Narrow Results

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

Corrole is a tetrapyrrolic ring-contracted porphyrin analog with one direct pyrrole-pyrrole link. The corrole macrocycle is electron-rich and easily stabilized in its one electron oxidized radical state, especially when coordinated to some specific metal ions. The corrole radicals and their complexes have demonstrated excellent air stability, amphoteric redox properties, and diversified spin configurations. Herein, we summarize the recent developments in the synthesis and properties of various corrole radicals, including $\pi$-monoradicals, antiferromagnetically and ferromagnetically coupled metal radicals, and $\pi$-diradicals. The regulation of spin, redox, and optical properties of corrole radicals by modifications to their periphery substituents, coordinated metals, and axial ligands is highlighted, which demonstrates their promising application prospects in spintronics, electronics, and energy-conversion materials.