nLab Lie bialgebra

Idea

For the moment, see the standard reference Chari and Pressley (1994).

Definition

A Lie bialgebra is a Lie algebra $\mathfrak{g}$ equipped with a skew-symmetric linear map (the cobracket)

$$\delta \colon {\mathfrak{g}}\to {\mathfrak{g}}\otimes {\mathfrak{g}},$$

such that

1. its dual linear map $\delta^\ast: \mathfrak{g}^\ast \otimes \mathfrak{g}^\ast \to \mathfrak{g}^\ast$ is a Lie bracket on the dual vector space $\mathfrak{g}^\ast$,

2. it is a cocycle for $[-,-]$ in that:

   $$\delta\big( [x,y] \big) = (ad_x \otimes id + id \otimes ad_x) \delta(y) - (ad_y \otimes id + id \otimes ad_y) \delta(x) \mathrlap{\,,}$$

   where $ad$ denotes the adjoint action of $\mathfrak{g}$ on itself.

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 by Etingof & Kazhdan.

References

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

• P. Etingof , D. Kazhdan: Quantization of Lie bialgebras (in five parts) (arXiv:q-alg/9506005), (arXiv:q-alg/9701038), (arXiv:q-alg/9610030), (arXiv:math/9801043), (arXiv:math/9808121).

• V. Chari and A. Pressley: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)

• Wikipedia, Lie biablgebra

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