A comment concerning cohomology and invariants of Lie algebras with respect to contractions and deformations

Campoamor-Stursberg, Rutwig

doi:10.1016/j.physleta.2006.10.050

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…• The contraction criterion formulated in Theorem 4.1 is a natural generalization of contraction criteria which involve standard cohomology cocycles -see e.g. [11,14]. • The invariant function ψ can be easily generalized and used as an invariant of an arbitrary anti-commutative or commutative algebra -it can, for example, describe all two-dimensional complex Jordan algebras and their contractions [17].…”

Section: Discussionmentioning

confidence: 99%

Twisted cocycles of lie algebras and corresponding invariant functions

Hrivnák

Novotný

2009

Linear Algebra and its Applications

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We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation -so called (α, β, γ)-derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants -we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.

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Section: Discussionmentioning

confidence: 99%

Twisted cocycles of lie algebras and corresponding invariant functions

Hrivnák

Novotný

2009

Linear Algebra and its Applications

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“…The spaces H 0 (g, g) and H 1 (g, g) are identified with the center Z(g) and the outer derivations Der(g)/IDer(g) of g, respectively [6]. To illustrate how the rigidity problem leads naturally to cohomological methods, we recall the notion of contraction of Lie algebras (see, e.g., [28][29][30][31] and references therein).…”

Section: Definitionmentioning

confidence: 99%

Some Features of Rank One Real Solvable Cohomologically Rigid Lie Algebras with a Nilradical Contracting onto the Model Filiform Lie Algebra Qn

Campoamor-Stursberg

García

2019

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The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec ( t ) = 1 , k , k + 1 , ⋯ , n + k − 3 , n + 2 k − 3 for k ≥ 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k ≤ 5 , some series of cohomologically rigid algebras for arbitrary values of k are determined.

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“…Contractions of Lie algebras have been investigated by many authors [Sa61], [He66], [LeN67], [We91] and continue to be a subject of active interest, particularly in connection with the somewhat inverse problem of deforming Lie algebras [FM05], [Bu07]. Note that contractions not only link two Lie algebras but also link some objects related to these Lie algebras such as representations, invariants, special functions and quantization mappings [MN72], [DR85], [CW99], [Cp07], [Ca09], and also coadjoint orbits, which provide the motivation for the present paper, as we will explain directly, below.…”

Section: Introductionmentioning

confidence: 99%