nLab
double Lie algebroid

Contents

Context

Higher Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

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

TA dρ TX A ρ X. \array{ T A &\stackrel{d \rho}{\to}& T X \\ \downarrow && \downarrow \\ A &\stackrel{\rho}{\to}& X } \,.
Example

For

ℳℴ𝓇 1 ℳℴ𝓇 0 𝒪𝒷𝒿 1 𝒪𝒷𝒿 0 \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

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

References

The notion originates somewhere around

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.

Last revised on March 25, 2013 at 22:17:10. See the history of this page for a list of all contributions to it.