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Non-commutative Hopf algebra of formal diffeomorphisms. (English) Zbl 1133.16025
The group $$G^{\text{inv}}$$ of formal power series $$f$$ with constant term 1 and complex coefficients under inversion has a coordinate Hopf algebra of polynomials in $$b_1,b_2,\dots$$, where $$b_n(f)=(1/n!)\times$$ the evaluation of the $$n$$-th derivative of $$f$$ at $$0$$. The coproduct on $$b_n$$ is the sum of $$b_k\otimes b_{n-k}$$ from $$k=0$$ to $$k=n$$ ($$b_0=1$$). The antipode $$S$$ with $$\langle S(b_n),f\rangle=\langle b_n,f^{-1}\rangle$$ is thus defined recursively. This Hopf algebra is commutative and cocommutative. Replacing the complexes by an algebra $$A$$ still makes $$G^{\text{inv}}(A)$$ a group, not necessarily commutative. One can associate two Hopf algebras to $$G^{\text{inv}}(A)$$. One is commutative but not necessarily cocommutative, and depends on $$A$$. The other, denoted $$H^{\text{inv}}$$, is cocommutative but not necessarily commutative, and does not depend on $$A$$ (i.e., it is functorial in $$A$$). $$G^{\text{inv}}(A)$$ cannot be reconstructed from $$H^{\text{inv}}$$ in the usual way using convolution products, but another recovery method is given using the free product of $$H^{\text{inv}}$$ with itself.
Most of the paper is devoted to the non-commutative Hopf algebras coming from formal diffeomorphisms with constant term $$0$$ and linear term $$x$$. When the coefficients are complex numbers, this is a group $$G^{\text{dif}}$$ under composition. The coordinate algebra of $$G^{\text{dif}}$$ is the (commutative) polynomial algebra in $$a_1,a_2,\dots$$, where $$a_n$$ is $$1/(n+1)!$$ times evaluation of the $$(n+1)$$-st derivative at $$0$$. The coproduct can be given by the usual duality, but the authors prefer a method using residues of Laurent series and the generating series $$A(x)$$ with constant term $$0$$, linear term $$x$$, and $$a_n$$ as coefficient of $$x^{n+1}$$. This involves interpreting $$1/(z-f(x))$$ as the infinite sum from $$n=0$$ of $$(f(x)^n)z^{-n-1}$$. When the complexes are replaced by an algebra $$A$$, composition of formal diffeomorphisms is still defined, but is not necessarily associative. Nevertheless, the authors associate a Hopf algebra $$H^{\text{dif}}(A)$$, neither commutative nor cocommutative, to $$G^{\text{dif}}(A)$$. As an algebra, it is the free algebra on $$a_1,a_2,\dots$$. The coproduct on $$a_n$$ is defined by residues, by saying that the coproduct on $$A(x)$$ is the residue of $$A(z)\otimes 1/(z-A(x))$$, and then extending multiplicatively to products of the generators. The associativity of the coproduct is proved using residues. An explicit description of the coproduct is given using polynomials $$((Q^m)^{(n)})(a=(a_1,\dots,a_m))$$, the sum of all products of $$n$$ $$a_j$$’s, where the $$j$$’s sum to $$m$$ (and are non-negative). Then the coproduct on $$a_n$$ is the sum from $$k=0$$ to $$k=n$$ of $$a_k\otimes ((Q^{(n-k)})^{(k)})(a)$$. $$(H^{\text{dif}})(A)$$ is a graded (degree of $$a_{j_1},a_{j_2},\dots,a_{j_n}$$ is $$j_1+j_2+\cdots+j_n$$) connected Hopf algebra, so the antipode $$S$$ can be described by a standard recursion formula. The authors give an explicit description of $$S$$ by giving a closed form of $$S(a_n)$$ which involves heavy use of binomial coefficients. This can be regarded as a non-commutative Lagrange inversion formula, since in the commutative case $$S$$ can be described by Lagrange inversion. The coefficients in the antipode formula can be used to label certain trees.
The authors consider semi-direct coproducts of various Hopf algebras. Since $$G^{\text{dif}}$$ acts (by composition) on $$G^{\text{inv}}$$, one has the semi-direct product group of $$G^{\text{dif}}$$ with $$G^{\text{inv}}$$. To this the authors associate a Hopf algebra (neither commutative nor cocommutative) which is the appropriate product of the coordinate Hopf algebra of $$G^{\text{dif}}$$ with $$H^{\text{inv}}$$. However, considering non-commutative coefficients $$A$$, $$(G^{\text{dif}})(A)$$ is not a group. But the authors still impose on the appropriate product of $$(H^{\text{dif}})(A)$$ and $$(H^{\text{inv}})$$ the structure of an algebra and a coalgebra (but not a bialgebra).
Finally, some relations to the renormalization functor of quantum field theory are given, as well as a brief indication of how to extend their results on formal diffeomorphisms to series in several variables.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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