Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees. (English) Zbl 1322.81074
A Hopf algebra \(\mathcal H_{rt}\) which is a generalization of \(\mathcal H(1)\) and contains \(\mathcal H_{rt}\), the Hopf algebra of rooted trees, is defined.
\(\mathcal H_{rt}\) is relevant to understanding of the perturbative regularization of Quantum Field Theory [A. Connes and H. Moscovici, Commun. Math. Phys. 198, No. 1, 199–246 (1998; Zbl 0940.58005)]. Known Hochschild one-cocycle of \(\mathcal H_{rt}\) plays an important role in understanding the combinatirics underlying the Dyson-Schwinger equation [C. Bergbauer and D. Kreimer, in: Physics and number theory. Zürich: European Mathematical Society Publishing House. 133–164 (2006; Zbl 1141.81024)]. This cocycle is described in §3). The authors say knowing about cohomology of \(H_{rt}\) will lead to a better understanding of connection of non-commutative geometry and renormalization.
\(\mathcal H(1)\) is \(n = 1\) case of \(\mathcal H(n)\), a family of Hopf algebras on \(n\)-dimensional flat manifold \(M\) intrduced by Connes and Moscovici [loc. cit.]. They act on the group of smooth functions on the frame bundles with local diffeomorphism on \(M\). They well studied in non-commutative geometry.
The key of the construction of \(\mathcal H(1)_{rt}\) is given in works of A. Cayley [Am. J. Math. 4, 266–268 (1881; JFM 13.0867.02)] and J. C. Butcher [The numerical analysis of ordinary differential equations. Runge-Kutta and general linear methods. Chichester etc.: John Wiley & Sons (1987; Zbl 0616.65072)] (reviewed in §2) that relate rooted trees to a particular first order differential equation. In §3, \(\mathcal H_{rt}\) is defined. It is a graded algebra graded by the number of vertexes of the tree. Then the natural growth operator \(N\) and generalized natural growth operator \(N_t\) are introduced and show \(\mathcal H_{rt}\) is generated by repeated application of operaotrs of the form \(N_t\) to the tree with a single vertex (Theorem 2).
In §4, \(\mathcal H(1)\) is shown to be the bicrossed product of \(\mathcal H_{CK}\), the algebra \(\mathbb Q=\{N^{i-1} (\bullet)|i \in \mathbb N\}\) and the universal enveloping algebra of \(\mathbb{Q}\), generated by two vector fields corresponding grading operator and natural growth operator. Then. in §5, the last one, showing the bicrossed product of \(H_{rt}\) and the universal enveloping algebra of \(\mathfrak{g}_{rt}\) , the Lie algebra generated by vector field \(X_t\), for \(t\) a rooted tree, is isomorphic the bicrossed product of \(\mathcal H_{CK}\) and the universal enveloping algebra of \(\mathbb{Q}\) (Theorem 9), \(\mathcal H(1)_{rt}\) is defined to be this bicrossed product.
\(\mathcal H_{rt}\) is relevant to understanding of the perturbative regularization of Quantum Field Theory [A. Connes and H. Moscovici, Commun. Math. Phys. 198, No. 1, 199–246 (1998; Zbl 0940.58005)]. Known Hochschild one-cocycle of \(\mathcal H_{rt}\) plays an important role in understanding the combinatirics underlying the Dyson-Schwinger equation [C. Bergbauer and D. Kreimer, in: Physics and number theory. Zürich: European Mathematical Society Publishing House. 133–164 (2006; Zbl 1141.81024)]. This cocycle is described in §3). The authors say knowing about cohomology of \(H_{rt}\) will lead to a better understanding of connection of non-commutative geometry and renormalization.
\(\mathcal H(1)\) is \(n = 1\) case of \(\mathcal H(n)\), a family of Hopf algebras on \(n\)-dimensional flat manifold \(M\) intrduced by Connes and Moscovici [loc. cit.]. They act on the group of smooth functions on the frame bundles with local diffeomorphism on \(M\). They well studied in non-commutative geometry.
The key of the construction of \(\mathcal H(1)_{rt}\) is given in works of A. Cayley [Am. J. Math. 4, 266–268 (1881; JFM 13.0867.02)] and J. C. Butcher [The numerical analysis of ordinary differential equations. Runge-Kutta and general linear methods. Chichester etc.: John Wiley & Sons (1987; Zbl 0616.65072)] (reviewed in §2) that relate rooted trees to a particular first order differential equation. In §3, \(\mathcal H_{rt}\) is defined. It is a graded algebra graded by the number of vertexes of the tree. Then the natural growth operator \(N\) and generalized natural growth operator \(N_t\) are introduced and show \(\mathcal H_{rt}\) is generated by repeated application of operaotrs of the form \(N_t\) to the tree with a single vertex (Theorem 2).
In §4, \(\mathcal H(1)\) is shown to be the bicrossed product of \(\mathcal H_{CK}\), the algebra \(\mathbb Q=\{N^{i-1} (\bullet)|i \in \mathbb N\}\) and the universal enveloping algebra of \(\mathbb{Q}\), generated by two vector fields corresponding grading operator and natural growth operator. Then. in §5, the last one, showing the bicrossed product of \(H_{rt}\) and the universal enveloping algebra of \(\mathfrak{g}_{rt}\) , the Lie algebra generated by vector field \(X_t\), for \(t\) a rooted tree, is isomorphic the bicrossed product of \(\mathcal H_{CK}\) and the universal enveloping algebra of \(\mathbb{Q}\) (Theorem 9), \(\mathcal H(1)_{rt}\) is defined to be this bicrossed product.
Reviewer: Akira Asada (Takarazuka)
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 16T05 | Hopf algebras and their applications |
| 57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |
| 58B34 | Noncommutative geometry (à la Connes) |
| 05C05 | Trees |
| 17B22 | Root systems |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
References:
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