Cohomology of Hom-Associative Algebras in Loday–Pirashvili Category with Applications

We introduce the concept of Hom-associative algebra structures in Loday–Pirashvili category. The cohomology theory of Hom-associative algebras in this category is studied. Some applications on deformation and abelian extension theory are given. We also introduce the notion of Nijenhuis operators to describe trivial deformations. It is proved that equivalent classes of abelian extensions are one-to-one correspondence to the elements of the second cohomology groups.