Bol transformation quasigroups and three-webs defined by them

G. A. Tolstikhina

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 3, pp. 81-86

Résumé

The notions of a local smooth quasigroup and a quasigroup of transformations are natural generalizations of the notions of the Lie group and the Lie transformation group. We define the quasigroup of transformations as an action $f$ of the local smooth $q$-dimensional quasigroup $Q(*)$ on the smooth $p$-dimensional manifold $Y$ $(1\leq p\leq q)$ given by a smooth function $$ f\colon Q\times Y\to Y,\quad z=f(a,y),\quad a\in Q,\quad y,z\in Y. $$ On the other hand, the equation $z=f(a,y)$ defines the three-web $QW(p,q,q)$ formed by a foliation of $p$-dimensional leaves $a=\mathrm{const}$ and two foliations of $q$-dimensional leaves $y=\mathrm{const}$ and $z=f(a,y)=\mathrm{const}$ on the manifold $Q\times Y$. Thus, we can use the three-web theory methods to study different classes of smooth local quasigroups of transformations. In the present paper, we investigate Bol quasigroups of transformations characterized by some condition on the function $f$.

Détail

Mots-clés : quasigroup, quasigroup of transformations, Bol quasigroup
Keywords: three-web, Bol three-web, three-web configuration, core of Bol three-web, locally symmetric space structure.