On the second cohomology group of simple Leibniz algebras

TL;DR

This paper investigates the second cohomology group of simple Leibniz algebras, proving triviality in specific cases, which advances understanding of their algebraic structure and cohomological properties.

Contribution

It provides new results on Leibniz 2-cocycles for simple Leibniz algebras and establishes the triviality of their second cohomology in certain cases.

Findings

Proves general results on Leibniz 2-cocycles for simple Leibniz algebras.

Establishes the triviality of the second Leibniz cohomology for simple Leibniz algebras with associated Lie algebra isomorphic to sl_2.

Advances understanding of the cohomological structure of simple Leibniz algebras.

Abstract

In this paper we prove some general results on Leibniz 2-cocycles for simple Leibniz algebras. Applying these results we establish the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose associated Lie algebra is isomorphic to $\mathfrak{sl}_2$.