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

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