Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests

Jinwei Wang,Dan Chen,Yuanyuan Zhang,Zhicheng Zhu

doi:10.1515/math-2020-0059

Abstract

Abstract In this paper, we decorate leaves and edges of planar rooted forests simultaneously and use a part of them to construct free nonunitary Rota-Baxter family algebras. As a corollary, we obtain the construction of free nonunitary Rota-Baxter algebras.

Highlights

Some combinatoric properties of Rota-Baxter algebras were studied by Rota [3] and Cartier [4]
Rota-Baxter algebra has become a new branch with broad connections to other objects in mathematics, such as pre-Lie algebras, pre-Poisson algebras [10,11], quantum field theory [12,13,14], Hopf algebras [15,16], commutative algebras [17,18], Loday’s dendriform algebras [9,19], and Aguiar’s associative analogue of the classical Yang-Baxter equation [20,21,22]
Patras [23, Proposition 9.1] introduced the first example of Rota-Baxter family about algebraic aspects of renormalization in quantum field theory, where a “Rota-Baxter family” appears: this terminology was suggested to the authors by

Summary

Introduction

An associative algebra R together with a k-linear operator P : R → R is called a Rota-Baxter algebra of weight λ, if. Patras [23, Proposition 9.1] (see [25, Theorem 3.7.2]) introduced the first example of Rota-Baxter family about algebraic aspects of renormalization in quantum field theory, where a “Rota-Baxter family” appears: this terminology was suggested to the authors by. Rota-Baxter family algebra arises naturally in renormalization of quantum field theory. Rooted trees/forests are a useful tool for studying many interesting algebraic structures. It appeared in the work of Arthur Cayley [27] in the 1850s considered rooted trees as a representation of combinatorial structures related to the free pre-Lie algebra. Pre-Lie structures on non-planar rooted trees lead to Hopf algebras of combinatorial nature, which appeared in the works in [15,30,33]. Typed decorated rooted forests appeared in a context of low-dimension topology [38] and in a context of the description of combinatorial species [39]

Motivation and layout of the paper

Planar rooted forests

Parallelly typed leaf-spaced decorated planar rooted forests

Free nonunitary Rota-Baxter family algebras