Abstract
We study the representations of two special Jordan triple systems (with respect to the product xyz + zyx): The first is the Jordan triple system of all symmetric by matrices over a field of characteristic zero, the second is the Jordan triple system of all Hermitian by matrices over the complex numbers . We show that the universal (associative) envelope of is isomorphic to , while the universal (associative) envelope of is isomorphic to . As corollaries, the Jordan triple system has two nontrivial finite-dimensional inequivalent irreducible representations, while the Jordan triple system has four nontrivial inequivalent finite-dimensional irreducible representations.
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