Lie Bialgebras of Generalized WITT Type, II

Abstract

In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H¹(𝒲, 𝒲 ⊗ 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.

Key Words:

• Lie algebras of generalized Witt type
• Lie bialgebras
• Yang-Baxter equation

Mathematics Subject Classification (2000):

• 17B62
• 17B05
• 17B37
• 17B66

ACKNOWLEDGMENTS

The authors would like to thank the referee for suggestions (especially the suggestion of significant simplification of the proof of Lemma 2.5), and correcting some errors in the previous versions. Supported by a NSF grant 10471091 of China, “One Hundred Talents Program” from University of Science and Technology of China.

Notes

Communicated by K. C. Misra.