Rota-Baxter algebras and shuffle products, rooted trees and operads

A Rota-Baxter algebra is an algebra with a linear endomorphism P that satisfies the relation P(x)P(y)=P(xP(y))+P(P(x)y)+lambda P(xy) for all x, y. Here lambda is a fixed constant. After a brief summary of its basic properties and main applications, we will focus on the construction of free Rota-Baxter algebras. In the commutative case, the construction is related to the shuffle product and quasi-shuffle product with applications to multiple zeta values and symmetric functions. In the noncommutative case, it is related to planar rooted trees with decorations. As an application, we study the adjoint functor of the functor from Rota-Baxter algebras to dendriform algebras.