Canonical connections of homogeneous Lie loops and 3-webs.

*(English)*Zbl 0588.53014A quasigroup (G,\(\mu)\) with the multiplication \(xy=\mu (x,y)\) for x,y\(\in G\) is called a loop if it has a unit element e in G. Let \(L_ x\) be the left translation of a loop G by \(x\in G\). G has the left inverse property (L.I.P.) if for every \(x\in G\) there exists a two-sided inverse \(x^{- 1}\) such that \(L_{x^{-1}}L_ x=I_ G\) (the identity map on G). Such a loop is called an L.I.P. loop. An L.I.P. loop is said to be homogeneous if all its left inner mappings given for any a,b\(\in G\) by \(L_{a,b}=L^{-1}_{ab} L_ a L_ b\) are automorphisms of G.

Let (G,\(\mu)\) be a differentiable loop of dimension r with unit element e. A differentiable 3-web of (G,\(\mu)\) on \(G\times G\) is given by the following three families of r-dimensional submanifolds of \(G\times G:\) vertical lines \(\{\) \(g\}\) \(\times G\), horizontal lines \(G\times \{g\}\) and transversal lines \(\{\) (u,v)\(| \mu (u,v)=g\}\), \(g\in G\). S. S. Chern [Abh. Math. Semin. Univ. Hamb. 11, 333-358 (1936; Zbl 0013.41802)] introduced an affine connection of an r-dimensional 3-web given on a (2r)-dimensional differentiable manifold. This connection is invariant under differentiable equivalences of 3-webs. The author calls it the Chern connection.

In the paper under review the author recalls the formulas for the torsion and curvature tensors of the Chern connection and applies them to 3-webs of differentiable L.I.P. loops and differentiable homogeneous loops (homogeneous Lie loops). He establishes the interrelation between the canonical connection of homogeneous Lie loops and the bilinear operations on the tangent spaces at the unit elements, induced by multiplications and left inner mappings of the loops. On this way the author finds the relation of tangent Lie triple algebras of homogeneous Lie loops and their Akivis algebras [see M. A. Akivis, Sib. Math. J. 17, 3-8 (1976); translation from Sib. Math. Zh. 17, 5-11 (1976; Zbl 0337.53018), or K. H. Hofmann and K. Strambach, The Akivis algebra of a homogeneous loop, Darmstadt: Preprint. Fachbereich Mathematik Technische Hochschule Darmstadt; No. 908 (1985)]. The last result can be also found in the above mentioned preprint.

Let (G,\(\mu)\) be a differentiable loop of dimension r with unit element e. A differentiable 3-web of (G,\(\mu)\) on \(G\times G\) is given by the following three families of r-dimensional submanifolds of \(G\times G:\) vertical lines \(\{\) \(g\}\) \(\times G\), horizontal lines \(G\times \{g\}\) and transversal lines \(\{\) (u,v)\(| \mu (u,v)=g\}\), \(g\in G\). S. S. Chern [Abh. Math. Semin. Univ. Hamb. 11, 333-358 (1936; Zbl 0013.41802)] introduced an affine connection of an r-dimensional 3-web given on a (2r)-dimensional differentiable manifold. This connection is invariant under differentiable equivalences of 3-webs. The author calls it the Chern connection.

In the paper under review the author recalls the formulas for the torsion and curvature tensors of the Chern connection and applies them to 3-webs of differentiable L.I.P. loops and differentiable homogeneous loops (homogeneous Lie loops). He establishes the interrelation between the canonical connection of homogeneous Lie loops and the bilinear operations on the tangent spaces at the unit elements, induced by multiplications and left inner mappings of the loops. On this way the author finds the relation of tangent Lie triple algebras of homogeneous Lie loops and their Akivis algebras [see M. A. Akivis, Sib. Math. J. 17, 3-8 (1976); translation from Sib. Math. Zh. 17, 5-11 (1976; Zbl 0337.53018), or K. H. Hofmann and K. Strambach, The Akivis algebra of a homogeneous loop, Darmstadt: Preprint. Fachbereich Mathematik Technische Hochschule Darmstadt; No. 908 (1985)]. The last result can be also found in the above mentioned preprint.

Reviewer: V.V.Goldberg