## LSU Historical Dissertations and Theses

#### Title

Jordan Algebras and Lie Semigroups.

1996

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Jimmie D. Lawson

#### Abstract

For a Euclidean Jordan algebra V with the corresponding symmetric cone $\Omega$, we consider the semigroup $\Gamma\sb{\Omega}$ of elements in the automorphism group $G(T\sb{\Omega})$ of the tube domain $V$ + $i\Omega$ which can be extended to $\Omega$ and maps $\Omega$ into itself. A study of this semigroup was first worked out by Koufany in connection to Jordan algebra theory and Lie theory of semigroups. In this work we give a new proof of Koufany's results and generalize up to infinite dimensional Jordan algebras, so called $JB$-algebras. One of the nice examples of the semigroup $\Gamma\sb{\Omega}$ is from the Jordan algebra $Sym(n,\IR$) of symmetric matrices. However, $V\sb{\sigma}$ the set of all self-adjoint operators on $\IR\sp{n}$ with respect to a non-degenerated symmetric bilinear form $\sigma$, is a non-Euclidean Jordan algebra with a cone $\Omega\sb{\sigma}$ which is isomorphic to $\Omega$ of the symmetric cone of $Sym(n,\IR)$. We get an isomorphism of the automorphism groups between two tube domains which also induces an isomorphism between two Lie semigroups. The Lorentzian cone, which is one of the irreducible symmetric cones, is an essential tool in the study of semigroups in Mobius and Lorentzian geometry. J. D. Lawson studied the Mobius and Lorentzian semigroups with an Ol'shanskii decomposition even in the infinite dimensional cases. We study these semigroups via a Jordan algebra theory.

9780591288674

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