The L∞-deformations of associative Rota–Baxter algebras and homotopy Rota–Baxter operators

Abstract

A relative Rota–Baxter algebra is a triple ( A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra ( A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra ( A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

A cohomological study of modified Rota-Baxter algebras

“…As a biproduct of the above two cochain complexes, we construct a new complex defining the cohomology of the modified Rota-Baxter algebra (A, R) with coefficients in the bimodule (M, S). We compare our cohomology of modified Rota-Baxter algebras with the cohomology of Rota-Baxter algebras recently introduced in [13,33]. More precisely, we show that the cohomology of a Rota-Baxter algebra (A, P ) is isomorphic to the cohomology of the modified Rota-Baxter algebra (A, R = λ id + 2P ).…”

“…This cohomology is obtained as a biproduct of the Hochschild cohomology of the underlying associative algebra A and the Hochschild cohomology of the induced associative algebra A R . We also compare our cohomology of a modified Rota-Baxter algebra with the cohomology of a Rota-Baxter algebra defined in [13,33].…”

“…See [6,7,14,17] for more details about deformations of various algebras and morphisms between them. Recently, various authors have considered formal deformations of Rota-Baxter operators, Rota-Baxter algebras and modified r-matrices motivated by the deformations of classical r-matrices [12,13,[23][24][25]32,33]. In this paper, we consider the formal deformation theory of a modified Rota-Baxter algebra (A, R).…”

Deformations and cohomology theory of weighted Rota-Baxter $3$-Lie algebras

“…They further determined the L ∞ -algebra that controls deformations of a relative Rota-Baxter Lie algebra and introduced the notion of a homotopy relative Rota-Baxter Lie algebra of weight zero. Das and Misha [10] developed the corresponding weight zero cohomology theory for relative Rota-Baxter associative algebras and Das also introduced cohomology theory for relative Rota-Baxter associative algebras and relative Rota-Baxter Lie algebras of arbitrary weight [8,9]. Wang and Zhou [24,25] defined cohomology theory for relative Rota-Baxter associative algebras of arbitrary weights, determined the underlying L ∞algebra and also showed that the dg operad of homotopy Rota-Baxter associative algebras is the minimal model of that of Rota-Baxter associative algebras.…”

A unified extension theory of Rota-Baxter algebras, dendriform algebras, and a fundamental sequence of Wells

“…Algebraic structures are better understood by their extension theory and cohomology theory. Recently, such theories for Rota-Baxter algebras and dendriform algebras are studied in [10][11][12][13]22,24,31]. However, in the above-mentioned references, only the abelian extension theories are considered.…”