Abstract
As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter algebras and Ω-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free Ω-Rota–Baxter algebra by typed, angularly decorated rooted trees.
Highlights
A Rota–Baxter algebra is an associative algebra equipped with a linear operator that generalizes the algebra of continuous functions with the integral operator
The concept of a Rota–Baxter algebra could be regarded as an algebraic framework of the integral analysis
Ω-Rota Baxter Algebras we mainly investigate some basic properties of Ω-Rota–Baxter algebras
Summary
A Rota–Baxter algebra is an associative algebra equipped with a linear operator that generalizes the algebra of continuous functions with the integral operator. The concept of Rota–Baxter algebra was introduced in 1960 by Glen Baxter [1] in his probability study of fluctuation theory, and studied in the 1960s and 1970s by Cartier and Gian-Carlo Rota [2,3,4] in connection with combinatorics This algebra remained inactive until 2000, when new motivations were found, coming from interesting applications in the prominent work of Connes and Kreimer [5] on the renormalization of perturbative quantum field theory, and from the close relationship with the associative Yang–Baxter equation [6] and the construction of free Rota–Baxter algebras related to the shuffle product [7,8]. The Rota–Baxter operator has appeared in a wide range of areas in mathematics and mathematical physics, such as number theory [9], pre-Lie and Lie algebra [6,10], Hopf algebras [11,12], operads [13], O-operators [14,15], Rota–Baxter groups and skew left braces [16,17], classical Yang–Baxter equations and associative Yang–Baxter equations [14,18]
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