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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.

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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