Mathematical analysis

Division algebra

Algebraic number

Economics

Hopf algebra

Quantum group

Pure mathematics

Mathematics

Representation theory of Hopf algebras

Morphism

Algebraic structure

Grossman

Algebra over a field

Subalgebra

Keynesian economics

Commutative property

Quasitriangular Hopf algebra

Noncommutative geometry

Symmetry Integrability and Geometry Methods and Applications

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

Algebraic Structures on Typed Decorated Rooted Trees