Algèbres De Lie Triple Sans Idempotent

Abstract

Idempotents play an important role in the investigation of nonassociative algebras structure (Albert in Trans Am Math Soc 69:503–527, 1950; Schafer in An introduction to nonassociative algebras Academic Press, New York, 1966; Zhevlakov et al. in Rings that are nearly associative. Academic Press, New York, 1982). However, the existence of such elements is not always guaranteed, specially when dealing with algebras that are defined by polynomial identities. Hence, in many cases, one has to assume the existence of idempotents (Arenas in Comm Algebra 35:675–688, 2007; Osborn in Trans Am Math Soc 16:1114–1120, 1965; Petersson in Math Zeitschr 98:104–118, 1967) in order to investigate algebraic structures via the classical Peirce decomposition. Lie triple algebras appear historically and independently in Osborn (Trans Am Math Soc 16:1114–1120, 1965) and Petersson (Math Zeitschr 97:1–15, 1967). In this paper, we deal mainly with Lie triple algebras which admit no idempotent. We prove that non nil algebras possess either an idempotent or a pseudo-idempotent. In the latter case, it is possible to obtain a Peirce decomposition relatively to the pseudo-idempotent and some relations on Peirce’s components. We notice that these results are analogous to that obtained by Osborn in (Trans Am Math Soc 16:1114–1120, 1965). When a Lie triple algebra has a non-trivial idempotent, we show that its heart is a Jordan subalgebra. Then we give a Lie triple non-conservative algebras classification for dimensions 2 and 3. Finally, we characterize train algebras of rank \(\le 3\) that are Lie triple algebras.