Averaging algebras of any nonzero weight

Abstract: In this paper, we first introduce the notion of a (relative) averaging operator of any nonzero weight $\lambda$. We show that such operators are intimately related to triassociative algebras introduced by Loday and Ronco. Next, we construct a differential graded Lie algebra and a $L_\infty$-algebra whose Maurer-Cartan elements are respectively relative averaging operators of weight $\lambda$ and relative averaging algebras of weight $\lambda$. Subsequently, we define cohomology theories for relative averaging operators and relative averaging algebras of any nonzero weight. Our cohomology is useful to study infinitesimal deformations of relative averaging algebras. Finally, we introduce and study homotopy triassociative algebras and homotopy relative averaging operators of any nonzero weight $\lambda$.