Atiyah classes of Lie bialgebras

TL;DR

This paper explores the Atiyah class associated with Lie bialgebras, providing a simple description based on their Lie algebra structures and examining specific cases for deeper insights.

Contribution

It introduces a straightforward description of the Atiyah class for Lie bialgebras and investigates its properties in special cases, expanding understanding in this area.

Findings

A simple description of the Atiyah class for Lie bialgebras is provided.

The first scalar Atiyah class is characterized by the Lie algebra structures.

Applications to specific cases reveal new properties of Atiyah classes in Lie bialgebras.

Abstract

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.