Atiyah classes of Lie bialgebras

The Atiyah class was originally introduced by M.F. Atiyah. It has many developments in recent years. One important case is the Atiyah classes of Lie algebra pairs. In this paper, we study the Atiyah class of the Lie algebra pair associated with a Lie bialgebra $(g,g^*)$. A simple description of the Atiyah class and the first scalar Atiyah class is given by the Lie algebra structures on $g$ and $g^*$. As its application, the Atiyah classes for some special cases are investigated.