Diassociative family algebras and averaging family operators

Abstract

Algebraic structures relative to a semigroup Ω (also called family algebraic structures) first appeared in the work of renormalization in quantum field theory. In this paper, we first consider Ω-diassociative algebras and diassociative family algebras as family analogues of diassociative algebras. We define the cohomologies of these algebras generalizing the cohomology of diassociative algebras introduced by Frabetti. We also introduce (relative) averaging family operators as the family analogue of averaging operators and show that they induce diassociative family algebras and Ω-diassociative algebras. Next, we define the cohomology of a (relative) averaging family operator as the cohomology of the induced Ω-diassociative algebra with coefficients in a suitable representation. Finally, as an application of our cohomology, we study formal one-parameter deformations of relative averaging family operators.