Journal of Lie Theory, Vol. 10, No. 2, pp. 375-381 (2000)

Compactification structure and conformal compressions of symmetric cones

Jimmie D. Lawson and Yongdo Lim

Jimmie d. Lawson
Department of Mathematics
Louisiana State University
Baton Rouge La 70803, U.S.A.
Yongdo Lim
Research Center
Kyungpook National University
Taegu 702-701, Korea

Abstract: In this paper we show that the boundary of a symmetric cone $\Omega$ in the standard real conformal compactification $\M$ of its containing euclidean Jordan algebra $V$ has the structure of a double cone, with the points at infinity forming one of the cones. We further show that ${\overline\Omega}^{\M}$ admits a natural partial order extending that of $\Omega$. Each element of the compression semigroup for $\Omega$ is shown to act in an order-preserving way on ${\overline\Omega}^{\M}$ and carries it into an order interval contained in ${\overline\Omega}^{\M}$.

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