Local algebras of a multidimensional three-web

Akivis, Maks A.

doi:10.1007/bf00969285

Cited by 46 publications

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“…We denote this loop by l A and call it the local loop of the quasigroup Q. The loop l A is also a local loop of the three-web W defined on the manifold M = Q × Q (see [4]). In (1) and in what follows, we denote by the "•" the new operation in the quasigroup Q defined in a neighborhood of the point A by formula (1).…”

Section: Local Loopsmentioning

confidence: 99%

“…Under these transformations, the coefficients of decompositions (4) are not transformed according to the tensor law. However, the quantities a i jk and b i jkl , which are expressed in terms of the coefficients of (4), are tensors.…”

Section: Commutators and Associators Of Local Loopsmentioning

confidence: 99%

“…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. Note also that some particular cases of Akivis algebras were considered by Yamaguti (see [57] and [58]) when he studied symmetric spaces.…”

Section: Local Algebras Of Differentiable Quasigroupsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], local algebras of multidimensional three-webs were defined and investigated. These local algebras are also local algebras of differentiable quasigroups connected with three-webs.…”

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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Section: Local Loopsmentioning

confidence: 99%

Section: Commutators and Associators Of Local Loopsmentioning

confidence: 99%

Section: Local Algebras Of Differentiable Quasigroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [4], local algebras of multidimensional three-webs were defined and investigated. These local algebras are also local algebras of differentiable quasigroups connected with three-webs.…”

Section: Introductionmentioning

confidence: 99%

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