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Torsion and curvature in smooth loops. (English) Zbl 0734.53013
Let G be a local Lie loop and \(L(G)=T_ e(G)\) be its tangent algebra at the identity e. The latter carries two operations: multiplication (x,y)\(\to [x,y]\) (called commutator bracket) and a trilinear operation \((x,y,z)\to <x,y,z>\) (called associator bracket) which vanishes if G is a Lie group. These two operations are linked by the Akivis identity: \[ \sum \{sgn(\sigma)<x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)}>:\;\sigma \in S_ 3\}=[[x,y],z]+[[y,z],x]+[[z,x],y]. \] This tangent algebra L(G) forms an Akivis algebra. If G is a Lie group, the Akivis identity is reduced to the classical Jacobi identity.
The authors give a procedure how to associate with a Lie loop G two left (right) canonical connections - they coincide if G is a Lie group. The first connection is curvature-free, and the torsion and curvature associated with the second left canonical connection and the Akivis algebra of the loop G are related as follows: \[ (i)\quad T(X,Y)_ e=[Y_ e,X_ e]\text{ and } (ii)\quad R(X,Y)(Z)_ e=-<X_ e,Y_ e,Z_ e>-<Y_ e,X_ e,Z_ e> \] for all smooth vector fields X, Y and Z. A similar result holds for the second right canonical connection. The authors compare this main result of the paper with the results obtained for the Chern canonical connection associated with a multidimensional three-web [see M. A. Akivis, Sib. Math. J. 17, 3-8 (1976); translation from Sib. Mat. Zh. 17, 5-11 (1976; Zbl 0337.53018); M. Kikkawa, Mem. Fac. Sci., Shimane Univ. 19, 37-50 (1985; Zbl 0588.53014); P. T. Nagy, Publ. Math. 32, 93-99 (1985; Zbl 0586.53005); L. V. Sabinin and P. O. Mikheev, Theory of smooth Bol loops. Texts of lectures. (1985; Zbl 0584.53001)] and for the connections associated with special local loops [see M. Kikkawa, Hiroshima Math. J. 5, 141-179 (1975; Zbl 0304.53037); L. V. Sabinin and P. O. Mikheev, Theory of smooth Bol loops (loc. cit.)].

53A60 Differential geometry of webs
22A99 Topological and differentiable algebraic systems
53B05 Linear and affine connections