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Elgendy, Hader A.

doi:10.1016/j.jalgebra.2013.02.028

Cited by 10 publications

(8 citation statements)

References 6 publications

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“…There are 36 consequences of identities (1)-(3) since we can substitute either of two operations. We must exclude these consequences together with all permutations of relations (13) and (15). The rest is similar to the proof of Lemma 5.1.…”

Section: Identities Of Degreementioning

confidence: 89%

“…Theorem 6.3. Every identity in degree 7 satisfied by the special commutator and translator in every associative triple system follows from (1)-(3), (13), (15), and (16)- (18), together with the new identities for the commutator and translator separately whose existence is demonstrated by Lemmas 6.1 and 6.2. That is, special comtrans algebras satisfy no new identities in which every term has both operations.…”

Section: Identities Of Degreementioning

confidence: 95%

“…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%

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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: Identities Of Degreementioning

confidence: 89%

Section: Identities Of Degreementioning

confidence: 95%

Section: Enveloping Algebras For 2 × 2 Matricesmentioning

confidence: 99%

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Special identities for comtrans algebras

Bremner

Elgendy

2018

Linear and Multilinear Algebra

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“…Lemma 32. Let A ∈T 7 with c 7 15 = 0 = c 7 25 . Then A is isomorphic to a unique algebra A ′ ∈T 7 with d 7 15 = 0 = d 7 25 and d 7 46 = 1, where the d k ij are the structure constants of A ′ .…”

Section: Identity Relationmentioning

confidence: 99%

“…Step n − 1. When we reach this final step, all c n+1 p,q (t) with p, q ≥ 2 and all a n+1,r (t) with r ≥ 3 have been defined without using the coefficient a n1 (t) and (7) has been shown to hold for all i, j ≥ 2. Hence, as a n1 (t) also has no role in (4) nor on the invertibility of A(t), we can redefine it at this point.…”

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confidence: 99%

The geometric classification of nilpotent ℭ𝔇-algebras

Kaygorodov

Khrypchenko

2020

J. Algebra Appl.

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We give a geometric classification of complex 4-dimensional nilpotent [Formula: see text]-algebras. The corresponding geometric variety has dimension 18 and decomposes into 2 irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid 4-dimensional complex nilpotent [Formula: see text]-algebras.

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Jordan quadruple systems

Bremner

Madariaga

2014

Journal of Algebra

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Abstract. We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad {a, b, c, d} = abcd + dcba as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are not consequences of the defining identities. We introduce four infinite families of finite dimensional Jordan quadruple systems, and construct the universal associative envelope for a small system in each family. We obtain analogous results for the anti-tetrad [a, b, c, d] = abcd − dcba. Our methods rely on computer algebra, especially linear algebra on large matrices, the LLL algorithm for lattice basis reduction, representation theory of the symmetric group, noncommutative Gröbner bases, and Wedderburn decompositions of associative algebras. IntroductionIn this paper we study the quadrilinear operations {a, b, c, d} = abcd + dcba and [a, b, c, d] = abcd − dcba in associative algebras. The first is the Jordan tetrad which plays an important role in the structure theory of Jordan algebras [28,31]. The second is the anti-tetrad, which seems not to have been studied until now.1.1. Motivation. In an associative algebra for n ≥ 2 we define the n-tad to be this n-ary multilinear operation: {a 1 , . . . , a n } = a 1 · · · a n + a n · · · a 1 . For n = 2 we obtain the Jordan product {a, b} = ab+ba satisfying commutativity and the Jordan identity: a}, {{{a, a}, b}, a} ≡ {{a, a}, {b, a}}. There are further "special" identities satisfied by the Jordan product in every associative algebra which do not follow from the defining identities; the simplest occur in degrees 8 and 9 and are called the Glennie identities [17,18]. A Jordan algebra is "special" if it can be represented as a subspace of an associative algebra closed under the Jordan product; otherwise, it is "exceptional". If a special Jordan algebra is finite dimensional then its universal associative enveloping algebra is also finite dimensional. For n = 3 we obtain the Jordan triple product abc + cba; in every associative algebra, this operation satisfies identities which define Jordan triple systems (JTS): b, a}, {{a, b, c}, d, e} ≡ {{a, d, e}, b, c} − {a, {b, e, d}, c} + {a, b, {c, d, e}}. In contrast to the Jordan identity these identities are multilinear. There are special identities in higher degree: identities satisfied by the Jordan triple product in every associative algebra but which do not follow from the defining identities [24,25]. For the classification of finite dimensional JTS, see [22,29,30] and for their universal associative envelopes, see [23]. Closely related to Jordan triple systems are the anti-Jordan triple systems (AJTS), see [16]. Finite dimensional simple AJTS have been classified [1]. These systems are defined by identities satisfied by the anti-Jordan triple product abc−cba in every associative algebra:Universal associative envelopes for one infinite family of simple AJTS have been constructed [15]. At the next step n = 4 we obtain the Jordan tetrad abcd + dcba, which arises in...

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

Cited by 10 publications

References 6 publications

Special identities for comtrans algebras

Special identities for comtrans algebras

The geometric classification of nilpotent ℭ𝔇-algebras

Jordan quadruple systems

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