Quadratic Hom-Lie triple systems

Baklouti, Amir

doi:10.1016/j.geomphys.2017.06.013

Cited by 20 publications

(8 citation statements)

References 19 publications

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“…However, it was the breakthrough research by Kanel-Belov et al [3] that began delving into the intricacies of pseudodifferential operators, emphasizing their invaluable role in mathematical physics. However, amid this flourishing area of study, the critical review in [4][5][6] cautioned the academic community to ensure rigorous proofs and validations. As the field continues to grow, the convergence of these diverse perspectives promises richer understandings and breakthroughs.…”

Section: Introductionmentioning

confidence: 99%

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

Assiry,

Mansour,

Baklouti

2023

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This paper performed an investigation into the s-embedding of the Lie superalgebra (→S1∣1), a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( SψD⊙) residing on the super-circle S1|1. We also introduce and rigorously define the central charge within the framework of (→S1∣1), leveraging the canonical central extension of SψD⊙. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→S1|1) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge.

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Section: Introductionmentioning

confidence: 99%

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

Assiry,

Mansour,

Baklouti

2023

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“…We list here some known results on this kind of algebraic structures: a Leibniz triple system decomposes as the orthogonal direct sum of well-described ideals if it admits a multiplicative basis [1]; Levi's theorem for Leibniz triple systems is determined [11]; the theory of centroid on Leibniz triple systems refers to [6]. In this paper, we will introduce the cohomology theory of Leibniz triple systems and use it to study the extension theory and 1-parameter formal deformation theory, which generalize partial results in [2,3,7,9,10,12,13,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

Cohomology of Leibniz Triple Systems and its applications

Wu¹,

Chen²,

Ma³

2021

Preprint

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In this paper, we introduce the first and third cohomology groups on Leibniz triple systems, which can be applied to extension theory and 1-parameter formal deformation theory. Specifically, we investigate the central extension theory for Leibniz triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Leibniz triple systems and the third cohomology group. We study the T * -extension of a Leibniz triple system and we determined that every even-dimensional quadratic Leibniz triple system (L, B) is isomorphic to a T * -extension of a Leibniz triple system under a suitable condition. We also give a necessary and sufficient condition for a quadratic Leibniz triple system to admit a symplectic form. At last, we develop the 1-parameter formal deformation theory of Leibniz triple systems and prove that it is governed by the cohomology groups.

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“…In the paper [42] Yau gave a general construction method of Hom-type algebras starting from usual algebras and a twisting self-map. For information on various types of Hom-algebras see [5,8,9,33,35,39].…”

Section: Introductionmentioning

confidence: 99%

Split regular Hom-Leibniz color 3-algebras

Kaygorodov

Popov

2019

Colloq. Math.

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We introduce and describe the class of split regular Hom-Leibniz color 3-algebras as the natural extension of the class of split Lie algebras, split Lie superalgebras, split Lie color algebras, split regular Hom-Lie algebras, split regular Hom-Lie superalgebras, split regular Hom-Lie color algebras, split Leibniz algebras, split Leibniz superalgebras, split Leibniz color algebras, split regular Hom-Leibniz algebras, split regular Hom-Leibniz superalgebras, split regular Hom-Leibniz color algebras, split Lie 3-algebras, split Lie 3-superalgebras, split Lie color 3-algebras, split regular Hom-Lie 3-algebras, split regular Hom-Lie 3superalgebras, split regular Hom-Lie color 3-algebras, split Lie triple systems, split Lie triple supersystems, split Lie triple color systems, split regular Hom-Lie triple systems, split regular Hom-Lie triple supersystems, split regular Hom-Lie triple color systems, split Leibniz 3-algebras, split Leibniz 3-superalgebras, split Leibniz color 3-algebras, split regular Hom-Leibniz 3-algebras, split regular Hom-Leibniz 3-superalgebras, and some other algebras. More precisely, we show that any of such split regular Hom-Leibniz color 3-algebras T is of the form T = U + j I j , with U a subspace of the 0-root space T 0 , and I j an ideal of T satisfying for j = k : [T, I j , I k ] + [I j , T, I k ] + [I j , I k , T ] = 0.Moreover, if T is of maximal length, we characterize the simplicity of T in terms of a connectivity property in its set of non-zero roots.

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Quadratic Hom-Lie triple systems

Cited by 20 publications

References 19 publications

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

Cohomology of Leibniz Triple Systems and its applications

Split regular Hom-Leibniz color 3-algebras

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