Rota-Baxter Algebras: Factorizing Feynman Rules

A Rota-Baxter algebra is an algebra with a linear operator R satisfying the identity $R(x)R(y)=R(R(x)y)+R(xR(y))-qR(xy)$, where q is a constant. It first occurred in the work of Glen Baxter in probability, and was popularized by the work of Gian-Carlo Rota and Pierre Cartier. They recently appeared in connection with the seminal work of Connes-Kreimer on renormalization theory in pQFT, Loday's dendriform operads, and associative analogs of the (modified) classical Yang-Baxter equation. We will review these results in some detail with an emphasis on the factorization problem underlying the Birkhoff decomposition of Hopf algebra characters.