nLab
tangent Lie algebroid

Definition

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

• the codiscrete groupoid $X\times X$ of $X$.

• the fundamental groupoid ${\Pi }_1(X)$ of $X$.

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

• the path ∞-groupoid ${\Pi }_{\mathrm{\infty }}(X)$.

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

So to some extent the tangent Lie algebroid is the tangent bundle $TX$ of $X$. More precisely, when using the definition of a Lie algebroid $E$ over $X$ as a diagram

$$\begin{array}{ccccc}E& & \stackrel{\rho }{\to }& & TX\\ & \searrow & & \swarrow \\ & & X\end{array}$$\array{ E &&\stackrel{\rho}{\to}&& T X \\ & \searrow && \swarrow \\ && X }

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

$$\begin{array}{ccccc}E=TX& & \stackrel{\rho =\mathrm{Id}}{\to }& & TX\\ & \searrow & & \swarrow \\ & & X\end{array}.$$\array{ E = T X &&\stackrel{\rho = Id}{\to}&& T X \\ & \searrow && \swarrow \\ && X } \,.

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

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

$$\mathrm{CE}(TX)=({\Omega }^{•}(X),d_{\mathrm{dR}}).$$CE(T X) = (\Omega^\bullet(X), d_{dR}) \,.

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

Relation to flat $\mathrm{\infty }$-Lie algebroid valued differential forms

For $𝔞$ any other Lie algebroid or Lie infinity-algebroid, a morphism of Lie infinity-algebroids

$$\omega :TX\to 𝔞$$\omega : T X \to \mathfrak{a}

is flat $𝔞$-valued differential form datum .

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

If $𝔤=𝔲(1)=ℝ$ is the abelian 1-dimensional Lie algebra, then a morphism $TX\to 𝔤$ is just a closed differential 1-form on $X$.

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

Related concepts

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

References

One of the earliest reference seems to be

• Ted Courant?, Tangent Lie algebroid (pdf)