CHARACTERIZATION OF DIFFERENTIATIONS IN SOME ALGEBRAS.

Abstract

Abstract: This article investigates the characterization of derivations and more general differentiations — including Jordan derivations, generalized derivations, and δ-derivations — in several classes of algebras: associative algebras, Lie algebras, Leibniz algebras, and nilpotent algebras. We develop a unified framework that allows one to identify when every differentiation of a given algebra is a derivation, and we determine conditions under which the derivation algebra coincides with the full differentiation algebra. Using cohomological methods, spectral techniques, and direct algebraic arguments, we establish new characterization theorems for low-dimensional nilpotent Lie algebras and for a broad family of semisimple associative algebras.