Leibniz algebras constructed by Witt algebras

Abstract

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.

Leibniz algebras as non-associative algebras

“…Then one can apply Theorem 2.3 in conjunction with [15, Proposition 1.1] to determine when the hemisemidirect products W ⋉ ℓ V α,β are simple (see [5, Theorems 3.1 and 3.2] for the "only if"-part). 5 Proposition 2.4. The hemi-semi-direct product W ⋉ ℓ V α,β is simple if, and only if, α ∈ Z or α ∈ Z and β = 0, 1.…”

“…Of course, the simplicity of these algebras is an immediate consequence of the "if"-part of Theorem 2.3. 5 In Remark 1 on p. 2062 of [5] the authors claim (without proof) that the Leibniz algebras they consider are simple. On the other hand, more generally, they also investigate non-split Leibniz extensions of W by the anti-symmetrization of V α,β which we will only discuss in the second part of this paper (see also [7] and [9] for non-split Leibniz extensions of simple Lie algebras by irreducible anti-symmetric Leibniz bimodules).…”

“…In part, the missing hypothesis in Example 2.3 of [18] and in Example 1 at the end of Section 2 of [12] as well as the lack of any proof have been the motivation for writing this paper, and especially, for proving the simplicity criterion Theorem 2.3. Moreover, as far as I can see, most of the existing papers on (semi-)simple Leibniz algebras (for example, [18], [12], [4], [5], [2]) do not use a conceptual approach, and the latter is exactly what we will present here. In particular, in Section 2 we discuss how our results can be employed to prove some of the statements in previous papers on (semi-)simple Leibniz algebras or sometimes how to complete the proofs of these statements.…”

On Leibniz cohomology

“…Remark. Very recently, Camacho, Omirov, and Kurbanbaev also proved that the second adjoint Leibniz cohomology of W vanishes (see [6,Theorem 4]) by explicitly showing that every adjoint Leibniz 2-cocycle (resp. Leibniz 2-coboundary) is an adjoint Chevalley-Eilenberg 2-cocycle (resp.…”