In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.
arXiv Mathematics e-prints
- Pub Date:
- March 2005
- Mathematics - Rings and Algebras;
- Mathematics - Combinatorics;
- Typos corrected and the last section on analog of Poincare-Birkhoff-Witt theorem deleted for a gap in the proof