Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight

Abstract

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ, which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an $L_{\infty }\left[1\right]$-algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ, the corresponding twisted $L_{\infty }\left[1\right]$-algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version.