We study the universal (associative) envelope of the Jordan triple system of all [Formula: see text] [Formula: see text] matrices with the triple product [Formula: see text] over a field of characteristic 0. We use the theory of non-commutative Gröbner–Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five finite-dimensional inequivalent irreducible representations of the Jordan triple system of all [Formula: see text] matrices.

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We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study.

We show that the universal associative enveloping algebra of the simple anti-Jordan triple system of all n × n matrices (n ≥ 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.If A is an associative algebra, A defines an anti-Jordan triple system A − relative to the product abc = abc − cba. Definition 1.2. A representation of an anti-Jordan triple system J is a homomorphism ρ: J → (End V ) − from J to the anti-Jordan triple system of endomorphisms of a vector space V . In other words, ρ is a linear mapping that satisfiesfor all a, b, c ∈ J. Two representations ρ 1 and ρ 2 of an anti-Jordan triple system J on the same vector space V are equivalent if there exists an invertible endomorphism T such that ρ 2 (a) = T −1 ρ 1 (a)T for all a ∈ J.In this paper we use the theory of non-commutative Gröbner bases to prove that the universal enveloping algebra of the simple anti-Jordan triple system of all n × n matrices is finite-dimensional. This theory was used by Bergman [3] to