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Adv. in Math. 200 (2006), 479-524,
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Non-commutative Hopf algebra of formal diffeomorphisms
This paper deals with two Hopf algebras which are the non-commutative
analogues of two different groups of formal power series. The first group
is the set of invertible series with the group law being multiplication
of series, while the second group is the set of formal diffeomorphisms
with the group law being composition of series.
The motivation to introduce these Hopf algebras comes from the study of
formal series with non-commutative coefficients. Invertible series with
non-commutative coefficients still form a group, and we interpret the
corresponding new non-commutative Hopf algebra as an alternative to the
natural Hopf algebra given by the co-ordinate ring of the group, which has
the advantage of being functorial in the algebra of coefficients.
For the formal diffeomorphisms with non-commutative coefficients, this
interpretation fails, because in this case the composition is not
associative anymore. However, we show that for the dual non-commutative
algebra there exists a natural co-associative co-product defining a
non-commutative Hopf algebra. Moreover, we give an explicit formula
for the antipode, which represents a non-commutative version of the
Lagrange inversion formula, and we show that its coefficients are related
to planar binary trees.
Then we extend these results to the semi-direct co-product of the
previous Hopf algebras, and to series in several variables.
Finally, we show how the non-commutative Hopf algebras of formal series
are related to some renormalization Hopf algebras,
which are combinatorial Hopf algebras motivated by the renormalization
procedure in quantum field theory, and to the renormalization functor
given by the double tensor algebra on a bi-algebra.
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