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

Abstract

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 6-dimensional Lie algebra $sl(2,C){⋉}_{{\mathrm{ad}}^{\ast }}sl(2,C{)}^{\ast }$. They also give rise to 3-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other 6-dimensional Lie algebras.

17 pages

References in corpus (2)

• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras

Cited by in corpus (9)

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• Rota-Baxter operators on Witt and Virasoro algebras
• Homogeneous conformal averaging operators on semisimple Lie algebras
• Spectrum of Rota-Baxter operators
• Rota-type operators on null-filiform associative algebras
• Rota--Baxter operators and skew left brace structures over Heisenberg Group
• Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra
• Rota-Baxter operators of nonzero weight on the matrix algebra of order three
• Double Lie algebras of a nonzero weight