Controlling algebra and homotopy theory of modified Rota–Baxter algebras

Develop the controlling algebra and homotopy analog of modified Rota–Baxter algebras, thereby resolving the two challenges that remained open in the existing theory of modified Rota–Baxter algebras.

Background

The introduction reviews prior work on the cohomology of modified Rota–Baxter algebras and distinguishes it from the controlling-algebra and homotopy-theoretic aspects. Before presenting the results of the paper, it identifies these two aspects as unresolved challenges in the literature.

The paper subsequently claims to address these challenges in the special case of modified Rota–Baxter algebras by constructing an $L_\infty[1]$-structure and introducing homotopy modified Rota–Baxter algebras. Thus, the passage is an explicitly stated open problem at the point where it is introduced, even though the paper presents results aimed at resolving it.