Leibniz algebra deformations of a Lie algebra

Fialowski, Alice; Mandal, Ashis Kumar

doi:10.1063/1.2981562

Cited by 32 publications

(28 citation statements)

References 11 publications

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“…Thus, we may look for n-Leibniz central extensions and deformations of FAs considering these as n-Leibniz ones and expect, in general, to find a richer structure. This has been observed explicitly for the n = 2 case [39] by looking at Leibniz deformations of the Heisenberg Lie algebra; also, for n = 3, a specific 3-Leibniz deformation of the simple Euclidean 3-Lie algebra A 4 has been given in [40]. Thus, a natural extension of our work above is to look for n-Leibniz deformations of simple n-Lie algebras to see whether this opens more possibilities.…”

Section: Whitehead Lemma For Fasmentioning

confidence: 70%

Topics on n-ary algebras

Azcárraga

Izquierdo

2011

J. Phys.: Conf. Ser.

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We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (≡ n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all n ≥ 2 FAs, n = 2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is led to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n − 1 entires of the n-Leibniz bracket.

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Section: Whitehead Lemma For Fasmentioning

confidence: 70%

Topics on n-ary algebras

Azcárraga

Izquierdo

2011

J. Phys.: Conf. Ser.

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“…But, since Lie algebras are also Leibniz, it is also possible to look for Leibniz deformations of Lie algebras when viewed as Leibniz ones. This may result in the appearance of more deformations, a fact recently discussed and observed in [9] for the nilpotent 3-dimensional Heisenberg algebra. In fact, and for a symmetric representation of L [1,2], there is a homomorphism [2] between the Leibniz and Lie algebra homologies as well as between the Lie algebra and Leibniz cohomologies for that representation.…”

Section: Introductionmentioning

confidence: 73%

On a class of n-Leibniz deformations of the simple Filippov algebras

Azcárraga

Izquierdo

2011

Journal of Mathematical Physics

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We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n>3 simple finite-dimensional Filippov algebras are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n=2 Filippov (i.e., Lie) algebras. The n=3 simple FAs, however, admit a non-trivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the $n\geq 3$ simple Filippov algebras do not admit non-trivial central extensions as n-Leibniz algebras of the above class.Comment: 19 pages, 30 refs., no figures. Some text rearrangements for better clarity, misprints corrected. To appear in J. Math. Phy

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“…[23] Since dim Z R (L T ) = n − 1, it follows at once that n − 1 ≤ dim Z R L . If we further suppose that L is a Lie algebra, then the structure tensor must be skew-symmetric, which further implies that the condition Z R L ⊆ Z L L is satisfied.…”

Section: Contractions Of Leibniz Algebrasmentioning

confidence: 97%

“…[20][21][22] and references therein). The notion of contraction of Leibniz algebras can be easily adapted, and the study of the geometry of orbits combined with deformation theory [23] allows, in particular, to determine those Lie algebras that are limits of Leibniz algebras. This also enables to separate the irreducible components of LE n generated by Lie algebras and those generated by non-Lie algebras.…”

Section: Contractions Of Leibniz Algebrasmentioning

confidence: 99%