Rota-Baxter operators on $\mathrm{sl(2,\mathbb {C})}$ sl ( 2 , C ) and solutions of the classical Yang-Baxter equation

We explicitly determine all Rota-Baxter operators (of weight zero) on $\mathrm{sl(2,\mathbb {C})}$ 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 $\mathrm{sl(2,\mathbb {C})}$ sl (...

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Veröffentlicht in:Journal of mathematical physics 2014-02, Vol.55 (2)
Hauptverfasser: Pei, Jun, Bai, Chengming, Guo, Li
Format: Artikel
Sprache:eng
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Zusammenfassung:We explicitly determine all Rota-Baxter operators (of weight zero) on $\mathrm{sl(2,\mathbb {C})}$ 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 $\mathrm{sl(2,\mathbb {C})}$ 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 $\mathrm{sl(2,\mathbb {C})}\ltimes _{{\rm ad}^{\ast }} \mathrm{sl(2,\mathbb {C})}^{\ast }$ sl ( 2 , C ) ⋉ ad * sl ( 2 , C ) * . They also give rise to three-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other six-dimensional Lie algebras.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4863898