Les algèbres de Hopf des arbres enracinés décorés, I

By Loïc Foissy. 2001

242 citations recorded by the source corpus.

Abstract We introduce a Hopf algebra of planar decorated rooted trees H D P,R which is non commutative and non cocommutative and generalizes the Hopf algebra of rooted trees H D R of Connes and Kreimer. We show that H D P,R satisfies a universal property in Hochschild cohomology and deduce that it is self-dual. We construct its coalgebra and Hopf algebra endomorphisms and show that the canonical epimorphism from H D P,R to H D R leads to the determination of the space of primitive elements of H D R , answering a question by D. Kreimer.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40897a42cb4d63d6cd78154c443433732a4420af

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Combinatorial Hopf algebras (2008, background): “Some of these results have been worked out explicitly in [18].”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, unknown): “For any y, y′, y′′ ∈ R, if all the triples (y, y′, y′′), (⌊y⌋R, y ′, y′′), (y, ⌊y′⌋R, y ′′), (y, y′, ⌊y′′⌋R), (⌊y⌋R, y ′, ⌊y′′⌋R),(14) (⌊y⌋R, ⌊y ′⌋R, y ′′), (y, ⌊y′⌋R, ⌊y ′′⌋R)(15) are associative triples for ·, then (⌊y⌋R, ⌊y ′⌋R, ⌊y ′′⌋R) is an associative triple for ·.”
Ramification of rough paths (2006, background): “Literature on combinatorial and algebraic properties of rooted trees is quite large, we prefer to single out the work of Hoffman [15] and the two papers of Foissy [16,17] on labeled rooted trees.”
Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations (2007, background): “This algebra is introduced simultaneously in [9,11].”
Rota–Baxter algebras and dendriform algebras (2005, methodology): “in 1995 with motivation from algebraic K-theory, and was further studied in connection with several areas in mathematics and physics, including operads [54], homology [34, 35], Hopf algebras [13, 47, 58, 65, 32], Lie and Leibniz algebras [35], combinatorics [6, 7, 33, 57], arithmetic [55] and quantu”
On products and duality of binary, quadratic, regular operads (2004, background): “In the next few years, their properties were studied by several authors in areas related to operads [35], homology [17, 18], Hopf algebras [7, 26, 42, 39], combinatorics [16, 37, 2, 3], arithmetic [34] and quantum field theory [16].”