WIAS Preprint No. 3087, (2024)

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


  • Schmitz, Leonard
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492


  • Signatures for images, matrix compositions, Hopf algebras, Lyndon words, two-parameter shuffles, quasi-shuffles




Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from B-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.

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