Local algebras of a differential quasigroup

Abstract

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

On Grassmannizable Group 3-Webs

“…However, the nature of the Lie group G, admitting the realization in the real projective space P r+1 described in this section, was not found either in [1] or in [3][4][5][6].…”

“…The converse is also true: the group G of parameters of similarity transformations of the affine space A r−1 is defined by (4), and it generates a Grassmannizable group 3-web GGW.…”

ENGLERT-TYPE SOLUTIONS OF d = 11 SUPERGRAVITY

“…These tensors are called the fundamental tensors of the geodesic loop G α . Note that these tensors are structure constants of the binary-ternary tangent algebra of the geodesic loop [11] (see also [12]). Comparing (11) and (12), we get…”

Spontaneous compactification and nonassociativity

“…To study geodesic loops in affinely connected spaces we may use a method which is usually employed to study the local structure of Lie groups. Despite the lack of associativity in geodesic loops, this method enables us to uniquely define binary [x, y] and ternary (x, y, z) operations in their tangent spaces T e and to construct the local algebras [12,13]. These operations are expressed in terms of the coordinates of the vectors x, y, z ∈ T e as follows:…”