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Finite dimensional comodules over the Hopf algebra of rooted trees. (English) Zbl 1017.16031
Let \({\mathcal H}_{\mathcal R}\) denote the Hopf algebra of rooted trees. The author proves some results about finite-dimensional comodules over \({\mathcal H}_{\mathcal R}\). The author first shows that if \(C\) is such a comodule of dimension \(n\) then for each \(i\in\{1,2,\dots,n\}\) there exists a submodule \(C^{(i)}\) of dimension \(i\) with \(C^{(1)}\subset\cdots\subset C^{(n)}=C\). Using this he provides a parametrization of such comodules by certain finite families of primitive elements and proves a classification results for such comodules arising from certain restricted families of primitive elements. He then proves that the Lie algebra of rooted trees is free. Next, the author considers the subalgebra of \({\mathcal H}_{\mathcal R}\) generated by ladders and shows that a basis of the primitive elements can be obtained by an inductive process. In the rest of the paper the author studies endomorphisms of \({\mathcal H}_{\mathcal R}\).

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
05C05 Trees
Full Text: DOI
[1] Kreimer, D., On the Hopf algebra structure of perturbative quantum field theories, 1998 · Zbl 1041.81087
[2] Connes, A.; Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, IHES/M/98/37 · Zbl 0940.58005
[3] Connes, A.; Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Comm. math. phys., 199, 203, (1998) · Zbl 0932.16038
[4] Broadhurst, D.J.; Kreimer, D., Renormalization automated by Hopf algebra, 1998 · Zbl 1049.81048
[5] Kreimer, D., On overlapping divergences, 1999 · Zbl 0977.81091
[6] Broadhurst, D.J.; Kreimer, D., Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees, 2000 · Zbl 0986.16015
[7] Panaite, F., Relating the cannes – kreimer grossman – larson Hopf algebras built on rooted trees, 2000 · Zbl 0959.16023
[8] Hoffman, M.E., Combinatorics of rooted trees and Hopf algebras, 2002
[9] Kreimer, D., Combinatorics of (perturbative) quantum field theory, 2000 · Zbl 0994.81080
[10] N.J.A. Sloane, On-line encyclopedia of integer sequences, sequence A000081 http://www.research.att.com/ njas/sequences
[11] Bourbaki, N., Groupes et algèbres de Lie, (1972), Hermann Paris, chapitre II, §3, corollaire 2, chapitres 2 et 3
[12] Sweedler, M., Hopf algebras, (1969), Benjamin New York
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