Leibniz Triple Systems

Bremner, Murray R.; Sánchez-Ortega, Juana

doi:10.1142/s021919971350051x

Cited by 17 publications

(24 citation statements)

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“…Let g and h be two elements in 1 . We say that g is connected to h if there exist g 1 , g 2 , · · · , g 2n+1 ∈ ± 1 ∪{1} such that…”

Section: Connections and Gradingsmentioning

confidence: 99%

“…Leibniz triple systems were defined in a functorial manner using the Kolesnikov-Pozhidaev algorithm, which took the defining identities for a variety of algebras and produced the defining identities for the corresponding variety of dialgebras [2]. In [1], Leibniz triple systems were obtained by applying the Kolesnikov-Pozhidaev algorithm to Lie triple systems. The study of gradings on Lie algebras begins in the 1933 seminal Jordan's work, with the purpose of formalizing Quantum Mechanics [3].…”

Section: Introductionmentioning

confidence: 99%

“…Since then, the interest on gradings on different classes of algebras has been remarkable in the recent years, motivated in Definition 2.6. [1] The standard embedding of a Leibniz triple system E is the twograded right Leibniz algebra L = L 0 ⊕ L 1 , L 0 being the K-span of {x ⊗ y, x, y ∈ E}, L 1 := E and where the product is given by…”

Section: Introductionmentioning

confidence: 99%

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On the structure of graded Leibniz triple systems

Cao

Chen

2016

Linear Algebra and its Applications

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We study the structure of a Leibniz triple system E graded by an arbitrary abelian group G which is considered of arbitrary dimension and over an arbitrary base field K. We show that E is of the formwith U a linear subspace of the 1-homogeneous component E 1 and any ideal I [j] of E, satisfying {I [j] , E,, where the relation ∼ in 1 = {g ∈ G\{1} : L g = 0}, defined by g ∼ h if and only if g is connected to h.

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“…Let g and h be two elements in 1 . We say that g is connected to h if there exist g 1 , g 2 , · · · , g 2n+1 ∈ ± 1 ∪{1} such that…”

Section: Connections and Gradingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the structure of graded Leibniz triple systems

Cao

Chen

2016

Linear Algebra and its Applications

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“…16 To continue in the proof of the non-Koszulness, we consider the two elements α n = σ∈Sh 1 (n 2 −1,n−1) ℓ • 1,σ ν, (9) β n = σ∈Sh 1 (n−1,n 2 −1) ν • 1,σ ℓ. (10) Note that ∂α n = σ∈Sh 1 (n 2 −1,n−1) ℓ • 1,σ ∂ν = = n! σ∈Sh 1 (n 2 −1,n−1)…”

Section: Remark 41mentioning

confidence: 99%

Veronese powers of operads and pure homotopy algebras

Dotsenko

Markl

Remm

2019

European Journal of Mathematics

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We define the mth Veronese power of a weight graded operad P to be its suboperad P [m] generated by operations of weight m. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads P [m] are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie k-algebras and Lie triple systems. In the case of Lie k-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg-Kapranov power series test is inconclusive.Organisation of the paper. The paper is organised as follows. In Section 1, we recall the key relevant definitions of the theory of operads. In Section 2, we define Veronese powers of weight graded operads, prove some results about them, and provide counterexamples to some celebrated properties of Veronese powers of associative algebras. In Section 3, we relate our work to research of polynomial 2010 Mathematics Subject Classification. 18D50 (Primary), 18G55, 33F10, 55P48 (Secondary).Key words and phrases. Operad, Veronese power, homological purity, Koszul duality, Koszulness, Zeilberger's algorithm. The second author was supported by the EduardČech Institute P201/12/G028 and RVO: 67985840. 1 identities, both classical and recent. In Section 4, we define operads for pure homotopy P-algebras, and explain how they are related to Veronese powers by Koszul duality. Finally, in Section 5, we demonstrate that the ungraded versions of strong homotopy Lie algebras sometimes used in the literature possess worse homotopical properties than the strong homotopy Lie algebras obtained from the minimal model of the Lie operad, and show that those algebras provide an economic example where the positivity criterion for Poincaré series does not settle the matter of non-Koszulness.Acknowledgements. Some of the key results of this paper were obtained in CINVESTAV (Mexico City); the first author and the second author wish to thank that institution for the excellent working conditions. The authors are also grateful to Murray Bremner for his interest in our work (he communicated to us [5] that arrived at the same definition of Veronese powers independently, when trying to define what an n-ary algebra over an operad P is), and to Eric Hoffbeck on comments on the first draft of the paper.

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“…A Triple system is a vector space g over a field K together with a K-trilinear map : g ⊗3 → g. Among the many examples known in the literature, one may mention 3-Lie algebras [1] and Lie triple systems [2] which are the generalizations of Lie algebras to ternary algebras, Jordan triple systems [2] which are the generalizations of Jordan algebras, and Leibniz 3-algebras [3] and Leibniz triple systems [4] which are generalizations of Leibniz algebras [5]. In this paper we enrich the family of triple systems by introducing the concept oftriple systems, presented as another generalization of Leibniz algebras with the particularity that, for all , ∈ g, the map…”

Section: Introductionmentioning

confidence: 99%