Cohomology and deformations in graded Lie algebras

Nijenhuis, Albert; Richardson, R. W.

doi:10.1090/s0002-9904-1966-11401-5

Cited by 312 publications

(244 citation statements)

References 11 publications

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“…Весовые векторы образованы ба-зисом Пуанкаре-Биркгофа-Витта алгебр U (n + c ) и U (a ⊥ ). Чтобы рассмотреть такое пространство как g-модуль, мы должны выполнить деформацию [15] алгебры n + c…”

Section: заключениеunclassified

Рекуррентные Свойства Ветвления И Резольвента Бернштейна - Гельфанда - Гельфанда

Ляховский¹,

Lyakhovsky²,

Nazarov³

2011

ТМФ

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Section: заключениеunclassified

Рекуррентные Свойства Ветвления И Резольвента Бернштейна - Гельфанда - Гельфанда

Ляховский¹,

Lyakhovsky²,

Nazarov³

2011

ТМФ

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“…With this formula in hand, and the rules we have derived for computing d a P(Wj, 3 Wk-j ) y it is now a routine matter to carry through the Kodaira-Spencer "deformation program, It as sketched, for example, in [4] ; it may be considered as an exercise for the reader.…”

Section: 7mentioning

confidence: 99%

“…As can be seen from Ref. 4, this necessitates studying the "multiplicative" structure on the cochains associated with Lie algebra c ohomology . We have delayed presenting this theory because of its complexity, but in this paper we can present a relatively simple independent exposition, and show how it is applied to the interesting deformation problems in a straightforward way.…”

mentioning

confidence: 99%

Analytic continuation of group representations. IV

Hermann

1967

Commun.Math. Phys.

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The main problem, deforming a subalgebra of a Lie algebra, is treated algebraically, requiring an extensive " development of methods of defining multiplications on Lie algebra cohomolo,gy cochains. Some applications to differential geometry are also presented.To be submitted to Communications in Math. Physics. In this paper, we shall develop the full algebraic formalism necessary to discuss this deformation problem. As can be seen from Ref. 4, this necessitates studying the "multiplicative" structure on the cochains associated with Lie algebra c ohomology . We have delayed presenting this theory because of its complexity, but in this paper we can present a relatively simple independent exposition, and show how it is applied to the interesting deformation problems in a straightforward way. There is considerable overlap in results with work done by A. Nijenhuis and R. Richardson [ 5,6,9,10] . However, the methods presented here are perhaps better adapted to the explicit calculations that are necessary to apply the theory to interesting problems of group representations and differential geometry. Our aim is to show that o! induces a bilinear map, which we also denote by Q, of Cr( 4,) X Cs( 9,) --+Crfs( $3), for each pair (r, s) of non-negative integers. Now, for I' = s = 0, Cr($,) = Vl, C'($I,) = V2, Cr+S(~3) = V3 . We require in this case that a! be the same as the map we are given. We will now proceed by induction on (1: + s), assuming that a! is defined on Cr' X C", for r'+s'

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“…Formal deformations of arbitrary rings and associative algebras, and related cohomology questions, were first investigated by Gerstenhaber [10]. Later, the notion of deformation was applied to Lie algebras by Nijenhuis and Richardson [16]. Because various fields in mathematics and physics exist in which deformations are used, we focus in the study of Leibniz superalgebras.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimal deformations of null-filiform Leibniz superalgebras

Khudoyberdiyev

Omirov

2013

Journal of Geometry and Physics

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Abstract. In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras N F n,m . One of them is a Leibniz algebra (that is m = 0) and the second one is a pure Leibniz superalgebra (that is m = 0) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra N F n . Similar results for the case of Leibniz superalgebras are obtained.

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Cohomology and deformations in graded Lie algebras

Cited by 312 publications

References 11 publications

Рекуррентные Свойства Ветвления И Резольвента Бернштейна - Гельфанда - Гельфанда

Рекуррентные Свойства Ветвления И Резольвента Бернштейна - Гельфанда - Гельфанда

Analytic continuation of group representations. IV

Infinitesimal deformations of null-filiform Leibniz superalgebras

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