Lie bialgebra

Summary

A Lie bialgebra is a Lie algebra that is also a Lie coalgebra, such that the cobracket is a $1$-cocycle for the adjoint action. Lie bialgebras were introduced by Drinfeld as the infinitesimal objects of Poisson–Lie groups and as the classical limits of quantum groups. For a finite-dimensional Lie bialgebra $\mathfrak{g}$, the dual $\mathfrak{g}^*$ is again a Lie bialgebra, and the two combine into the Drinfeld double.