2 00 5 Rota - Baxter Algebras , Dendriform Algebras and Poincaré - Birkhoff - Witt Theorem

Rota-Baxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative Rota-Baxter algebras relate to double shuffle relations for multiple zeta values. The interest in the non-commutative setting arose in connection with the work of Connes and Kreimer on the Birkhoff decomposition in renormalization theory in perturbative quantum field theory. We construct free non-commutative Rota-Baxter algebras and apply the construction to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We also prove an analog of the Poincaré-Birkhoff-Witt theorem for universal enveloping algebra in the context of dendriform trialgebras. In particular, every dendriform dialgebra and trialgebra is a subalgebra of a Rota-Baxter algebra. We explicitly show that the free dendriform dialgebras and trialgebras, as represented by planar trees, are canonical subalgebras of free Rota-Baxter algebras.