nLab Lie bialgebra

Idea

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Quantization

Idea

Quantum groups were introduced independently by Drinfeld and Jimbo around 1984. One of the most important examples of quantum groups are deformations of

universal enveloping algebras. These deformations are closely related to Lie bialgebras. In particular, every deformation of a universal enveloping algebra induces

a Lie bialgebra structure on the underling Lie algebra. In (Drinfeld) Drinfeld asked if the converse of this statement holds:

“Does there exist a universal quantization for Lie bialgebras?”

This was answered to the positive in (Etingof-Kazhdan).

References

• Vladimir Drinfeld, On some unsolved problems in quantum group theory, Lecture Notes in Math., 1510, Springer, Berlin, 1992.

• P. Etingof , D. Kazhdan, Quantization of Lie bialgebras , I, Selecta Math. 2 (1996) n. 1, 1-41.

• Wikipedia, Lie biablgebra
• Štefan Sakáloš, Pavol Ševera, On quantization of quasi-Lie bialgebras, arxiv/1304.6382