Remarks on the construction of Lie-Yamaguti algebras from Leibniz algebras (Q2901400)

There is the well known correspondence between Lie groups and Lie algebras. For smooth loops, that is generalized Lie groups (associative loops are groups), the tangent algebras are Sabinin algebras; these algebras first were called hyper--algebras [\textit{P. O. Mikheev} and \textit{L. V. Sabinin}, Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 357--430 (1990; Zbl 0721.53018)]. In general Sabinin algebras possess infinitely many operations of different signatures. In 1973 Akivis for a smooth loop constructed an algebra with just one binary operation, coming from the torsion, just one ternary operation, coming from the curvature and one relation apart from the anti--commutativity of the binary operation [\textit{M. A. Akivis}, Sib. Math. J. 17, 3--8 (1976); translation from Sibir. Mat. Zh. 17, 5--11 (1976; Zbl 0337.53018)]. Originally Akivis called these algebras W--algebras. Another way to obtain Akivis algebras consists in considering commutator and associatior of an arbitrary non-associative algebra as operations.NEWLINENEWLINEFor many important classes of smooth loops the Sabinin algebra is an Akivis algebra. For example for smooth Moufang loops one has Malcev algebras, for diassociative loops on has binary Lie algeras, for Bol loops one has Bol algebras and for smooth loops equivalent to reductive spaces one has Lie-Yamaguti algebras.NEWLINENEWLINE\textit{J.-L. Loday} introduced the so--called Leibniz algebras [Enseign. Math., II. Sér. 39, No. 3--4, 269--293 (1993; Zbl 0806.55009)]. Such an algebra is close to a Lie algebra but is not anti--commutative. The author considers the Akivis algebras obtained from (left) Leibniz algebras. He discusses connections with Lie Triple Systems and Lie-Yamaguti algebras. An algebra is called Malcev-admissible if its commutator algebra is a Malcev algebra. The Leibniz algebras which are Malcev-admissible algebras are characterized in Lemma 4.2 and Proposition 4.6. Besides this short note offers a useful introduction to algebraical generalized Lie theory, an area of investigation where recently great progress has been made.