Smooth Bol Loops and Bol Algebras

Abstract

In this chapter we develop the theory of smooth Bol loops. See [L.V. Sabinin 91b], [L.V. Sabinin, P.O. Miheev 85b, 90]. Our interest in Bol loops was stimulated by their applications to differential geometry and the theory of homogeneous spaces. Our approach uses, in particular, the fact that any local C3-smooth Bol loop (and the corresponding homogeneous space) admits an infinitesimal treatment in terms of finite-dimensional symmetric Lie algebras (g,s) with g = g- +g+, where s ∈ Aut g, s2 = id, g- = {ξ ∈ g; s ξ = — ξ}, g+ = {ξ ∈ g; s ξ = ξ}, and additional subalgebras h in g such that g = g- + h.