Rota-Baxter operators on sl (2,C) and solutions of the classical Yang-Baxter equation

@article{Pei2013RotaBaxterOO,
title={Rota-Baxter operators on sl (2,C) and solutions of the classical Yang-Baxter equation},
author={Jun Pei and Chengming Bai and Li Guo},
journal={Journal of Mathematical Physics},
year={2013},
volume={55},
pages={021701}
}
• Published 4 November 2013
• Mathematics
• Journal of Mathematical Physics
We explicitly determine all Rota-Baxter operators (of weight zero) on sl (2,C) under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in sl (2,C), confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the six-dimensional Lie algebra sl (2,C)⋉ ad * sl (2,C)*. They also give rise to three…
26 Citations
• Mathematics
Algebra and Logic
• 2022
Rota–Baxter operators for associative algebras appeared in Baxter’s paper [1] as part of the study of integral operators emerging in probability theory and mathematical statistics. Independently, in
• Mathematics
• 2017
Our aim is to classify the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebra whose derived algebra’s dimension is 2. We explicitly determine all Rota-Baxter operators (of weight
• Mathematics
Mediterranean Journal of Mathematics
• 2018
We prove that all Rota–Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota–Baxter operators on the simple odd-dimensional Jordan algebra of
• Mathematics
Communications in algebra
• 2019
All the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra are determined by directly solving the defining equations of the operators.
• Mathematics
• 2019
In this paper, we study Rota–Baxter operators and super $$\mathcal {O}$$O-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Then we give
• Mathematics
• 2021
In this paper, we construct free Lie Rota-Baxter superalgebra by using Gröbner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov
• Mathematics
• 2017
We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov Ω-words on a set X and prove that the set of all the nonassociative Lyndon–Shirshov Ω-words forms a linear basis of the free Lie
Rota-Baxter operators were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. The polynomial algebra
Rota-Baxter operators were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. The polynomial algebra

References

SHOWING 1-10 OF 21 REFERENCES

• Mathematics
• 2011
Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an
In this paper, the different operator forms of the classical Yang–Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric
All constant solutions of the classical Yang-Baxter equation (CYBE) are listed for the function with values in sl(2) and sl(3) and an algorithm which allows one to obtain all constant solutions for a
In 1984 Drinfeld conjectured that any rational solution X(u, upsilon) of the classical Yang-Baxter equation (CYBE)' with X taking values in a simple complex Lie algebra g is equivalent to one of the
• Mathematics
• 2010
We generalize the classical study of (generalized) Lax pairs, the related $${\mathcal O}$$ -operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian
In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs
Left-symmetric algebras have close relations with many important fields in mathematics and mathematical physics. Their classification is very complicated due to the nonassociativity. In this article,
• Mathematics
• 2000
Abstract:This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure
A definition of pre-Poisson algebras is proposed, combining structures of pre-Lie and zinbiel algebra on the same vector space. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra