Non-commutative Hopf algebra of formal diffeomorphisms

Abstract

The subject of this paper are 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.

A One-Parameter Deformation of the Noncommutative Lagrange Inversion Formula

“…This deformation of the noncommutative Faà di Bruno Hopf algebra of [1] has been recently discovered by Foissy ([2]) in his investigation of combinatorial Dyson-Schwinger equations in the Connes-Kreimer algebra. As a Hopf algebra, H 0 is the algebra of noncommutative symmetric functions, and the noncommutative Faà di Bruno Hopf algebra is the case γ = 1.…”

“…However, the analogue of the Faá di Bruno algebra still exists in this context. It is investigated in [1] in view of applications in quantum field theory. In [1], one finds in particular a combinatorial formula for its antipode.…”

“…It is investigated in [1] in view of applications in quantum field theory. In [1], one finds in particular a combinatorial formula for its antipode. This formula is rederived by Novelli and Thibon [10], who also show that it is equivalent to the noncommutative Lagrange formula of Gessel and Pak-Postnikov-Retakh.…”

“…Namely, we compute explicitely the coefficients α I (I n) in their expansion in the basis of complete noncommutative symmetric functions S I . As in the original version, we end up with a sum of products of binomial coefficients over a Catalan set, that is (1) α I = (−1) l(I) ρ∈A l(I)…”

“…25 p. 132, [7] Section 2.4 and [8]). Similarly, for various noncommutative analogues of the Lagrange inversion formula (see [4], [11] and [1]), Novelli and Thibon give in [10] interpretations in terms of noncommutative symmetric functions.…”