Two interacting coordinate Hopf algebras of affine groups of formal series on a category. (English) Zbl 1342.16029
Let \(R\) be any ring. There are two group structures on formal series in one variable over \(R\); the first one is \(G=1+xR[[x]]\) with the product, the second one is \(H=x+x^2R[[x]]\), 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.
Reviewer: Loïc Foissy (Calais)
MSC:
| 16T05 | Hopf algebras and their applications |
| 16T30 | Connections of Hopf algebras with combinatorics |
| 14L17 | Affine algebraic groups, hyperalgebra constructions |
| 20J99 | Connections of group theory with homological algebra and category theory |
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
Keywords:
locally finite categories; coordinate Hopf algebras of affine groups; formal series; smash coproductsReferences:
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