nLab
Lie infinity-algebroid

Idea

$L_{\mathrm{\infty }}$-algebroids or Lie $\mathrm{\infty }$-algebroids are to Lie ∞-groupoids as Lie algebras are to Lie groups. Hence $L_{\mathrm{\infty }}$-algebroids are a horizontal categorification and vertical categorification of Lie algebras: they encompass $L_{\mathrm{\infty }}$-algebras as well as Lie algebroids.

A Lie $\mathrm{\infty }$-algebroid is a complex $g$ of modules over a ring $A_0$, satisfying some (complicated) conditions. It turns out that it is easier to formulate these conditions in terms of the Chevalley–Eilenberg algebra $\mathrm{CE}(g)$ of $g$. This is a certain type of graded super commutative algebra constructed from $g$.

Definition in terms of differential algebra

Definition (Lie $\mathrm{\infty }$ Algebroid)

A Lie $\mathrm{\infty }$-algebroid $g$ over a manifold $X_0$ is the following data:

• A commutative algebra $A_0$, thought of as the ring of functions on a manifold $X_0$.

• a non-positively graded complex $g$ of (finitely generated projective) $A_0$-modules

  $$\dots \to g_{-n}\to g_{-n+1}\to \dots \to g_0.$$\dots \to g_{-n}\to g_{-n+1} \to \dots \to g_0.

• A degree 1 derivation Q on the Chevalley-Eilenberg algebra $\mathrm{CE}(g)={\wedge }_A^{•}{g}^{*}=S^{•}{g}^{*}[1]$ of $g$, such that $Q^2=0$.

The Chevalley-Eilenberg algebra $\mathrm{CE}(g)$ of a Lie $\mathrm{\infty }$-algebroid is a commutative semifree dga, and any such (of non negative degree) is of the form $\mathrm{CE}(g)$ for some Lie $\mathrm{\infty }$ algebroid.

A morphism of Lie $\mathrm{\infty }$ algebroids $g\to h$ is a linear map $\varphi :\mathrm{CE}(h)\to \mathrm{CE}(g)$ of the Chevalley-Eilenberg algebras in the opposite direction, respecting the grading and the differentials.

Notice that the first few terms of ${\wedge }^{•}{g}^{*}$ in degree 0, 1 and 2, respectively, are

$${\wedge }^{•}{g}^{*}=A_0\oplus g_0^{*}[1]\oplus (g_0^{*}[1]\wedge g_0^{*}[1])\oplus g_{-1}^{*}[1]\oplus \cdots .$$\wedge^\bullet g^* = A_0 \oplus g_0^*[1] \oplus (g_0^*[1]\wedge g_0^*[1]) \oplus g_{-1}^*[1] \oplus \cdots \,.

Example

Take $g=\Gamma (TX)$ to be the $C^{\mathrm{\infty }}(X)$-module of tangent vector fields on $X$, then ${\wedge }_{C^{\mathrm{\infty }}(X)}^{•}{g}^{*}={\wedge }_{C^{\mathrm{\infty }}(X)}^{•}{\Omega }^1(X)={\Omega }^{•}(X)$.

Remark

There is plenty of room to fuss about the grading conventions.

The Chevalley–Ehresmann functor on Lie $\mathrm{\infty }$-algebroids

A Lie $\mathrm{\infty }$-algebroid $g$ is a complex of $A_0$-modules, so corresponds to a bunch of vector bundles $E_i\to X_0$ over the underlying manifold $X_0$, where $g_i=\Gamma (E_i)$, and the differential of the complex gives fibrations

$$\dots \to E_{-n}\to E_{-n+1}\to \dots \to E_0\to X_0$$\dots \to E_{-n}\to E_{-n+1}\to \dots \to E_0\to X_0

So one can think of a Lie $\mathrm{\infty }$-algebroid as a rather intricate classical geometric structure: vector bundles with extra stuff. These form a category. But taking the Chevalley–Ehresmann algebra gives a functor

$$\mathrm{Lie}\mathrm{\infty }\mathrm{Algebroids}\stackrel{\mathrm{CE}(-)}{\to }\mathrm{sDGCAs}$$Lie \infty Algebroids \stackrel{CE(-)}{\to} sDGCAs

from the category of Lie $\mathrm{\infty }$-algebroids to the category of semi free Differential Graded Commutative algebras (sDGCAs).

Each SDGCA is the algebras of functions on a supermanifold, more precisely an NQ-supermanifold. So the CE functor gives a remarkable connection between classical geometry and super geometry.

Remarks

• The definition of a Lie $\mathrm{\infty }$-structure in terms of a dual differential graded-commutative algebra is the higher graded version of the definition of Lie algebras in terms of Lie coalgebras.

• In the physics literature $\mathrm{CE}(g)$ is called a BRST complex.

Special Cases

• a Lie $\mathrm{\infty }$-algebroid over a point, $X=\mathrm{pt}$ is an $L_{\mathrm{\infty }}$-algebra;

• an Lie $\mathrm{\infty }$-algebroid $g$ with generators for $\mathrm{CE}(g)$ concentrated in the first $n$ degrees is a Lie $n$-algebroid;

• an Lie $\mathrm{\infty }$-algebroid the differential of whose CE algebra is “co-binary”, i.e. $d:{𝔤}^{*}\to g^{*}\oplus g^{*}\wedge g^{*}$, is strict.

So in particular

• a Lie 1-algebroid is a Lie algebroid;

• a Lie 1-algebroid over the point is a Lie algebra;

• a Lie $n$-algebroid over a point is a Lie n-algebra.

Further examples

• A Courant algebroid is not exactly an $L_{\mathrm{\infty }}$-algebroid, but it is encoded by a Lie $3$-algebra.

When I read the literature a Courant algebraoid seems to be a Lie 2-algebroid with as extra structure a symplectic form of ghostnumber 2. So it is just a special case of a Lie $\mathrm{\infty }$-algebroid, no?

• Standard examples of exterior differential systems are Chevalley–Eilenberg algebras of $L_{\mathrm{\infty }}$-algebroids.

Further resources

There should be an internalization of this into the context of generalized smooth spaces and generalized smooth algebras. This is discussed at generalized smooth L-infinity algebroid.

References

The term “Lie $\mathrm{\infty }$-algebroid” or ”$L_{\mathrm{\infty }}$-algebroid” as such is not as yet established in the literature, as most authors working with these objects think of them entirely in terms of NQ-supermanifolds and either ignore the relation to Lie theory or take it more or less for granted.

Possibly the first explicit appearance of the idea of $\mathrm{\infty }$-Lie algebroids recognized in their full Lie theoretic meaning is

• Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv)

which uses “NQ-supermanifolds”. Of course, as this article also points out, in hindsight one finds that much of this is already implicit in the much older theory of Sullivan models? in rational homotopy theory, which is concerned with modelling spaces by qDGCAs. That these spaces can be regarded as ∞-groupoids and as Lie ∞-groupoids in particular is clear in hindsight, but was possibly first explicitly realized in the above reference. See also Lie integration.

Related entries

• Lie ∞-algebroid representation