Free generators and Hoffman's isomorphism for the two-parameter shuffle algebra

TL;DR

This paper extends signature transforms to multi-parameter data, introducing new algebraic isomorphisms and a Hoffman exponential for two-parameter shuffle algebras, linking them to classical shuffle structures.

Contribution

It introduces three classes of isomorphisms for two-parameter shuffle and quasi-shuffle algebras using free Lyndon generators and a novel Hoffman exponential.

Findings

Established isomorphisms between two-parameter shuffle and quasi-shuffle algebras

Connected two-parameter shuffle algebra to classical shuffle algebra via Hopf algebraic methods

Developed a two-parameter Hoffman exponential for algebraic transformations

Abstract

Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from $\mathbf{B}_\infty$-algebras and a novel two-parameter Hoffman exponential, we provide three classes of isomorphisms between the underlying two-parameter shuffle and quasi-shuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, one-parameter) shuffle algebra over the extended alphabet of connected matrix compositions.