Dirac generating operators and Manin triples

TL;DR

This paper establishes a characterization of Lie bialgebroids through Dirac generating operators, linking algebraic structures on a vector bundle and its dual with differential operators and their squares.

Contribution

It introduces a new approach using Dirac generating operators to characterize Lie bialgebroids, providing new identities relating Lie algebroid structures.

Findings

The square of the Dirac operator equals a function if and only if the pair is a Lie bialgebroid.

The pair (A, A*) is a Lie bialgebroid iff the Dirac operator is a Dirac generating operator.

New identities relating Lie algebroid structures are established.

Abstract

Given a pair of (real or complex) Lie algebroid structures on a vector bundle $A$ (over $M$) and its dual $A^*$, and a line bundle $\module$ such that $\module\otimes\module=(\wedge^{\TOP} A^*\otimes\wedge^{\TOP} T^*M)$, there exist two canonically defined differential operators $\bdees$ and $\bdel$ on $\sections{\wedge A\otimes\module}$. We prove that the pair $(A,A^*)$ constitutes a Lie bialgebroid if, and only if, the square of $\bdirac =\bdees+\bdel$ is the multiplication by a function on $M$. As a consequence, we obtain that the pair $(A,A^*)$ is a Lie bialgebroid if, and only if, $\bdirac$ is a Dirac generating operator as defined by Alekseev & Xu \cite{AlekseevXu}. Our approach is to establish a list of new identities relating the Lie algebroid structures on $A$ and $A^*$ (Theorem \ref{Thm:C}).