Free generators and Hoffman's isomorphism for the two-parameter shuffle algebra
Leonard Schmitz, Nikolas Tapia
TL;DR
This work develops a two-parameter analogue of shuffle and quasi-shuffle Hopf algebras for matrix compositions, introducing free Lyndon generators and a two-parameter Hoffman exponential to produce three distinct isomorphisms between the two-parameter shuffle and quasi-shuffle structures. It connects the two-parameter framework to the classical one-parameter shuffle over connected matrix compositions by a shared generating set and COP (CFL) factorization, framed through $\mathbf{B}_\infty$-algebra ideas. A central contribution is the explicit two-parameter Hoffman map $\Phi$ with an inverse, and a coproduct-preserving isomorphism that yields a transparent proof of the two-parameter bialgebra relation via a Hopf-algebraic viewpoint. These developments offer new tools for two-parameter signature theory and set the stage for higher-parameter generalizations and applications to data-stream analysis and mapping-space signatures.
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.