nLab double Lie algebroid

Idea

A double Lie algebroid is the Lie algebroid-analog of a Lie groupoid-double groupoid; essentially a Lie algebroid object internal to the category of Lie algebroids.

Examples

Example

For $(A \stackrel{\rho}{\to} X, [-,-] : \wedge^2\Gamma(A) \to \Gamma)$ a Lie algebroid, its tangent double Lie algebroid $T A$ is the degreewise tangent Lie algebroid of the base space $X$ and of the total space $A$, respectively

$$\array{ T A &\stackrel{d \rho}{\to}& T X \\ \downarrow && \downarrow \\ A &\stackrel{\rho}{\to}& X } \,.$$

Example

For

$$\array{ \mathcal{Mor}_1 &\stackrel{\to}{\to}& \mathcal{Mor}_0 \\ \downarrow && \downarrow \\ \mathcal{Obj}_1 &\stackrel{\to}{\to}& \mathcal{Obj}_0 }$$

a double Lie groupoid, applying Lie differentiation degreewise yields a double Lie algebroid

$$\array{ Lie(\mathcal{Mor}) \\ \downarrow \\ Lie(\mathcal{Obj}) } \,.$$

Related concepts

• double Lie algebroid, Lie algebroid-groupoid, double Lie groupoid

• foliation of a Lie algebroid

References

The notion originates somewhere around

• Kirill Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid (arXiv:math/9808081)

Further discussion is in

• Kirill Mackenzie, Double Lie algebroids and second-order geometry. I. Adv. Math. 94 (1992), no. 2, 180–239

  Double Lie algebroids and second-order geometry. II. Adv.Math. 154 (2000), no. 1, 46–75. dg-ga/9712013).

A textbook account is in

• Kirill Mackenzie, General theory of Lie groupoids and Lie algebroids Cambridge Univ. Press, Cambridge, 2005.MR2157566.