.net/10220/4702
• Institution: Nanyang Technological University

T-algebras are nonassociative algebras defined by Vinberg in the early 1960s for the purpose of studying homogeneous cones. Vinberg defined a cone K(A) for each T-algebra A and proved that every homogeneous cone is isomorphic to one such K(A). We relate each T-algebra A with a space of linear operators in such a way that K(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.