On Engel's Theorem for Leibniz Algebras

Abstract

I give a simpler proof of the generalisation of Engel's Theorem to Leibniz algebras.

An irreducible component of the variety of Leibniz algebras having trivial intersection with the variety of Lie algebras

“…As a generalization of Lie algebras, it is expected that Leibniz algebras share at least some of the fundamental properties of the former structures. To a certain extent, this actually holds, [4,5] albeit strong differences are soon encountered. [6] Possibly the best studied case is that of nilpotent Leibniz algebras, [7,8] as well as the classification problem in low dimensions.…”

“…In analogy, the Engel theorem is generalized for Leibniz algebras, [4] hence providing some criteria for the solubility of Leibniz algebras. As a special case, if L is a solvable non-nilpotent Leibniz algebra, then N R ⊇ L 2 , and hence the nilradical must coincide with L 2 .…”

Leibniz algebras constructed by Witt algebras

“…From this equality together with (7) we get Substituting instead of parameters {i, j, k} the following values (1, −2, 1), (1, −2, −1), (−1, 2, −1), (−1, 2, 1), (1, 2, −1), (−1, −2, 1)…”

“…Considering (9) for i = ±1, j := ±i, k := ∓1, j and applying (7) we conclude γ i,−1 = (i + 1)γ 2,1 , i ∈ Z + \ {1}, γ −i,1 = −(i + 1)γ 2,1 , i ∈ Z + \ {1},…”

On a class of Leibniz algebras

“…Applying the precedent proposition to the adjoint representation (Ad, ad, L) of the Leibniz algebra L and using Engel's Theorem [2], we get the:…”