Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations. (English) Zbl 1158.16020
Let \(K\) be a field of characteristic zero, \(\mathcal H_{CK}\) the Connes-Kreimer Hopf algebra of rooted trees. \(\mathcal H_{CK}\) is commutative as an algebra, and has a Hopf subalgebra isomorphic to the Hopf algebra of functions on the group under composition of formal diffeomorphisms, also called the Faà di Bruno Hopf algebra \(\mathcal H_{FdB}\).
The author has introduced a non-commutative version \(\mathcal H_{NCK}\) of \(\mathcal H_{CK}\) [Bull. Sci. Math. 126, No. 4, 249-288 (2002; Zbl 1013.16027)], which contains a non-commutative version of \(\mathcal H_{FdB}\) whose Abelianization is isomorphic to \(\mathcal H_{FdB}\). The paper under review considers a family of subalgebras \(\mathcal A_{N,P}\) of \(\mathcal H_{NCK}\), where \(P\) is a formal series in \(K[\![h]\!]\) with constant term 1. There is a unique solution \(X_P\) in the completion of \(\mathcal H_{NCK}\) of the equation \(X_P=B^+(P(X_P))\), where \(B^+\) is the operator of grafting on a root. \(\mathcal A_{N,P}\) is the subalgebra of \(\mathcal H_{NCK}\) generated by the homogeneous components of \(X_P\).
The main theorem of the paper gives necessary and sufficient conditions for \(\mathcal A_{N,P}\) to be a Hopf subalgebra of \(\mathcal H_{NCK}\), which involve the existence of two scalars \(a\) and \(b\) in \(K\) satisfying certain conditions involving \(P\). Thus a two parameter family \(\mathcal A_{N,a,b}\) of Hopf subalgebras of \(\mathcal H_{NCK}\) is obtained. Equalities between the \(\mathcal A_{N,a,b}\), and their isomorphism classes are determined. There are three isomorphism classes: one for \(a=0\), \(b=1\); one for \(a=1\), \(b=-1\); and one for \(a=1\), \(b\) different from \(-1\). This last class consists of non-commutative versions of \(\mathcal H_{FdB}\). By Abelianizing, analogous results hold for \(\mathcal H_{CK}\). A final section makes some observations about the free Faà di Bruno algebra on a fixed finite number of variables.
The author has introduced a non-commutative version \(\mathcal H_{NCK}\) of \(\mathcal H_{CK}\) [Bull. Sci. Math. 126, No. 4, 249-288 (2002; Zbl 1013.16027)], which contains a non-commutative version of \(\mathcal H_{FdB}\) whose Abelianization is isomorphic to \(\mathcal H_{FdB}\). The paper under review considers a family of subalgebras \(\mathcal A_{N,P}\) of \(\mathcal H_{NCK}\), where \(P\) is a formal series in \(K[\![h]\!]\) with constant term 1. There is a unique solution \(X_P\) in the completion of \(\mathcal H_{NCK}\) of the equation \(X_P=B^+(P(X_P))\), where \(B^+\) is the operator of grafting on a root. \(\mathcal A_{N,P}\) is the subalgebra of \(\mathcal H_{NCK}\) generated by the homogeneous components of \(X_P\).
The main theorem of the paper gives necessary and sufficient conditions for \(\mathcal A_{N,P}\) to be a Hopf subalgebra of \(\mathcal H_{NCK}\), which involve the existence of two scalars \(a\) and \(b\) in \(K\) satisfying certain conditions involving \(P\). Thus a two parameter family \(\mathcal A_{N,a,b}\) of Hopf subalgebras of \(\mathcal H_{NCK}\) is obtained. Equalities between the \(\mathcal A_{N,a,b}\), and their isomorphism classes are determined. There are three isomorphism classes: one for \(a=0\), \(b=1\); one for \(a=1\), \(b=-1\); and one for \(a=1\), \(b\) different from \(-1\). This last class consists of non-commutative versions of \(\mathcal H_{FdB}\). By Abelianizing, analogous results hold for \(\mathcal H_{CK}\). A final section makes some observations about the free Faà di Bruno algebra on a fixed finite number of variables.
Reviewer: Earl J. Taft (New Brunswick)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 05C05 | Trees |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
Citations:
Zbl 1013.16027Online Encyclopedia of Integer Sequences:
Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).Weights of Connes-Moscovici Hopf subalgebra.
References:
| [1] | Anshelevich, Michael; G. Effros, Edward; Popa, Mihai, Zimmermann type cancellation in the free Faà di Bruno algebra, J. Funct. Anal., 237, 1, 76-104 (2006) · Zbl 1123.05095 |
| [2] | Bergbauer, Christoph; Kreimer, Dirk, Hopf algebras in renormalization theory: locality and Dyson-Schwinger equations from Hochschild cohomology, IRMA Lect. Math. Theor. Phys., 10, 133-164 (2006) · Zbl 1141.81024 |
| [3] | Broadhurst, David J.; Kreimer, Dirk, Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees, Comm. Math. Phys., 215, 1, 217-236 (2000) · Zbl 0986.16015 |
| [4] | Brouder, Christian, Trees, renormalization and differential equations, BIT, 44, 3, 425-438 (2004) · Zbl 1072.16033 |
| [5] | Connes, Alain; Kreimer, Dirk, Hopf algebras, Renormalization and Noncommutative geometry, Comm. Math. Phys., 199, 1, 203-242 (1998) · Zbl 0932.16038 |
| [6] | Connes, Alain; Moscovici, Henri, Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys., 198, 1, 199-246 (1998) · Zbl 0940.58005 |
| [7] | Hoffman, Michael E., Combinatorics of rooted trees and Hopf algebras, Trans. Amer. Math. Soc., 355, 9, 3795-3811 (2003) · Zbl 1048.16023 |
| [8] | Foissy, Loïc, Finite-dimensional comodules over the Hopf algebra of rooted trees, J. Algebra, 255, 1, 85-120 (2002) · Zbl 1017.16031 |
| [9] | Loïc Foissy, O., Les algèbres de Hopf des arbres enracinés, I, Bull. Sci. Math., 126, 193-239 (2002) · Zbl 1013.16026 |
| [10] | Foissy, Loïc, Les algèbres de Hopf des arbres enracinés, II, Bull. Sci. Math., 126, 249-288 (2002) · Zbl 1013.16027 |
| [11] | Holtkamp, Ralf, Comparison of Hopf algebras on trees, Arch. Math. (Basel), 80, 4, 368-383 (2003) · Zbl 1056.16030 |
| [12] | Kreimer, Dirk, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., 2, 2, 303-334 (1998) · Zbl 1041.81087 |
| [13] | Kreimer, Dirk, On overlapping divergences, Comm. Math. Phys., 204, 3, 669-689 (1999) · Zbl 0977.81091 |
| [14] | Kreimer, Dirk, Combinatorics of (perturbative) quantum field theory, Phys. Rep., 4-6, 387-424 (2002) · Zbl 0994.81080 |
| [15] | Kreimer, Dirk, Dyson-Schwinger Equations: From Hopf Algebras to Number Theory, Fields Inst. Commun., vol. 50 (2007), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1370.81128 |
| [16] | Kreimer, Dirk; Yeats, Karen, An étude in non-linear Dyson-Schwinger equations, Nuclear Phys. B Proc. Suppl., 160, 116-121 (2006) |
| [17] | Malvenuto, Clauda; Reutenauer, Christophe, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177, 3, 967-982 (1995) · Zbl 0838.05100 |
| [18] | W. Milnor, John; C. Moore, John, On the structure of Hopf algebras, Ann. of Math. (2), 81, 211-264 (1965) · Zbl 0163.28202 |
| [19] | P. Stanley, Richard, Enumerative Combinatorics, vol. 1, Cambridge Stud. Adv. Math., vol. 49 (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0889.05001 |
| [20] | Stanley, Richard P., Enumerative Combinatorics, vol. 2, Cambridge Stud. Adv. Math., vol. 62 (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0928.05001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
