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 n by n (n ≥ 2) matrices over a field
...of characteristic zero, the second is the Jordan triple system
of all Hermitian n by n (n ≥ ~2) 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.
We show that the universal enveloping algebra of the comtrans algebra of all n × n (n > 2) matrices over a field of characteristic 0 with respect to the trilinear operations:
(the special commutator) ...and
(the special translator) is finite-dimensional. We explicitly determine the decomposition of the universal envelope into matrix algebras. We also show that the universal enveloping algebra of a comtrans algebra is not necessary be finite-dimensional.
The purpose of this article is to show the link between Jordan quadruple systems with quadripotents and Jordan algebras. We also extend the notions of the orthogonality, primitivity, and minimality ...of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system. We refine the Peirce decomposition of a Jordan quadruple system with respect to a quadripotent to be with respect to a system of orthogonal quadripotents and get the multiplication rules of the Peirce spaces. We show that the notions of primitive and minimal quadripotents coincide in a Jordan quadruple system.
We extend the notion of compatibility of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system and get a criterion for compatibility of quadripotents in a Jordan ...quadruple system.
Poisson triple systems Bremner, Murray R.; Elgendy, Hader A.
Linear & multilinear algebra,
05/2023, Volume:
71, Issue:
7
Journal Article
Peer reviewed
We introduce Poisson triple systems, which are vector spaces with 3 trilinear operations satisfying 9 polynomial identities of degree 5. We show that every Poisson triple system has a universal ...enveloping Poisson algebra. Finally, we briefly discuss operadic aspects of Poisson triple systems.
We describe the universal associative enveloping algebra of a Lie-Jordan algebra with a unit over a field of characteristic 0. We show that if L is a Lie-Jordan algebra with a unit e over a field
of ...characteristic 0, then the (unital) universal associative envelope of L is isomorphic to
where
The center of the universal associative envelope is also described. We conclude the paper with some open problems for further research.
Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gröbner basis for the comtrans operad, and state a conjecture on its dimension ...formula. We study multilinear polynomial identities for the special commutator
and special translator
in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Gröbner bases to construct the universal associative envelope for the special comtrans algebra of
matrices.
Let $\mathcal{J}_{\field}$ be the Jordan triple system of all $p \times q$ ($p\neq q$; $p,q >1)$ rectangular matrices over a field $\field$ of characteristic 0 with the triple product $\{x,y,z\}= x ...y^t z+ z y^t x $, where $y^t$ is the transpose of $y$. We study the universal associative envelope $\mathcal{U}(\mathcal{J}_{\field})$ of $\mathcal{J}_{\field}$ and show that $\mathcal{U}(\mathcal{J}_{\field}) \cong M_{p+q \times p+q}(\field)$, where $M_{p+q\times p+q} (\field)$ is the ordinary associative algebra of all $(p+q) \times (p+q)$ matrices over $\field$. It follows that there exists only one nontrivial irreducible representation of $\mathcal{J}_{\field}$. The center of $\mathcal{U}(\mathcal{J}_{\field})$ is deduced.
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.
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