Rota-Baxter Leibniz Algebras and Their Constructions

Liangyun Zhang,Linhan Li,Huihui Zheng

doi:10.1155/2018/8540674

Liangyun Zhang, Linhan Li + Show 1 more

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https://doi.org/10.1155/2018/8540674

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Abstract

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

Highlights

We introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras
The Leibniz algebra [1] was mentioned by Bloh at the first time, which was called a D-algebra in 1965
In Loday’s work, he was mainly interested in the properties of the corresponding homology theory on “group level” (“Leibniz K-Theory”)

Summary

Introduction

The Leibniz algebra [1] was mentioned by Bloh at the first time, which was called a D-algebra in 1965. Leibniz algebras are a well-established algebraic structure generalizing Lie algebras with their own structure and homology theory. A Rota-Baxter algebra is an associative algebra A with a linear operator P on A that satisfies the Rota-Baxter identity. Our main aims are to introduce the concept of Rota-Baxter Leibniz algebras and to obtain a large number of Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra, as well as construct all Rota-Baxter operators of weight 0 and −1 on solvable and nilpotent non-Lie Leibniz algebras of dimension ≤3. We construct all Rota-Baxter operators of weight 0 and −1 on Advances in Mathematical Physics solvable and nilpotent non-Lie Leibniz algebras of dimension ≤3. Throughout the paper, all algebras, linear maps, and tensor products are taken over the complex field C unless otherwise specified

Rota-Baxter Leibniz Algebras

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