Hopf algebras, renormalization and noncommutative geometry. (Q2710280)

This paper deals with a close relation between \({\mathcal H}_R\), the Hopf algebra raised by the combinatorics of the subtraction procedure inherent to perturbative renormalization [\textit{D. Kreimer}, Adv. Theor. Math. Phys., 2, 303--334 (1998; Zbl 1041.81085) hereafter refered to as [1]] and \({\mathcal H}_T\), the Hopf algebra appeared in the study of transverse index theorem [\textit{A. Connes} and \textit{H. Moscovici}, Commun. Math. Phys., 198, 199--246 (1998; Zbl 0940.58005) hereafter refered to as [2]]. Explanation is mainly focussed to the definition of \({\mathcal H}_R\) and its application to renormalization theory.NEWLINENEWLINE The computation of the local index formula for transversely hypoelliptic operators is governed by a Hopf algebra \({\mathcal H}_T\) associated to each integer codimension (cf. [2]). In Section 1, the definition of \({\mathcal H}_T\) for the case of dimension one is reviewed. It is the linear span of \(Y,X, \delta_n, n\geq 1\) with the relations and coproduct \(\Delta\) NEWLINE\[NEWLINE\begin{aligned} [Y,X] & =[Y,\delta_n]=n \delta_n,[\delta_n,\delta_m] =0,\;[X,\delta_n]= \delta_{n+1},\\ \Delta Y & =Y \otimes 1+1 \otimes Y,\;\Delta X=X\otimes 1+1\otimes X+\delta_1 \otimes Y,\\ \Delta\delta_1 & =\delta_1\otimes 1+1 \otimes \delta_1,\;\Delta(h_1,h_2)=\Delta h_1\Delta h_2.\end{aligned}NEWLINE\]NEWLINE It is shown that the Hopf algebra \({\mathcal H}^1\), as inductive limit of \({\mathcal H}_n\), the Hopf algebra consisting of polynomials of \(\delta_1, \dots, \delta_n\), is the dual of the enveloping algebra \({\mathcal U}({\mathcal A}^1)\), where \({\mathcal A}^1\) is the Lie algebra of formal vector fields on the line vanishing at order 2 at 0. Generators of \({\mathcal A}^1\) are \(Z_k=\frac {x^{k+1}} {(k+1)!} \frac{\partial} {\partial x}\) which relate to \(\delta_k\) by \(\langle Z_k,P \rangle= (\frac{\partial P} {\partial \delta_k}) (0)\). Then adjoining \(Z_0=x\frac{\partial}{\partial x}\) and \(Z_{-1}= \frac {\partial} {\partial x}\) to \({\mathcal A}^1\), a Lie algebra \({\mathcal A}\), which is used later (Section 3. Theorem 8) is constructed. Algebraic rules of \({\mathcal H}_T\) are the expressions of the group law of composition of diffeomorphisms of \(\mathbb{R}\) in terms of the coordinate \(\delta_n\) which are given by the Taylor expansion of \(-\log(\psi' (x))\) at \(x=0\). The antipode in \({\mathcal H}_T\) is, modulo a change of variable, the same as the operation of inversion of a formal power series under the composition law. These are stated as Theorem 8 without proof (cf. [2]).NEWLINENEWLINE\({\mathcal H}_R\) is the algebra of polynomials over \(\mathbb{Q}\) in rooted trees. A rooted tree \(t\) is a connected and simply-connected set of oriented edges and vertices such that there is one distinguished vertex with no incoming edge. This vertex is called the root of \(t\). Every edge connects two vertices. The fericity \(f(v)\) of a vertex \(v\) is the number of edges outgoing from \(v\). Before to define \({\mathcal H}_T\), rooted trees relate to the subtraction of the divergent subintegrals (renormalization) in the calculation of divergent integral, is explained taking NEWLINE\[NEWLINEx_t(c)= \int^\infty_0 \frac{1}{y_1+c} \prod^m_{i=2} \frac{1} {y_i+y_{j(i)}} y_m^{-\varepsilon} \,dy_1,\;c>0,NEWLINE\]NEWLINE as example. The integrals NEWLINE\[NEWLINE x_{t_1} (c)= \int^\infty_0 \frac{y^{-\varepsilon}} {y+c}\,dy,\;x_{t_2} (c)= \int^\infty_0 \frac{y^{-\varepsilon} x_{t_1}(y)} {y+c} \,dy,NEWLINE\]NEWLINE are corresponds to the rooted tree with one vertex (root) and no edge, and rooted tree with two vertices and one edge. The difference of the integrals NEWLINE\[NEWLINE\int^\infty_0 \frac{y^{-\varepsilon} x_{t_2} (y)}{y+c} \,dy, \quad \int^\infty_0 \frac{y^{-\varepsilon} x_{t_1}(y)x_{t_2} (y)} {y+c}\,dy, NEWLINE\]NEWLINE corresponds to the difference of two rooted trees with three vertices, rooted trees with \(f(r)=1\) and \(f(r)=2\). These correspondences are explained using figure 1 of Section 2.NEWLINENEWLINE The coproduct of \({\mathcal H}_R\) is defined as follows: for any rooted tree \(t\), with root \(r\), we have \(f(r)\) trees \(t_1,\dots,t_{f(r)}\). \(B_-\) is the operator which removes the root \(r\) from a tree \(t\): \(B_-(t):t\to B(t)= t_1\cdots t_{f(r)}\). \(B_+\) is the operation which maps a monomial of \(n\) rooted tree to a new rooted tree \(t\) which has a root \(r\) with \(f(r)=n\) which connects to the \(n\) roots of \(t_1, \dots, t_n\): \(B_+:t_1 \dots t_n\to B_+(t_1\dots t_n)=t\). If \(t\) is the tree without edge, then it is defined \(B_-(t)=1\), \(B_+(1)=t\). The coproduct \(\Delta\) is defined by the equations NEWLINE\[NEWLINE\begin{aligned} \Delta(e) & =e \otimes e,\;\Delta(t_1\dots t_n)= \Delta(t_1)\dots \Delta(t_n),\\ \Delta(t) & =t\otimes e+(id\otimes B_+) \biggl[\Delta\bigl(B_-(t)\bigr)\biggr]. \end{aligned}NEWLINE\]NEWLINE These definitions are explained in Section 2 together with figure 2-figure 6. The rest of this Section explains how to relate this Hopf algebra structure to the subtraction of a divergent subintegral (renormalization) in the calculation of the integral and the forest formula of Zimmermann [cf. \textit{J. C. Collis}, Renormalization, Cambridge (1984)] is recovered by using antipode (subsection of Section 2). To search the relation between \({\mathcal H}_T\) and \({\mathcal H}_R\), the operator \(N\) which maps a tree \(t\) with \(n\) vertices to a sum \(N(t)\) of \(n\) trees \(t_i\), each having \(n+1\) vertices, by attaching one more outgoing edge and vertex to each vertex of \(t\) and define \(\delta_k=N^k(e)\). After calculating \(\Delta (\delta_n)\), the coproduct \(\Delta\delta_T\), for the symbol \(\delta_T\), \(T\) a rooted tree, is defined. Let \({\mathcal H}_n\) be the Hopf algebra generated by \(\delta_T\) where \(T\) is a tree of degree \(\leq n\). Then it is shown the \({\mathcal H}=\cup{\mathcal H}_n\) is the solution of a universal problem in Hochschild cohomology (Theorem 2).NEWLINENEWLINE To define Lie algebra \({\mathcal L}^1\) such that \({\mathcal H}_R\) is the dual of its envelopping algebra, the notion admissible cut defined in Section 2 is used. By using this notion, a number \(n(T_1,T_2;T)\) for a rooted tree \(T\) is defined and the product \(Z_{T_1}*Z_{T_2}\) is defined to be \(\sum_T n(T_1, T_2;T)Z_T\). \({\mathcal L}^1\) is the Lie algebra spanned by \(Z_T\), elements indexed by rooted tree, with the bracket by the *-product. Adjoin \(Z_0\) and \(Z_{-1}\) to \({\mathcal L}^1\) as elements \([Z_{-1},Z_1]=Z_0\), \([Z_0,Z_T]=\deg (T) Z_T\) and \([Z_1,Z_T]=\sum n(T; T_1)Z_{T_!}\), \(n(T;T_!)\) \(n(T;T_!)\) is the number of times the tree \(T\) is obtained by adjoining an edge and vertex to \(T_1\), \({\mathcal L}^1\) is extended to a Lie algebra \({\mathcal L}\). Then it is shown there exists a surjective Lie algebra homomorphism \(\Theta:{\mathcal L}\to{\mathcal A}\) such that NEWLINE\[NEWLINE\Theta(Z_T)= n(T)Z_n,\quad \Theta(Z_i)= Z_i,\;i=0,1,NEWLINE\]NEWLINE where \(n(T)\) is the number of times \(\delta_T\) occurs in \(N^{\deg(T)-1} (\delta_1)\) (Theorem 8). Section 2 does not deal with overlapping divergence. In the Appendix, taking \(\varphi^3\) theory as an example it is explained how to handle overlapping divergence in this framework.NEWLINENEWLINEFor the entire collection see [Zbl 0935.00053].