Abelian Extensions and Crossed Modules of Modified λ-Differential Left-Symmetric Algebras

In this paper, we define the cohomology of a modified -differential left-symmetric algebra with coefficients in a suitable representation. We also introduce the notion of modified -differential left-symmetric 2-algebra. We classify linear deformations and abelian extensions of modified -differential left-symmetric algebras using the second cohomology group and classify skeletal modified -differential left-symmetric 2-algebra using the third cohomology group as our propose cohomology applications. Moreover, we prove that strict modified -differential left-symmetric 2-algebras are equivalent to crossed modules of modified -differential left-symmetric algebras.