Free Rota-Baxter family algebras and Free (tri)dendriform family algebras

Abstract

In this paper, we first construct the free Rota-Baxter family algebra generated by some set $X$ in terms of typed angularly $X$-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.

Reference precised in the introduction. This text uses in an essential way the description of the free dendriform and tridendriform family algebras given in arXiv:1909.08946

References in corpus (3)

• A Lie theoretic approach to renormalization
• Hopf algebraic Renormalization of Kreimer's toy model
• Generalized dendrifom algebras and typed binary trees

Cited by in corpus (1)

• Dendriform Algebras Relative to a Semigroup