Imbedding of Lie triple systems into Lie algebras

Smirnov, O. N.

doi:10.1016/j.jalgebra.2011.06.011

Cited by 11 publications

(13 citation statements)

References 12 publications

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“…The part of the present paper related to Jordan pairs is parallel to the content from [13]. It is a 3-graded version of the theory of central extensions of Lie algebras (e.g., see [2,Sect.…”

Section: Preliminariesmentioning

confidence: 98%

Categories of Jordan Structures and Graded Lie Algebras

Caveny

Smirnov

2013

Communications in Algebra

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In the paper we describe the subcategory of the category of Zgraded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the TKK construction. For instance, we prove that L = L −1 ⊕ L 0 ⊕ L 1 can be constructed from a Jordan pair if and only if L 0 = [L −1 , L 1 ] and the second graded homology group H gr 2 (L) is trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras.

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Section: Preliminariesmentioning

confidence: 98%

Categories of Jordan Structures and Graded Lie Algebras

Caveny

Smirnov

2013

Communications in Algebra

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“…is a Lie triple system (see for instance [15,18,20] for the definition and the properties of Lie triple systems).…”

Section: Remarkmentioning

confidence: 99%

“…Proposition 3.7 describing the Lie algebra structure of the para-Kähler Lie algebra associated to a quasi S -matrix will play a crucial role in Sections 6-7. It shows also (see Remark 2 (b)) that a quasi S -matrix on a left symmetric algebra U defines a Lie triple system on U * (see [15,18,20] for the definition and properties of Lie triple systems). In Section 4, we develop some general methods to build new examples of para-Kähler Lie algebras.…”

mentioning

confidence: 96%

On para-Kähler and hyper-para-Kähler Lie algebras

Benayadi

Boucetta

2015

Journal of Algebra

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We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain some new results. The study of para-Kähler and hyper-para-Kähler is intimately linked to the study of left symmetric algebras and, in particular, those admitting invariant symplectic forms. In this paper, we give many new classes of left symmetric algebras and a complete description of all associative algebras admitting an invariant symplectic form. We give also all four dimensional hyper-para-Kähler Lie algebras.

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“…This Lie algebra, called the standard imbedding of the Lts m, does in general not depend functorially on m; see [11,21] for a detailed study of functorial properties related to this and other constructions. Note that, in terms of the Lie algebra g, we can write…”

Section: From Lie Triple Systems To Lie Algebras With Involutionmentioning

confidence: 99%

Symmetric bundles and representations of Lie triple systems

Bertram¹,

Didry²

2009

J Generalized Lie Theory Appl

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We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing "unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.2000 MSC: 17A01, 17B10, 53C35.

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Imbedding of Lie triple systems into Lie algebras

Cited by 11 publications

References 12 publications

Categories of Jordan Structures and Graded Lie Algebras

Categories of Jordan Structures and Graded Lie Algebras

On para-Kähler and hyper-para-Kähler Lie algebras

Symmetric bundles and representations of Lie triple systems

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