The $L_{\infty }$-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 et al., we construct an $L_{\infty }[1]$-algebra whose Maurer-Cartan elements are precisely relative Rota-Baxter algebras. By a standard twisting, we define a new $L_{\infty }[1]$-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 coboundary 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.

Version updated; 25 pages

References in corpus (4)

• Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators
• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Review of deformation theory II: a homotopical approach
• Review of deformation theory I: Concrete formulas for deformations of algebraic structures

Cited by in corpus (2)

• Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras