Some Studies About Baxter Algebras

In this thesis, we study some problems about Baxter algebras. The study of Baxter algebras originated from 1960s, which attracted Rota, Baxter, Miller and Carrier because of its important application in Algebras and Combinatorics, leading to their further researches. First, we introduce the definition of Baxter algebras and some important examples. We study two constructions of free Baxter algebras obtained by Rota and Carrier and the recent construction of free baxter algebras obtained by Guo and Keigher in terms of mixable shuffles. We proof that the construction of Carrier is isomorphic to that of Guo and Keigher in category of Baxter algebras.Later, we apply the tree of Combinatics to study Baxter algebras, computing the coefficients of the expansion of the Rota-Baxter identity. And construct a Baxter operator with the weight A on the rooted trees as well as the Baxter algebras.Finally, applying the construction of Baxter algebras to study the constructions of Hopf algebras, we get a new class of Hopf algebras-divided power Hopf algebras with the construction of Baxter algebras.