Gröbner bases in universal enveloping algebras of Leibniz algebras

Insua, Manuel A.; Ladra, M.

doi:10.1016/j.jsc.2007.07.020

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…It is an open problem to extend these results further to the quadri-algebras of Aguiar and Loday [3], and to the Koszul dual of quadri-algebras introduced by Vallette [102, §5.6]. For Leibniz algebras, which are the analogues of Lie algebras in the setting of dialgebras, see Loday and Pirashvili [79], Aymon and Grivel [4], Casas et al [36], Insua and Ladra [68]. For pre-Lie algebras, which are the analogue of Lie algebras in the setting of dendriform algebras, see Bokut et al [16].…”

Section: Bibliographical Remarksmentioning

confidence: 99%

Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

Bremner

2013

Preprint

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The underlying motivation is to apply the theory of noncommutative Gröbner bases in free associative algebras to the construction of universal associative envelopes for nonassociative structures defined by multilinear operations. Trilinear operations were classified by the author and Peresi in 2007. In her Ph.D. thesis of 2012, Elgendy studied the universal associative envelopes of nonassociative triple systems obtained by applying these trilinear operations to the 2-dimensional simple associative triple system. In these notes I use computer algebra to extend some aspects of her work to the 4-dimensional and 6-dimensional simple associative triple systems.

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Section: Bibliographical Remarksmentioning

confidence: 99%

Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

Bremner

2013

Preprint

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“…where J A − stands for the Jacobian of A − . If A is flexible, then the function S alternates in its second and third arguments, i.e., A satisfies S(x, y, z) ≡ −S(x, z, y), and so (12) applies to get (13) 2S(x, y, z) ≡ J A − (x, y, z).…”

Section: On the Search For A Nonlinear Di-malcev Identitymentioning

confidence: 99%

“…It is well known that every associative algebra A gives rise to a Lie algebra A − , when the associative product xy is replaced by the commutator [x, y] = xy − yx. And conversely, the famous Poincaré-Birkhoff-Witt Theorem [13] states that any Lie algebra is isomorphic to a subalgebra of A − for some associative algebra A. Loday and Pirashvili [22] proved that the Poincaré-Birkhoff-Witt Theorem remains true for Leibniz algebras, a "noncommutative" version of Lie algebras; see also Aymon and Grivel [1], Insua and Ladra [12] for other approaches. The role played by associative algebras is taken now by associative dialgebras, introduced by Loday [20,21] in the last decade of the 20 th century.…”

Section: Introductionmentioning

confidence: 99%

On the definitions of nucleus for dialgebras

Sánchez-Ortega

2013

Journal of Algebra

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Communicated by Alberto ElduqueIn memory of Jean-Louis Loday MSC: primary 17D10 secondary 17A20, 17A30, 17A32, 17D05 Keywords:Nonassociative algebras and dialgebras Nucleus Ternary derivations Computer algebra Malcev dialgebras were introduced recently by Bremner, Peresi and Sánchez-Ortega. In the present paper, we continue their study by introducing the notion of the generalized alternative dinucleus of a 0-dialgebra. A general conjecture about the speciality of Malcev dialgebras in terms of this dinucleus is formulated. In the last section, we introduce the appropriate generalization of the associative nucleus for dialgebras, and prove an analogue of Kleinfeld's theorem for the setting of dialgebras.

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“…For the original proof, see [35]. For different approaches, see Aymon and Grivel [1], Insua and Ladra [28].…”

Section: Dialgebrasmentioning

confidence: 99%

Algebras, dialgebras, and polynomial identities

Bremner¹

2012

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This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc − bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raúl Felipe, Luiz A. Peresi, and Juana Sánchez-Ortega. AlgebrasThroughout this talk the base field F will be arbitrary, but we usually exclude low characteristics, especially p ≤ n where n is the degree of the polynomial identities under consideration. The assumption p > n allows us to assume that all polynomial identities are multilinear and that the group algebra FS n is semisimple.Definition 1.1. An algebra is a vector space A with a bilinear operationUnless otherwise specified, we write ab = µ(a, b) for a, b ∈ A. We say that A is associative if it satisfies the polynomial identity (ab)c ≡ a(bc).Throughout this paper we will use the symbol ≡ to indicate an equation that holds for all values of the arguments; in this case, all a, b, c ∈ A.Theorem 1.2. The free unital associative algebra on a set X of generators has basis consisting of all words of degree n ≥ 0,with the product defined on basis elements by concatenation and extended bilinearly,

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Gröbner bases in universal enveloping algebras of Leibniz algebras

Cited by 7 publications

References 11 publications

Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

On the definitions of nucleus for dialgebras

Algebras, dialgebras, and polynomial identities

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