Mathematics – Quantum Algebra
Scientific paper
2001-05-25
Mathematics
Quantum Algebra
LaTeX, 73 pages (paper in French)
Scientific paper
In \cite{Kreimer1,Connes,Broadhurst,Kreimer2}, a commutative, non cocommutative Hopf algebra H_R of (decorated) rooted trees was introduced. It is related to the Hopf algebra H_CM introduced in \cite{Moscovici}. Its dual Hopf algebra is the enveloping algebra of the Lie algebra of rooted trees L^1. In this paper, we introduce a non commutative, non cocommutative Hopf algebra H_PR of decorated planar rooted trees. We first show that H_PR is self dual. We then use this result to construct a non degenerate bilinear form (,) on H_PR, which respects the Hopf algebra structure of H_PR. Moreover, we give a combinatorial expression for the bilinear form (,). This allows us to give a direct formula for a basis of the primitive elements of H_PR. In the next sections, we show that H_PR is isomorphic to the Hopf algebra of planar binary trees introduced in \cite{Frabetti, Brouder}, and construct a subalgebra of formal non commutative diffeomorphisms. We then classify the Hopf algebra endomorphisms and coalgebra endomorphisms of H_PR. In the last sections, we give results about tensorial coalgebras and apply them to H_PR and H_R. We show how to construct the primitive elements of H_R from the primitive elements of H_PR and finally prove that L^1 is a free Lie algebra.
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