Algèbres De Lie Triple Sans Idempotent

Derivations and dimensionally nilpotent derivations in Lie triple algebras

“…Definition 2.2. A pseudo-idempotent of A is a non-zero element e such that there is t = 0 in L satisfying e 2 = e + t and et = 1 2 t. Theorem 2.3 ( [2]). Every Lie triple non nil-algebra contains either a non-zero idempotent, or a pseudo-idempotent.…”

“…So e 0 e 3 = 12 e 3 and a 3,4 = 0 ⇒ γ 124 = 0 because of (4.3) and finally γ 113 = a 2,3 = 0. In the same way e2 2 A e 0 (0) or e 2 ∈ A e 0 (1/2), because A e 0 (1/2) 2 ⊆ A e 0 (0) + A e 0 (1) and A e 0 (0) 2 ⊆ A e 0 (0). But e 0 e 2 = 1 2 e 2 + a 2,4 e 4 ⇒ e 2 ∈ A e 0 (1/2) so a 2,4 = 0 = γ 114 because of (4.1).…”

Evolution algebras that are almost Jordan

“…(1.1) Almost Jordan algebras have been studied by Osborn [8], Petersson [11], Sidorov [12,13], Hentzel and Peresi [7], Bayara and al. [2,3]. In [7], the authors showed the existence of an ideal L, generated by the associators (x 2 , x, x), such that L 2 = 0, and then established that A/L is a Jordan algebra.…”

“…In [7], the authors showed the existence of an ideal L, generated by the associators (x 2 , x, x), such that L 2 = 0, and then established that A/L is a Jordan algebra. In [3], the authors showed that every non-nil almost Jordan algebra contains either a non-zero idempotent or a pseudo-idempotent. In [5], the authors studied derivations of almost Jordan algebras, distinguishing between those containing non-zero idempotents and those containing pseudo-idempotents.…”

Derivations and dimensionally nilpotent derivations in Lie triple algebras

“…[2], Proposition 4.3). Let L = L e (1)⊕L e (1/2)⊕L e (0) and A = A e (1)⊕A e (1/2)⊕A e (0) be the respective Peirce decomposition of L and A, relative to the pseudo-idempotent e, satisfying e 2 = e + t with t ∈ L 1/2 fixed.…”