2008/07/31 by Walter D. van Suijlekom · 29 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Division algebra #Feynman diagram #Gauge theory #Hopf algebra #Mathematical physics #Mathematics #Physics #Pure mathematics #Renormalization #Representation theory of Hopf algebras #hep-th #math-ph #math.MP #msc:16W30 #msc:81T13 #msc:81T15
paper · pdf · doi:10.1007/s00220-009-0829-x
published in Communications in Mathematical Physics 290(1), 291-319 (Springer Science+Business Media) · 30 pages; minor corrections, to appear in CMP
arxiv created 2009/02/09 · openalex publication_date 2009/05/22 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the structure of renormalization Hopf algebras of gauge theories. We identify certain Hopf subalgebras in them, whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms. This can be understood physically as wave function renormalization and renormalization of the coupling constants, respectively. After taking into account the Slavnov–Taylor identities for the couplings as generators of a Hopf ideal, we find Hopf subalgebras in the corresponding quotient as well. In the second part of the paper, we explain the origin of these Hopf ideals by considering a coaction of the renormalization Hopf algebras on the Batalin-Vilkovisky (BV) algebras generated by the fields and couplings constants. The so-called classical master equation satisfied by the action in the BV-algebra implies the existence of the above Hopf ideals in the renormalization Hopf algebra. Finally, we exemplify our construction by applying it to Yang–Mills gauge theory.