The compression semigroup of the dual Vinberg cone

TL;DR

This paper studies the semigroup related to the dual Vinberg cone, establishing its decompositions and showing it lacks the contraction property under the cone's Riemannian metric.

Contribution

It provides the first detailed analysis of the semigroup's structure, including its polar decompositions and metric properties, for the dual Vinberg cone.

Findings

Established the triple and Ol'shanskib1i polar decompositions.

Proved the semigroup does not have the contraction property.

Enhanced understanding of the geometric and algebraic structure of the dual Vinberg cone.

Abstract

We investigate the semigroup associated to the dual Vinberg cone and prove its triple and Ol'shanski\u{\i} polar decompositions. Moreover, we show that the semigroup does not have the contraction property with respect to the canonical Riemannian metric on the cone.