The theory of smooth hyporeductive and pseudoreductive loops

For Bol loops and reductive loops one can construct an infinitesimal theory similar to Lie group theory by associating with a loop a certain binary-ternary algebra with identities, namely the Bol algebra for a Bol loop and the triple Lie algebra for a reductive loop. It is also possible to construct a proper infinitesimal theory for hyporeductive loops that generalize Bol loops and reductive loops. This can be achieved by associating with these loops a hyporeductive algebra with two binary and one ternary operation and the system of identities [see L. V. Sabinin, in {it Variational methods in modern geometry (Russian)}, 50--69, Univ. Druzhby Narodov, Moscow, 1990; [msn] MR1130903 (92g:22008) [/msn]; Dokl. Akad. Nauk SSSR {bf 314} (1990), no.~3, 565--568; [msn] MR1094021 (92d:22002) [/msn]; in {it Webs and quasigroups (Moscow, 1989)}, 129--137, Tver. Gos. Univ., Tverʹ, 1991; see [msn] MR1140959 (92f:53003) [/msn]]. The latter algebra generalizes both the Bol algebra and the triple Lie algebra. In the paper under review the author constructs the infinitesimal theory for smooth hyporeductive and pseudoreductive loops.