We study complex finite-dimensional Leibniz algebra bimodule over [Formula: see text] that as a Lie algebra module is split into a direct sum of two simple [Formula: see text]-modules. We prove that in this case there are only two nonsplit Leibniz [Formula: see text]-bimodules and we describe the actions.

No 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.

We investigate almost inner derivations of some finite-dimensional nilpotent Leibniz algebras. We show the existence of almost inner derivations of Leibniz filiform non-Lie algebras differing from inner derivations, we also show that the almost inner derivations of some filiform Leibniz algebras containing filiform Lie algebras do not coincide with inner derivations

Recall that in [12] it is obtained the criteria solvability of the equationZp and Qp for p > 3. Since any p-adic number x has a unique form x = p k x * , where x * ∈ Z * p and k ∈ Z, in [12] it is also shown that from the criteria in Z * p it follows the criteria in Zp and Qp. In this paper we provide the algorithm of finding the solutions of the equation x 3 + ax = b in Z * 3 with coefficients from Q 3 .