Deformations and homotopy theory of Rota-Baxter algebras of any weight

Abstract

This paper studies formal deformations and homotopy theory of Rota-Baxter algebras of any weight. We define an L ∞ -algebra, which controls simultaneous deformations of associative products and Rota-Baxter operators. As a consequence, we develop a cohomology theory of Rota-Baxter algebras of any weight and justify it by interpreting lower degree cohomology groups as formal deformations and abelian extensions. The notion of homotopy Rota-Baxter algebras is introduced and it is shown that the operad governing homotopy Rota-Baxter algebras is a minimal model of the operad of Rota-Baxter algebras.

Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators

“…In [30] Wang and Zhou defined the cohomology of a weighted Rota-Baxter associative algebra with coefficients in a Rota-Baxter bimodule. In this subsection, we show that our cohomology is related to the cohomology of [30] by suitable skew-symmetrization.…”

“…We denote this associative algebra by A R . Moreover, if (M, R) is a Rota-Baxter bimodule over the λ-weighted Rota-Baxter associative algebra (A, R), then it has been observed in [30] that M carries a bimodule structure over the associative algebra A R with left and right actions…”

“…This motivates the present author to study cohomology and deformations of weighted Rota-Baxter operators on both associative and Lie algebras [10]. However, the simultaneous deformations of (associative) algebras and weighted Rota-Baxter operators are considered in a recent preprint of Wang and Zhou [30]. More precisely, they considered weighted Rota-Baxter associative algebras and define cohomology of them with coefficients in a suitable Rota-Baxter bimodule.…”

“…Our aim in this paper is to apply the approach of [30] to weighted Rota-Baxter Lie algebras. More precisely, we first consider representations of weighted Rota-Baxter Lie algebras and provide various constructions of representations.…”

Deformations, cohomologies and integrations of relative difference Lie algebras

“…Therefore, it is natural to extend the study of deformations and cohomologies of relative difference operators and LieDer pairs to the context of relative difference Lie algebras. Another motivation for such a study due to that difference operators are formal inverses of Rota-Baxter operators, while the deformation and cohomology theories of the latter are of much interest recently [1,2,14,24,28]. There is a general principle for the deformation theory of an algebraic structure proposed by Deligne, Drinfeld and Kontsevich: on the one hand, for a given algebraic structure, there should be a differential graded Lie algebra (or an L ∞ -algebra, called the controlling algebra) whose Maurer-Cartan elements characterize deformations of this object.…”