Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations
By Loïc Foissy. 2007
76 citations recorded by the source corpus.
We consider the combinatorial Dyson–Schwinger equation X=B+(P(X)) in the non-commutative Connes–Kreimer Hopf algebra of planar rooted trees HNCK, where B+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra AN,P of HNCK. We describe all the formal series P such that AN,P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of HNCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faa di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faa di Bruno algebra into the Connes–Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faa di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7276c727e43b403d70903245b463289166362407
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Citations in the corpus
- Modules of the 0-Hecke algebra and quasisymmetric Schur functions (2014, background): “For example, NSym is isomorphic [22] to Solomon’s descent algebra of Type A [21, 50] and QSym is isomorphic to the Hopf algebra of ladders [20] and the algebra of smooth functions on Young diagrams [3].”