Universal Associative Envelopes of (<i>n</i>+ 1)-Dimensional<i>n</i>-Lie Algebras

Bremner, Murray R.; Elgendy, Hader A.

doi:10.1080/00927872.2011.558549

Cited by 8 publications

(9 citation statements)

References 16 publications

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“…Whereas the multiplicative structure of higher arity in rings makes easier contact with binary algebra (see e.g. [17] or [12], for example) largely based on the fact that, as above, multiplicative pairs can be used in these examples just as we did here for commutative ternary groups, ternary addition produces phenomena of a more unusual kind and has been, to the knowledge of the authors, treated less thoroughly. We therefore keep multiplication in the fields binary for the moment and stick to ternary addition.…”

Section: Introductionmentioning

confidence: 95%

Structure of unital 3-fields

Duplij

Werner

2020

Math Semesterber

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We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring $$\mathcal{R}$$ R having $$\mathbb{Z}\diagup 2\mathbb{Z}$$ Z ╱ 2 Z as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of $$\mathcal{R}$$ R . We include a number of illustrative examples and prove that the structure of a finite 3‑field is not connected to any binary field.

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Section: Introductionmentioning

confidence: 95%

Structure of unital 3-fields

Duplij

Werner

2020

Math Semesterber

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“…In this section we recall the basic definitions and results in the theory of noncommutative Gröbner bases in free associative algebras following [4,7]. = e i,j e j,i e i,t − e i,t e j,i e i,j − e i,t (t < j), R (i,j,k,t) 3 = e i,j e k,i e t,k − e t,k e k,i e i,j + e t,j (t < i), R (i,j,k) 4 = e i,j e k,i e i,k − e i,k e k,i e i,j + e i,j (k < j), R (i,j,k,t,r,s) 5 = e i,j e k,t e r,s − e r,s e k,t e i,j (r < i, j = k or t = r, s = k or t = i) , R (i,j,k,t,s) 6 = e i,j e k,t e i,s − e i,s e k,t e i,j (s < j, j = k or t = i, s = k or t = i) .…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section we recall the basic definitions and results in the theory of noncommutative Gröbner bases in free associative algebras following [4,7]. Definition 2.1.…”

Section: Preliminariesmentioning

confidence: 99%

“…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%

“…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.…”

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

Elgendy

2013

Communications in Algebra

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We construct universal associative envelopes for the nonassociative triple systems arising from the trilinear operations of Bremner and Peresi applied to the 2-dimensional simple associative triple system. We use noncommutative Gröbner bases to determine monomial bases, structure constants, and centers of the universal envelopes. We show that the infinite dimensional envelopes are closely related to the down-up algebras of Benkart and Roby. For the finite dimensional envelopes, we determine the Wedderburn decompositions and classify the irreducible representations.

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Universal Associative Envelopes of (n+ 1)-Dimensionaln-Lie Algebras

Cited by 8 publications

References 16 publications

Structure of unital 3-fields

Structure of unital 3-fields

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

Universal Associative Envelopes of Nonassociative Triple Systems

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