nLab Lie bialgebroid

Idea

(…something like “Lie algebroid internal to Lie algebroids”, but subtle…)

Definition

Given a Lie algebroid $A$ together with the structure of a Lie algebroid $A^*$ on the dual of the vector bundle underlying $A$, the interpretation of a Lie algebroids as a supermanifold as described at Lie infinity-algebroid induces two notions of differentials $d_A$ and $d_{A^*}$ and two notions of Schouten brackets.

A pairs $(A,A^*)$ of Lie algebroids is a Lie bialgebroid if these differentials are derivations of the corresponding Schouten brackets.

See for instance definition 2.2.2 in Roytenberg99.

Remarks

• Every Lie bialgebroid $(A,A*)$ induces the structure of a Courant algebroid on $E := A \oplus A^*$. This is theorem 2.3.3 in Roytenberg99. For $A = T X$ the tangent Lie algebroid of a manifold $X$, this is the crucial fact underlying generalized complex geometry.

References

• Roytenberg99 Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds (arXiv)