A review on comodule-bialgebras (Q1733963)

This paper is a review of recent uses of comodule-bialgebras in various domains of mathematics, including combinatorics, analysis or quantum field theories. The main examples are based on words (related to Ecalle's mould calculus) or on trees (related to Ecalle's arborified moulds or to B-series). Other examples are based on cycle-free graphs, on posets or on finite topological spaces. It is also explained how these objects are used for the study of regularity structures. The paper ends with an example (based on trees) of a infinite family of comodule-bialgebras, all on the same bialgebra. For the entire collection see [Zbl 1410.60006].