Isoclinism in lie algebras with derivations

Abstract

A LieDer pair is a Lie algebra equipped with a derivation. In this paper, we extend the concept of isoclinism to the framework of LieDer pair, which encompasses various generalizations of Lie algebras. By definition, every isomorphism induces an isoclinism, yet the converse generally fails to hold. To investigate this concept, we introduce the notion of a factor set, which serves as a key tool in describing the structure of isoclinism classes. Our principal theorem asserts that any two finite-dimensional Lie Der pairs of equal dimension are isomorphic whenever they are isoclinic. The proof is based on the construction of stem algebras together with a decomposition result valid inside each isoclinism class.