Two interacting coordinate Hopf algebras of affine groups of formal series on a category.
Let \(R\) be any ring. There are two group structures on formal series in one variable over \(R\); the first one is \(G=1+xRx\) with the product, the second one is \(H=x+x^2Rx\), with the composition. Moreover, \(H\) acts on \(G\) giving a semidirect product. These three groups are affine, so have a Hopf algebra of coordinates. This is generalized to locally finite categories. If \(C\) is such a category, it admits a large algebra, with a Cauchy-type product, and the set \(G\) of formal series in this large algebra such that the constant term is \(1\) is a group for the product. Under technical conditions, a group \(H\) of substitutions also exists. It is proved that both of these groups are affine, so have a coordinate Hopf algebra, which are free commutative algebras. Moreover, the action of \(H\) over \(G\), inducing a semidirect product, induces, at the level of their coordinate Hopf algebras, a smash coproduct.
coordinate Hopf algebras of affine groups
scientific article; zbMATH DE number 6519782
scientific article