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Classical Yang-Baxter equation and left invariant affine geometry on lie groups

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Abstract.

Let G be a Lie group, T*G=Lie(G)*⋊G its cotangent bundle considered as a Lie group, where G acts on Lie(G)* via the coadjoint action. Each solution r of the Classical Yang Baxter Equation on G, corresponds to a connected Lie subgroup H of T*G such that Lie(H) is a Lagrangian graph in Lie(G)⊕Lie(G)* and H carries a left invariant affine structure. If r is invertible, the Poisson Lie tensor π given by r on G is polynomial of degree at most 2 and every double Lie group of (G,π) is endowed with an affine and a complex structures ∇ and J, both left invariant and given by r, such that ∇J=0.

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Authors and Affiliations

  1. The University of Liverpool, Department of Mathematical Sciences, M&O Building, Peach Street, Liverpool, L69 7ZL, UK

    Andre Diatta & Alberto Medina

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  1. Andre Diatta
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  2. Alberto Medina
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Correspondence to Andre Diatta.

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Mathematics Subject Classification (2000): 53D17, 53A15, 17B62

Acknowledgements. The first author was partially supported by Enterprise Ireland. He wishes to thank the mathematical department of NUI Maynooth for their kind welcome, during his stay.

Revised version: 5 April 2004

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Diatta, A., Medina, A. Classical Yang-Baxter equation and left invariant affine geometry on lie groups. manuscripta math. 114, 477–486 (2004). https://doi.org/10.1007/s00229-004-0475-8

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