On modified Rota-Baxter Hom-Lie algebras

Abstract:

      Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified $r$-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the concept and representations of modified Rota-Baxter Hom-Lie algebras. We then develop the cohomology of modified Rota-Baxter Hom-Lie algebras with coefficients in a suitable representation. As applications, we study formal deformations and abelian extensions of modified Rota-Baxter Hom-Lie algebras in terms of second cohomology groups.