Geometry and Algebra of Multidimensional Three-Webs

Akivis, Maks A.; Shelekhov, A. M.

doi:10.1007/978-94-011-2402-7

Cited by 54 publications

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“…Книга [29] содержит детальное изложение теории 3-тканей. Эту теорию на-чали развивать Г. Бол, Ш. Ш. Черн, В. Бляшке и продолжили М. А. Акивис и его школа.…”

Section: скобка куранта в пространстве сечений γ(T (M )) определяетсяunclassified

Однородные Пара-Кэлеровы Многообразия Эйнштейна

Alekseevskiĭ¹,

Alekseevsky²,

Медори³

et al. 2009

Uspekhi Mat. Nauk

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Section: скобка куранта в пространстве сечений γ(T (M )) определяетсяunclassified

Однородные Пара-Кэлеровы Многообразия Эйнштейна

Alekseevskiĭ¹,

Alekseevsky²,

Медори³

et al. 2009

Uspekhi Mat. Nauk

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“…The right-hand side of (11) is the Jacobian of the vectors ξ, η and ζ. This is the reason that relation (11) is called the generalized Jacobi identity (see [12], Section 2.4 or [8], Section 2.4). The term "generalized" is used here since in the case when a loop Q is a Lie group (i.e., when associativity holds), the identity (11) becomes the Jacobi identity.…”

Section: Local Algebras Of Differentiable Quasigroupsmentioning

confidence: 99%

“…They also used the name Akivis identity for the generalized Jacobi identity. Note that in [10] Akivis and Shelekhov used the term "triple system" for the Akivis algebra. In [4], Akivis used the term "local W -algebra" for the triple system connected with a three-web.…”

Section: Local Algebras Of Differentiable Quasigroupsmentioning

confidence: 99%

“…In this case, formula (4) takes the form z = u + v. Thus, the commutators and the associators of local algebras vanish, and the local algebras become trivial. Three-webs defined by such quasigroups are parallelizable, and the Thomsen closure condition (T ) (see Figure 3) holds on them (see [15], [12], [8], [24]). …”

Section: Closed G-structures Defined By Differentiable Quasigroupsmentioning

confidence: 99%

“…After World War II the investigations initiated in [16] and [18] were developed further by Akivis (see, for example, [12], [8] and [9], where one can find further references). It should be noted that the algebraic theory of quasigroups was intensively developed by Belousov's school in Kishinev (on algebraic theory of quasigroups; see [13] and [17]).…”

Section: Introductionmentioning

confidence: 99%

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Local algebras of a differential quasigroup

Akivis

Goldberg

2006

Bull. Amer. Math. Soc.

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Abstract. The authors provide a summary of results in the theory of differential quasigroups and their local algebras and indicate the relationship of these results to recent work on this subject. IntroductionA concept close to the notion of a quasigroup appeared first in combinatorics, when near the end of the eighteenth century (in 1779 and 1782) Euler published two papers [20] and [21] on pairs of mutually orthogonal Latin squares. Euler's quasigroup operation was considered on a discrete set. The notion of quasigroup was introduced by Suschkewitsch [55]. The term "quasigroup" was first used by Moufang [34], who considered certain quasigroups with a unit. (Later on such quasigroups were named loops.) The first publication on differentiable (and even analytic) quasigroups is the Mal'cev paper [31].Differentiable quasigroups are a subject of study in both algebra and differential geometry. In geometry, the theory of differentiable quasigroups is connected with the theory of multidimensional three-webs, and in algebra, it is connected with the theory of Lie groups.From the latter point of view, quasigroups were first considered in a paper of Mal'cev [31]. Sometime later, in [1], Akivis found a canonical decomposition for analytic loops similar to the Campbell-Hausdorff decomposition for Lie groups. In [4], Akivis constructed local algebras of multidimensional three-webs which are also local algebras of differentiable quasigroups associated with those webs. Hofmann and Strambach (see, for example, [25], [26]) named these algebras Akivis algebras.The differentiable quasigroups differ from the Lie groups only by the fact that multiplication in quasigroups is not associative. As a result, in addition to the operation of commutation, local algebras of differentiable quasigroups have an additional ternary operation, namely the operation of association, which is connected with commutation by the so-called generalized Jacobi identity. This new operation is defined in a third-order neighborhood of a differentiable quasigroup.It is well-known that every Lie algebra is isomorphic to a subalgebra of commutators of a certain associative algebra. A similar theorem was recently proven by Shestakov (see [50], [51] and [52]). He proved that an arbitrary Akivis algebra can

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Conformal Differential Geometry and Its Generalizations

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Geometry and Algebra of Multidimensional Three-Webs

Abstract: Geoletry and algebra of lultldllenslonal three-webs I by Maks A. Aklvls and Alexander M. Shelekhov. p. c •. --(Mathelatlcs and lts appllcatlons (Sovlet serles) v. 82)Translated frol the Russlan.Includes blbllographlcal references and Index.

Cited by 54 publications

References 0 publications

Однородные Пара-Кэлеровы Многообразия Эйнштейна

Однородные Пара-Кэлеровы Многообразия Эйнштейна

Local algebras of a differential quasigroup

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