Combinatorial Deformations of Algebras: Twisting and Perturbations

TL;DR

This paper explores algebraic deformations using twisted shifted dual laws, interpreting parameters as tensor structure deformations and coproduct perturbations, with applications to Feynman-Bender diagram algebras.

Contribution

It provides a clear interpretation of the parameters involved in algebraic deformations and systematically details the constructions of twisted and perturbed algebraic structures.

Findings

Deformation parameters interpreted as tensor and coproduct modifications

Systematic construction of twisted shifted dual laws

Application to algebra of Feynman-Bender diagrams

Abstract

The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.