| Summary: | A Lie bialgebra is a vector space endowed simultaneously with the structure of a Lie algebra and the structure of a Lie coalgebra, and some compatibility condition. Moreover, Lie brackets have skew symmetry. Because of the close relation between Lie bialgebras and quantum groups, it is interesting to consider the Lie bialgebra structures on the Lie algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">L</mi></semantics></math></inline-formula> related to the Virasoro algebra. In this paper, the Lie bialgebras on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">L</mi></semantics></math></inline-formula> are investigated by computing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>Der</mi><mo>(</mo></mrow><mi mathvariant="fraktur">L</mi><mrow><mo>,</mo><mo> </mo></mrow><mi mathvariant="fraktur">L</mi><mo>⊗</mo><mi mathvariant="fraktur">L</mi><mo>)</mo></mrow></semantics></math></inline-formula>. It is proved that all such Lie bialgebras are triangular coboundary, and the first cohomology group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>1</mn></msup><mo>(</mo><mi mathvariant="fraktur">L</mi><mrow><mo>,</mo><mo> </mo></mrow><mi mathvariant="fraktur">L</mi><mo>⊗</mo><mi mathvariant="fraktur">L</mi><mo>)</mo></mrow></semantics></math></inline-formula> is trivial.
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