Algebraic Birkhoff decomposition is a fundamental result in Hopf algebra approach to renormalization in pertubative quatum field theory developoed by A. Connes and D. Kreimers. In this work, we give two generalizations of algebraic Birkhoff decomposition. The first generalization replaces the condition A = A₊ ⊕ A_- by A = ⊕_i=0ⁿ A_i, consequently replacing the factorization of certain groups as a unique product into two subgroups by a unique product into n subgroups. The second one is the dual version of the decomposition in which connected coalgebras and Hopf algebras in the original decomposition are replaced by algebras with descending filtration. This generalization can be applied to any algebras with a descending filtration and with a direct sum decomposition of finitely many...[ Read more ]

Algebraic Birkhoff decomposition is a fundamental result in Hopf algebra approach to renormalization in pertubative quatum field theory developoed by A. Connes and D. Kreimers. In this work, we give two generalizations of algebraic Birkhoff decomposition. The first generalization replaces the condition A = A₊ ⊕ A_- by A = ⊕_i=0ⁿ A_i, consequently replacing the factorization of certain groups as a unique product into two subgroups by a unique product into n subgroups. The second one is the dual version of the decomposition in which connected coalgebras and Hopf algebras in the original decomposition are replaced by algebras with descending filtration. This generalization can be applied to any algebras with a descending filtration and with a direct sum decomposition of finitely many subalgebras which is compatible with the descending filtration. At the end of this thesis, examples using the generalizations of algebraic Birkhoff decomposition are given.