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Lie bialgebra structures on the extended affine Lie algebra$\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Liebialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to betriangular coboundary. This result is obtained by employing some techniques,which may also work for more general extended affine Lie algebras, to prove thetriviality of the first cohomology group of $\widetilde{sl_2(\mathbb{C}_q)}$with coefficients in the tensor product of its adjoint module, namely,$H^1(\widetilde{sl_2(\mathbb{C}_q)},\widetilde{sl_2(\mathbb{C}_q)}\otimes\widetilde{sl_2(\mathbb{C}_q)})=0$.

Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$