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) |
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Hauptverfasser: | , , |
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. |
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ISSN: | 0022-2488 1089-7658 |
DOI: | 10.1063/1.4863898 |