Lie bialgebra structures on the extended affine Lie algebra sl₂(Cq)

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Lie bialgebra structures on the extended affine Lie algebra sl₂(Cq) are investigated. In particular, all Lie bialgebra structures on sl₂(Cq) are shown to be triangular coboundary. This result is obtained by employing some techniques, which may also work for more general extended affine Lie algebras, to prove the triviality of the first cohomology group of sl₂(Cq) with coefficients in the tensor product of its adjoint module, namely, H¹(sl₂(Cq),sl₂(Cq)⊗sl₂(Cq))=0.