Quotients in Graded Lie Algebras. Martindale-like Quotients for Kantor Pairs and Lie Triple Systems

Garcı́a, Esther; Lozano, Miguel Gómez

doi:10.1007/s10468-011-9303-5

Cited by 6 publications

(9 citation statements)

References 7 publications

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“…Definition 3.11. [3] A filter F on a Lie algebra is a nonempty family of nonzero ideals such that for any I 1 , I 2 ∈ F there exists I ∈ F such that I ⊆ I 1 ∩ I 2 . Moreover, F is a power filter if for any I ∈ F there exists K ∈ F such that K ⊆ [I, I].…”

Section: Systems Of Quotients Of a Lie Triple Systemmentioning

confidence: 99%

“…Throughout this paper, we let F be a field of arbitrary characteristic. For background material on Lie triple systems the reader is referred to [3][4][5][6][7]. Our notation and terminology are standard as may be found in [2,10,11] [1] that if T is a LTS, then the standard embedding of T is the Z 2 -graded Lie algebra L(T ) = L 0 ⊕ L 1 , L 0 being the F-span of {L(x, y) : x, y ∈ T }, denoted by L(T, T ), where L(x, y) denotes the left multiplication operator in T , L(x, y)(z) := [x, y, z]; L 1 := T and where the product is given by…”

Section: Introductionmentioning

confidence: 99%

“…16 [3]. Let T be a subsystem of a LTS S and let L(T ) and L(S) be standard embeddings of T and S, respectively.…”

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confidence: 99%

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Systems of Quotients of Lie Triple Systems

Chen

Lin

2014

Communications in Algebra

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In this paper, we introduce the notion of system of quotients of Lie triple systems and investigate some properties which can be lifted from a Lie triple system to its systems of quotients. We relate the notion of Lie triple system of Martindale-like quotients with respect to a filter of ideals and the notion of system of quotients, and prove that the system of quotients of a Lie triple system is equivalent to the algebra of quotients of a Lie algebra in some sense, and these allow us to construct the maximal system of quotients for nondegenerate Lie triple systems.

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Section: Systems Of Quotients Of a Lie Triple Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Systems of Quotients of Lie Triple Systems

Chen

Lin

2014

Communications in Algebra

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“…Martindale rings of quotients were introduced by Martindale in 1969 for prime rings in [12], which was designed for applications to rings satisfying a generalized polynomial identity. In [6], García and Gómez defined Martindalelike quotients for Lie triple systems with respect to power filters of sturdy ideals and constructed the maximal system of quotients in the nondegenerate cases. More research about quotients of Lie systems refer in references [8,10].…”

Section: Introductionmentioning

confidence: 99%

Algebras of quotients of Hom-Lie algebras

Yao,

Chen

2020

Preprint

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In this paper, we introduce the notion of algebras of quotients of Hom-Lie algebras and investigate some properties which can be lifted from a Hom-Lie algebra to its algebra of quotients. We also give some necessary and sufficient conditions for Hom-Lie algebras having algebras of quotients. We also examine the relationship between a Hom-Lie algebra and the associative algebra generated by inner derivations of the corresponding Hom-Lie algebra of quotients. Moreover, we introduce the notion of dense extensions and get a proposition about Hom-Lie algebras of quotients via dense extensions.

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“…Martindale rings of quotients were introduced by Martindale in 1969 for prime rings in [16], which was designed for applications to rings satisfying a generalized polynomial identity. In [12], García and Gómez defined Martindale-like quotients for Lie triple systems with respect to power filters of sturdy ideals and constructed the maximal system of quotients in the nondegenerate cases. More research about quotients of Lie systems refer in references [14,19].…”

Section: Introductionmentioning

confidence: 99%

Algebras of quotients and Martindale-like quotients of Leibniz algebras

Yao¹,

Ma²,

Tang³

et al. 2020

Preprint

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In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold for a Leibniz algebras but also can been lifted to its algebras of quotients are investigated. Secondly, for any semiprime Leibniz algebra, its maximal algebra of quotients is constucted and a Passman-like characterization of the maximal algebra is described. Thirdly, the relationship between a Leibniz algebra and the associative algebra which is generated by left and right multiplication operators of the corresponding Leibniz algebras of quotients are examined. Finally, the definition of dense extensions and some vital properties about Leibnia algebras via dense extensions are introduced.

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Quotients in Graded Lie Algebras. Martindale-like Quotients for Kantor Pairs and Lie Triple Systems

Cited by 6 publications

References 7 publications

Systems of Quotients of Lie Triple Systems

Systems of Quotients of Lie Triple Systems

Algebras of quotients of Hom-Lie algebras

Algebras of quotients and Martindale-like quotients of Leibniz algebras

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