nLab
tangent Lie algebroid

Context

\infty-Lie theory

∞-Lie theory

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

Definition

Let XX by a differentiable manifold. The tangent Lie algebroid TXT X of XX is the Lie algebroid that corresponds – in the sense of Lie theory – to

When the tangent Lie algebroid is regarded as a Lie ∞-algebroid it corresponds to

The space of objects of TXT X is XX itself and its elements in degree 1 are the vectors on XX, i.e. the elements in the tangent bundle of XX. These are to be thouhght of as the infinitesimal paths in XX.

So to some extent the tangent Lie algebroid is the tangent bundle TXT X of XX. More precisely, when using the definition of a Lie algebroid EE over XX as a diagram

E ρ TX X \array{ E &&\stackrel{\rho}{\to}&& T X \\ & \searrow && \swarrow \\ && X }

where ρ\rho is the map called the anchor map, the tangent Lie algebroid is that whose anchor map is the identity map

E=TX ρ=Id TX X. \array{ E = T X &&\stackrel{\rho = Id}{\to}&& T X \\ & \searrow && \swarrow \\ && X } \,.

Therefore the tangent Lie algebroid of XX is usually denoted TXT X, just as the tangent bundle itself.

The Chevalley-Eilenberg algebra of TXT X is correspondingly fundamental: it is the deRham dg-algebra of differential forms on XX:

CE(TX)=(Ω (X),d dR). CE(T X) = (\Omega^\bullet(X), d_{dR}) \,.

So regarded as an NQ-supermanifold the tangent Lie algebroid is the shifted tangent bundle ΠTX\Pi T X equipped with its canonical odd vector field.

Relation to flat \infty-Lie algebroid valued differential forms

For 𝔞\mathfrak{a} any other Lie algebroid or Lie infinity-algebroid, a morphism of Lie infinity-algebroids

ω:TX𝔞 \omega : T X \to \mathfrak{a}

is flat 𝔞\mathfrak{a}-valued differential form datum .

For instance if 𝔞=𝔤\mathfrak{a} = \mathfrak{g} is an ordinary Lie algebra then a morphism TX𝔤T X \to \mathfrak{g} is a flat Lie algebra valued differential form on XX, i.e. an element AΩ 1(X)𝔤A \in \Omega^1(X) \otimes \mathfrak{g} such that dA+[,](AA)=0d A + [-,-](A \wedge A) = 0.

If 𝔤=𝔲(1)=\mathfrak{g} = \mathfrak{u}(1) = \mathbb{R} is the abelian 1-dimensional Lie algebra, then a morphism TX𝔤T X \to \mathfrak{g} is just a closed differential 1-form on XX.

More on \infty-Lie algebroid-valued differential forms is (eventually) at ∞-Lie algebroid valued differential forms.

The higher-order version of tangent Lie algebroids are jet bundle D-schemes.

References

One of the earliest reference seems to be

Revised on July 11, 2013 23:11:39 by Urs Schreiber (82.113.106.241)