Universal Associative Envelopes of Nonassociative Triple Systems

Elgendy, Hader A.

doi:10.1080/00927872.2012.749409

Cited by 10 publications

(6 citation statements)

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“…In this final section we use noncommutative Gröbner bases [5] to construct the universal associative enveloping algebras of the nonassociative triple systems A C , A T , and A CT , obtained by applying the special commutator, special translator, and both together, to the associative triple system A = (a ij ) of 2 × 2 matrices. This method has previously been used for the universal envelopes of triple systems [16] obtained by applying trilinear operations [8] to the 2 × 2 matrices with a 11 = a 22 = 0, and for infinite families of simple anti-Jordan triple systems [15,17]. This approach to the representation theory of comtrans algebras is based on the special commutator and special translator in an associative triple system, and therefore differs essentially from the approach of Im, Shen & Smith [22,23,31].…”

Section: Enveloping Algebras For 2 × 2 Matricesmentioning

confidence: 99%

Special identities for comtrans algebras

Bremner

Elgendy

2018

Linear and Multilinear Algebra

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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 [x, y, z] = xyz − yxz and special translator x, y, z = xyz − yzx 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 2 × 2 matrices.2010 Mathematics Subject Classification. Primary 17A40. Secondary 15-04, 15A21, 15A69, 15B36, 18D50, 20C30, 68W30.Key words and phrases. Comtrans algebras, trilinear operations, polynomial identities, lattice basis reduction, representation theory of the symmetric group, algebraic operads, Gröbner bases, universal associative enveloping algebras.The commutator alternates in the first two arguments (1), the translator satisfies the Jacobi identity (2), and the operations are related by the comtrans identity (3). Example 1.2. If T is a Lie triple system with bracket [x, y, z] then letting both commutator and translator equal [x, y, z] gives T the structure of a comtrans algebra T CT . If T is obtained from a Lie algebra L by [x, y, z] = [[x, y], z] then Shen & Smith [32, Theorem 3.2] have shown that L is simple if and only if T CT is simple. Example 1.3. Let A m,n denote the vector space of m × n matrices over F. Fix matrices p (n × n) and q (m × m) over F. Define a trilinear operation on A m,n by (x, y, z) = xpy t qz. Setting [x, y, z] = (x, y, z) − (y, x, z) and x, y, z = (x, y, z) − (y, z, x) gives A m,n the structure of a comtrans algebra [32, Example 2.1]. Definition 1.4. An associative triple system [25] is a vector space A with a trilinear operation xyz satisfying (vw(xyz)) = (v(wxy)z) = (vw(xyz)). The special commutator and special translator in A are [x, y, z] = xyz − yxz and x, y, z = xyz − yzx.

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Section: Enveloping Algebras For 2 × 2 Matricesmentioning

confidence: 99%

Special identities for comtrans algebras

Bremner

Elgendy

2018

Linear and Multilinear Algebra

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“…First, we prove the following lemma. Also, following [11], we note that if the product of Table 1 in [11] is defined by abc = (cā)b, then the fourth family with q = 0 (i.e. abc − acb + bca) reduces to (cā)b − (cb)a + (ab)c, which is called an anti-structurable algebra of a special case in (−1, −1)-FKTSs [14].…”

Section: B)mentioning

confidence: 99%

Symmetry of Lie algebras associated with (ε, δ)-Freudenthal-Kantor triple system

Kamiya

Ôkubo

2015

Proceedings of the Edinburgh Mathematical Society

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“…Let i, j, k, ℓ ∈ Ω. Then in A, we have (5 ) e i,j • e k,ℓ = δ j,k δ i,ℓ δ i,1 + δ i,1 δ j,1 e 1,j e j,1 + δ j,1 δ i,1 + δ i,1 δ j,1 e i,1 e 1,i…”

Section: The Structure Constants Of Aunclassified

“…The classification of finite-dimensional simple anti-Jordan triple systems over an algebraically closed field of characteristic 0 was given by Bashir [1,Theorem 6]. give a new proof of the PBW theorem and was used recently by Elgendy [5] and Elgendy and Bremner [7] to construct universal associative envelopes of nonassociative triple systems and universal envelopes of the (n+1)-dimensional n-Lie algebras respectively. This paper is structured as follows.…”

Section: Introductionmentioning

confidence: 99%

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The universal associative envelope of the anti-Jordan triple system ofn×nmatrices

Elgendy

2013

Journal of Algebra

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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

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Universal Associative Envelopes of Nonassociative Triple Systems

Cited by 10 publications

References 14 publications

Special identities for comtrans algebras

Special identities for comtrans algebras

Symmetry of Lie algebras associated with (ε, δ)-Freudenthal-Kantor triple system

The universal associative envelope of the anti-Jordan triple system ofn×nmatrices

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