Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

Benayadi, Saı̈d; Hidri, Samiha

doi:10.1080/00927872.2015.1085550

Cited by 11 publications

(7 citation statements)

References 19 publications

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“…Since a Lie bracket is skew-symmetric, we obtain that left ad-invariant, right ad-invariant, and associative-invariant nondegenerate symmetric bilinear forms on a Lie algebra are the same. See [4] for more details.…”

Section: Definition 23 ([4 Definition 2])mentioning

confidence: 99%

Generalized metricn-Leibniz algebras and generalized orthogonal representationof metric Lie algebras

Tang¹,

Song²

2018

Turk J Math

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We introduce the notion of a generalized metric n-Leibniz algebra and show that there is a one-to-one correspondence between generalized metric n-Leibniz algebras and faithful generalized orthogonal representations of metric Lie algebras (called Lie triple data). We further show that there is also a one-to-one correspondence between generalized orthogonal derivations (resp. generalized orthogonal automorphisms) on generalized metric n-Leibniz algebras and Lie triple datas. 0 Keyword: Generalized metric n-Leibniz algebra, metric Lie algebra, generalized orthogonal representation, generalized orthogonal derivation, generalized orthogonal automorphism 0 MSC:17B10, 17B40, 17B99

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Section: Definition 23 ([4 Definition 2])mentioning

confidence: 99%

Generalized metricn-Leibniz algebras and generalized orthogonal representationof metric Lie algebras

Tang¹,

Song²

2018

Turk J Math

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“…In this purpose, many results about quadratic Lie algebras (that is Lie algebras endowed with a nondegenerate, symmetric and invariant bilinear form) have been extended to Lie superalgebras and Leibniz (super)algebras (see for example, [1] [2] [3] [4] [5] [6]). For instance, in [7] the authors generalize the notion of double extension and describe quadratic Lie superalgebras; quadratic Leibniz algebras are also studied in [1] and in [8].…”

Section: Introductionmentioning

confidence: 99%

Simple and Pseudo Quadratic Leibniz Superalgebras

Pouye,

Kpamegan,

Todjihoundé

2023

JAMP

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Compactness of subspaces of a 2  -graded vector space is introduced and used to study simple Leibniz superalgebras. We introduce left and right super-invariance of bilinear forms over superalgebras. Pseudo-quadratic Leibniz superalgebras are Leibniz superalgebras endowed with a non degenerate, supersymmetric and super-invariant bilinear form. In this paper, we show that every nondegenerate, supersymmetric and super-invariant bilinear form over a Leibniz superalgebra induce a Lie superalgebra over the underlying vector space. Then by using double extension extended to Leibniz superalgebras, we study pseudo-quadratic Leibniz superalgebras and the induced Lie superalgebras. In particular, we generalize some results on Leibniz algebras to Leibniz superalgebras.

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“…Recently, in order to investigate commutative non-associative algebras, authors in [5] introduce the so-called Jacobi-Jordan algebras that are commutative algebras satisfying the Jacobi identity. Those algebras were first defined in [12] and since then they have been studied in various papers [3,4,6,8] under different name such as Jordan algebras of nil rank 3, Mock-Lie algebras, Lie-Jordan algebras or pathological algebras. It turns out that the commutativity and Jacobi identity satisfied by the product of an algebra (A, * ) induce two relations x * (y * z) = −(x * y) * z−y * (x * z) and x * (y * z) = −(x * y) * z−(x * z) * y for all x, y, z ∈ A, called respectivelly left Lie-type identity and right Lie-type identity.…”

Section: Introductionmentioning

confidence: 99%

Extensions, crossed modules and pseudo quadratic Lie type superalgebras

Pouye

Kpamegan²

2022

Extr. Math.

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Extensions and crossed modules of Lie type superalgebras are introduced and studied. We construct homology and cohomology theories of Lie-type superalgebras. The notion of left super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.

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Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

Cited by 11 publications

References 19 publications

Generalized metricn-Leibniz algebras and generalized orthogonal representationof metric Lie algebras

Generalized metricn-Leibniz algebras and generalized orthogonal representationof metric Lie algebras

Simple and Pseudo Quadratic Leibniz Superalgebras

Extensions, crossed modules and pseudo quadratic Lie type superalgebras

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