  We show that the universal associative enveloping algebra of the simple
anti-Jordan triple system of all $n \times n$ matrices $(n \ge 2)$ over an
algebraically closed field of characteristic 0 is finite dimensional. We
investigate the structure of the universal envelope and focus on the monomial
basis, the structure constants, and the center. We explicitly determine the
decomposition of the universal envelope into matrix algebras. We classify all
finite dimensional irreducible representations of the simple anti-Jordan triple
system, and show that the universal envelope is semisimple. We also provide an
example to show that the universal enveloping algebras of anti-Jordan triple
systems are not necessary to be finite-dimensional.
