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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.
For n even, we prove Pozhidaevs conjecture on the existence of associative enveloping algebras for simple n-Lie algebras. More generally, for n even and any (n+1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(
We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.
An $ntimes n$ complex matrix $A$ is called coninvolutory if $bar AA=I_n$ and skew-coninvolutory if $bar AA=-I_n$ (which implies that $n$ is even). We prove that each matrix of size $ntimes n$ with $n>1$ is a sum of 5 coninvolutory matrices and each m
We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric group S_3. The classification is then given by the stu
$n$-ary algebras of the first degeneration level are described. A detailed classification is given in the cases $n=2,3$.