Kernel characterization for general Hopf-algebra-valued signatures

Characterize the kernel of the signature map for general connected graded Hopf-algebra-valued rough paths directly in terms of tree-like equivalence of the original Hopf-algebra-valued paths.

Background

The paper extends its inverse-limit and tree-topological framework from free nilpotent groups to character groups of connected graded Hopf algebras. A general lifting theorem is available, but the authors note that the geometric theory’s kernel characterization does not presently have a general Hopf-algebraic counterpart. Existing results apply only to certain Hopf algebras after an isomorphism to a shuffle Hopf algebra, and that coordinate change obscures the direct meaning of tree-likeness in the original model.

References

To the best of our knowledge no general replacement for \cref{prop:Kernel-signature-characterisation}, i.e.~a characterization of the kernel of the corresponding signature by tree-like equivalence, is known in the literature.

— Metric Geometry of the Signature Group for $p$-Variation Rough Paths  (2609.10875 - Medwed et al., 9 Sep 2026) in Section 5, Remark \ref{rem:hopfrp}, “Transfer to Character Groups”