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Lie infinity-algebroid

Idea

An $\mathrm{\infty }$-Lie algebroid is the infinitesimal approximation to an ∞-Lie groupoid.

$\mathrm{\infty }$-Lie algebroids are to ∞-Lie groupoids as Lie algebras are to Lie groups.

• Lie group - Lie algebra

• Lie groupoid - Lie algebroid

• ∞-Lie group - ∞-Lie algebra

• ∞-Lie groupoid - $\mathrm{\infty }$-Lie algebroid .

One obtains $\mathrm{\infty }$-Lie groupoids from $\mathrm{\infty }$-Lie algebroids by Lie integration.

In terms of synthetic differential geometry an $\mathrm{\infty }$-Lie algebroid may be thought of as an ∞-Lie groupoid all whose k-morphism-spaces over a given object are infinitesimal spaces.

Since in typical convenient models for synthetic differential geometry these infinitesimal spaces are represented by formal duality in terms of their smooth algebras of functions, it follows that when ∞-groupoids are incarnated as simplicial smooth spaces, an $\mathrm{\infty }$-Lie algebroid

$$𝔞=(\cdots {𝔞}_2\overrightarrow{\overrightarrow{\to }}{𝔞}_1\overrightarrow{\to }{𝔞}_0)$$\mathfrak{a} = \left( \cdots \mathfrak{a}_2 \stackrel{\to}{\stackrel{\to}{\to}} \mathfrak{a}_1 \stackrel{\to}{\to} \mathfrak{a}_0 \right)

may be modeled by cosimplicial smooth algebras

$$C^{\mathrm{\infty }}(𝔞)=(\cdots C^{\mathrm{\infty }}({𝔞}_2)\overleftarrow{\overleftarrow{\leftarrow }}{C}^{\mathrm{\infty }}({𝔞}_1)\overleftarrow{\leftarrow }{C}^{\mathrm{\infty }}({𝔞}_0)).$$C^\infty(\mathfrak{a}) = \left( \cdots C^\infty(\mathfrak{a}_2) \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}} C^\infty(\mathfrak{a}_1) \stackrel{\leftarrow}{\leftarrow} C^\infty(\mathfrak{a}_0) \right) \,.

Under the monoidal Dold-Kan correspondence, these map by the normalized cochain complex functor $N$ to cochain dg-algebras in non-negative degree:

$$\mathrm{CE}(𝔞):=N{C}^{\mathrm{\infty }}(𝔞).$$CE(\mathfrak{a}) := N C^\infty(\mathfrak{a}) \,.

This dg-algebra is usefully thought of as the Chevalley-Eilenberg algebra of the $\mathrm{\infty }$-Lie algebroid. If $𝔞=𝔤$ is an ordinary Lie algebra, then this is indeed the ordinary Chevalley-Eilenberg algebra of that Lie algebra.

In the literature on Lie algebroids, however, $\mathrm{CE}(𝔞)$ often goes by different names, such as “canonical complex” or “the complex that computes Lie algebroid cohomology”. In the literature on what we here identify as $\mathrm{\infty }$-Lie algebroids, the algebras $\mathrm{CE}(𝔞)$ are often thought of as algebras of functions on NQ-supermanifolds.

Accordingly, there is a bit of room for different approaches of how to define the (∞,1)-category of ∞-Lie algebroids. A very general abstract nPOV perspective proceeds via the notion of function algebras on ∞-stacks:

here the (∞,1)-topos $H$ of the given notion of ∞-Lie groupoids is taken to be equipped with a specific line-object $R$, and the (∞,1)-category $L$ of ∞-Lie algebroids is the reflective (∞,1)-subcategory that localize $H$ at those morphism that induce isomorphisms in the $R$-cohomology internal to $H$

$$L\overleftarrow{\hookrightarrow }H.$$\mathbf{L} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H} \,.

In the case that $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ is the (∞,1)-category of (∞,1)-sheaves on a site $C$ like the opposite category ${\mathrm{Alg}}_{ℝ}^{\mathrm{op}}$ of commutative (and suitably “small”) algebras, or the site $C=𝕃$ of smooth loci, the opposite $C^{\mathrm{\infty }}{\mathrm{Alg}}^{\mathrm{op}}$ of (suitably small) smooth algebra, the line object may be taken to be the real line in the corresponding in ternal incarnation, and one finds then that

$$L\simeq ([\Delta ,C^{\mathrm{\infty }}\mathrm{Alg}{]}^{\mathrm{op}}{)}^{\circ }$$\mathbf{L} \simeq ([\Delta, C^\infty Alg]^op)^\circ

is the (∞,1)-category presented by the the opposite of the model structure on cosimplicial commutative (smooth) algebras.

At least for the underlying plain algebras this is equivalent, by the monoidal Dold-Kan correspondence, to the $(\mathrm{\infty }\mathrm{,1})$-category presented by the opposite of the standard model structure on dg-algebras $({\mathrm{dgAlg}}^{\mathrm{op}}{)}^{\circ }$, for graded commutative cochain dg-algebras in non-negative degree. This is of course the category in which much of classical rational homotopy theory takes place, and indeed it has been noticed that much of classical rational homotopy theory may be understood as being about $\mathrm{\infty }$-Lie theory. Notably the Sullivan construction of a topological space from a dg-algebra may be thought of as essentially being the Lie integration of the $\mathrm{\infty }$-Lie algebroid corresponding to the dg-algebra to an ∞-groupoid. This can straightforwardly be refined to an integration to an ∞-Lie groupoid.

Special Cases

• a $\mathrm{\infty }$-Lie algebroid over the point, $𝔞=*$ is an L-∞-algebra;

• an $n$-truncated $\mathrm{\infty }$-Lie algebroid is a Lie $n$-algebroid;

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

So in particular

• a 1-Lie algebroid is a Lie algebroid;

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

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

• a BRST-complex? is the Chevalley-Eilenberg algebra of an action-$\mathrm{\infty }$-Lie algebroid of the action of an $L_{\mathrm{\infty }}$-algebra, see Lie ∞-algebroid representation;

  more generally, the complexes appearing in BV-BRST formalism are derived $\mathrm{\infty }$-Lie algebroids, whose Chevalley-Eilenberg algebra may have generators in negative degree.

• a Courant algebroid is a certain Lie 2-algebroid.

• more generally an n-symplectic manifold is a Lie $n$-algebroid.

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

Examples

Tangent Lie algebroid

The following example is in a way the archetypical example on which all others are modeled in a sense.

For $X$ any smooth manifold, there is a standard notion of the Lie algebroid which is the tangent Lie algebroid of $X$. As an $\mathrm{\infty }$-Lie algebroid, this is the simplicial smooth locus given by the infinitesimal singular simplicial complex

$$TX=(\cdots X^{({\Delta }_{\inf }^2)}\overrightarrow{\overrightarrow{\to }}{X}^{({\Delta }_{\inf }^1)}\overrightarrow{\to }X)$$T X = \left( \cdots X^{(\Delta^2_{inf})} \stackrel{\to}{\stackrel{\to}{\to}} X^{(\Delta^1_{inf})} \stackrel{\to}{\to} X \right)

of $X$. By the central observation on combinatorial differential forms in synthetic differential geometry by Anders Kock, the Chevalley-Eilenberg algebra of this is indeed isomorphic to the de Rham complex of $X$:

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

For more details on the computations involved see Spaces of infinitesimal k-simplices at infinitesimal object.

In particular for $X={ℝ}^n$ a Cartesian space, we have

$$T{ℝ}^n=(\cdots {ℝ}^n\times \tilde{D}(2,n)\overrightarrow{\overrightarrow{\to }}{ℝ}^n\times \tilde{D}(1,n)\overrightarrow{\to }{ℝ}^n),$$T \mathbb{R}^n = \left( \cdots \mathbb{R}^n \times \tilde D(2,n) \stackrel{\to}{\stackrel{\to}{\to}} \mathbb{R}^n \times \tilde D(1,n) \stackrel{\to}{\to} \mathbb{R}^n \right) \,,

where $\tilde{D}(k,n)$ is the smooth locus of infinitesimal k-simplices based at the origin in ${ℝ}^n$.

Lie algebra

Let $G$ be a Lie group with Lie algebra $𝔤$. We describe how $𝔤$ looks when regarded as a special case of an $\mathrm{\infty }$-Lie algebroid.

Write

$$BG=(\cdots G\times G\overrightarrow{\overrightarrow{\to }}G\overrightarrow{\to }*)$$\mathbf{B}G = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right)

for the delooping groupoid of $G$, regarded as an an ∞-Lie groupoid modeled by a simplicial smooth space.

We claim that a morphism

$$\omega :TU\to BG$$\omega : T U \to \mathbf{B}G

from the tangent Lie algebroid of some $U\in$ CartSp is flat Lie-algebra valued form and how that can be used to find the Lie algebra $𝔤$ as the infinitesimal sub-$\mathrm{\infty }$-groupoid

$$b𝔤\hookrightarrow BG$$\mathbf{b}\mathfrak{g} \hookrightarrow \mathbf{B}G

inside $BG$.

Since $BG$ is 2-coskeletal (being the nerve of a groupoid) a morphism $TU\to BG$ is fixed already under its 2-truncation

$$\begin{array}{ccc}U\times \tilde{D}(2,n)& \stackrel{{\omega }_2}{\to }& G\times G\\ \downarrow \downarrow \downarrow & & {\downarrow }^{p_2}{\downarrow }^{\cdot }{\downarrow }^{p_1}\\ U\times \tilde{D}(1,n)& \stackrel{{\omega }_1}{\to }& G\\ \downarrow \downarrow & & \downarrow \downarrow \\ U& \to & *\end{array}.$$\array{ U \times \tilde D(2,n) &\stackrel{\omega_2}{\to}& G \times G \\ \downarrow \downarrow \downarrow && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} \\ U \times \tilde D(1,n) &\stackrel{\omega_1}{\to}& G \\ \downarrow \downarrow && \downarrow \downarrow \\ U &\to& * } \,.

It is clear that ${\omega }_1$ factors through the inclusion $\tilde{D}(1,\dim (G))\hookrightarrow G$ that sends the unique point of $\tilde{D}(1,\dim (G))$ to the neutral element (by respect for the degeneracy maps). Then from that one finds that ${\omega }_2$ factors through the inclusion $\tilde{D}(2,\dim (G))\hookrightarrow G\times G$ that sends the unique point of $\tilde{D}(2,\dim (G))$ to $(e,e)\in G\times G$. And evidently these two factorizations are universal, in that every other factorization will uniquelyy factor through these

$$\begin{array}{ccccc}U\times \tilde{D}(2,n)& \stackrel{{\omega }_2}{\to }& \tilde{D}(2,\dim (G))& \hookrightarrow & G\times G\\ \downarrow \downarrow \downarrow & & {\downarrow }^{p_2}{\downarrow }^{\cdot }{\downarrow }^{p_1}& & {\downarrow }^{p_2}{\downarrow }^{\cdot }{\downarrow }^{p_1}\\ U\times \tilde{D}(1,n)& \stackrel{{\omega }_1}{\to }& \tilde{D}(1,\dim (G))& \hookrightarrow & G\\ \downarrow \downarrow & & \downarrow \downarrow & & \downarrow \downarrow \\ U& \to & *& \to & *\end{array}.$$\array{ U \times \tilde D(2,n) &\stackrel{\omega_2}{\to}& \tilde D(2,dim(G)) &\hookrightarrow& G \times G \\ \downarrow \downarrow \downarrow && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} && \downarrow^{p_2} \downarrow^{\cdot} \downarrow^{p_1} \\ U \times \tilde D(1,n) &\stackrel{\omega_1}{\to}& \tilde D(1, dim(G)) &\hookrightarrow& G \\ \downarrow \downarrow && \downarrow \downarrow && \downarrow \downarrow \\ U &\to& * &\to& * } \,.

The universal object found this way we claim is the Lie algebra $𝔤$ in its incarnation as an infinitesimal $\mathrm{\infty }$-Lie groupoid

$$\begin{array}{rl}b𝔤& :=\mathrm{InitialObject}(TU\downarrow {𝕃}^{{\mathrm{Delta}}^{\mathrm{op}}}\downarrow BG)\\ & =(\cdots \tilde{D}(2,\dim (G))\overrightarrow{\overrightarrow{\to }}\tilde{D}(1,\dim (G))\overrightarrow{\to }*)\end{array}.$$\begin{aligned} b \mathfrak{g} &:= InitialObject( T U\downarrow \mathbb{L}^{Delta^{op}}\downarrow \mathbf{B}G) \\ & = \left( \cdots \tilde D(2,dim(G)) \stackrel{\to}{\stackrel{\to}{\to}} \tilde D(1,dim(G))\stackrel{\to}{\to} * \right) \end{aligned} \,.

Proposition

The normalized cochain complex of the cosimplicial alghebra of functions on thi $b𝔤$ is isomorphic to the ordinary Chevalley-Eilenberg algebra $({\wedge }^{•}{𝔤}^{*},[-,-{]}^{*})$ of $𝔤$.

Proof

By the discussion at Spaces of infinitesimal k-simplices we have that for $C^{\mathrm{\infty }}(\tilde{D}(k,\dim (G)){)}_{\mathrm{top}}\subset C^{\mathrm{\infty }}(\tilde{D}(k,n))$ the subspace of those functions that are in the joint kernel of the co-degeneracy maps is naturally isomorpic to ${\wedge }^k({ℝ}^{\dim (G)}{)}^{*}$, so that we have a natural isomorphism of vector spaces

$$N{C}^{\mathrm{\infty }}(b𝔤{)}_k\simeq {\wedge }^k{𝔤}^{*}.$$N C^\infty(b \mathfrak{g})_k \simeq \wedge^k \mathfrak{g}^* \,.

By the fact that everything is 2-coskeletal it suffices to check that the differential in first degree

$$N{C}^{\mathrm{\infty }}(\tilde{D}(1,\dim (G)))\stackrel{p_1^{*}+p_2^{*}-(\cdot {)}^{*}}{\to }N{C}^{\mathrm{\infty }}(\tilde{D}(2,\dim (G)))$$N C^\infty(\tilde D(1,dim(G))) \stackrel{p_1^* + p_2^* - (\cdot)^*}{\to} N C^\infty(\tilde D(2,dim(G)))

is indeed the dual of the Lie bracket. But the product ${\cdot }_G:G\times G\to G$ restricted along $\tilde{D}(2,\dim (G))\hookrightarrow G\times G$ to the infinitesimal space $\tilde{D}(2,\dim (G))$ linearizes in each of its arguments: for $(\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{y})$ \in \tilde D(2,dim(G))$ we have

$$\stackrel{\rightharpoonup }{x}{\cdot }_G\stackrel{\rightharpoonup }{y}=\stackrel{\rightharpoonup }{x}\cdot {\nabla }_x{\cdot }_G(\mathrm{0,0})+\stackrel{\rightharpoonup }{y}\cdot {\nabla }_y{\cdot }_G(\mathrm{0,0})+\stackrel{\rightharpoonup }{x}\cdot {\nabla }_x\stackrel{\rightharpoonup }{y}\cdot {\nabla }_y{\cdot }_G(\mathrm{0,0}).$$\vec x \cdot_G \vec y = \vec x \cdot \nabla_x \cdot_G (0,0) + \vec y \cdot \nabla_y \cdot_G (0,0) + \vec x \cdot \nabla_x \vec y \cdot \nabla_y \cdot_G(0,0) \,.

Since thr origin here corresponds to the neutral element of $G$ and since with one of its arguments the neutral element the operaton ${\cdot }_G$ is the identity, and since the double derivative produces the Lie bracket (keeping in mind that $x^i{y}^j+x^j{y}^i=0$ in $\tilde{D}(2,\dim (G))$), this is

$$\cdots =\stackrel{\rightharpoonup }{x}+\stackrel{\rightharpoonup }{y}+[\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{y}].$$\cdots = \vec x + \vec y + [\vec x, \vec y] \,.

Accordingly the alternating sum of co-face maps is

$$\begin{array}{rl}d& =p_1^{*}+p_2^{*}-{\cdot }_G^{*}\\ & =p_1^{*}+p_2^{*}-(p_1^{*}+p_2^{*}+[-,-{]}^{*})\\ & =-[-,-{]}^{*}\end{array}$$\begin{aligned} d &= p_1^* + p_2^* - \cdot_G^* \\ & = p_1^* + p_2^* - ( p_1^* + p_2^* + [-,-]^*) \\ & = - [-,-]^* \end{aligned}

as it should be for the Chevalley-Eilenberg algebra of a Lie algebra.

The infinitesimal reasoning involved in this proof is discussed in section 6.8 of

• Anders Kock, Synthetic geometry of manifolds (pdf)

Tangent Lie algebroid of a Lie algebra

We can form the tangent $\mathrm{\infty }$-Lie algebroid of any $\mathrm{\infty }$-Lie algebroid $𝔞$ as

$$T𝔞={\int }^{[n]\in \Delta }\Delta [n]\cdot T{𝔞}_n={\int }^{[n]\in \Delta }\Delta [n]\cdot ({𝔞}_n{)}^{({\Delta }_{\inf }^{•})}.$$T \mathfrak{a} = \int^{[n] \in \Delta} \Delta[n] \cdot T \mathfrak{a}_n = \int^{[n] \in \Delta} \Delta[n] \cdot (\mathfrak{a}_n)^{(\Delta^\bullet_{inf})} \,.

We want to claim now that for $𝔤$ a Lie algebra, we have a canonical isomorphism

$$\mathrm{CE}(Tb𝔤)\simeq W(𝔤)$$CE(T b \mathfrak{g}) \simeq W(\mathfrak{g})

that identifies the Chevalley-Eilenberg algebra of the tangent Lie algebra of $bg$ with the Weil algebra of $𝔤$.

The key observation for this is that in the bisimplicial object

$$\begin{array}{ccccccc}& & ⋮& & ⋮& & ⋮\\ & \cdots & \tilde{D}(3,\dim (G){)}^{({\Delta }_{\inf }^2)}& & \tilde{D}(3,\dim (G){)}^{({\Delta }_{\inf }^1)}& & \tilde{D}(3,\dim (G))\\ & & \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ & \cdots & \tilde{D}(1,\dim (G){)}^{({\Delta }_{\inf }^2)}& & \tilde{D}(1,\dim (G){)}^{({\Delta }_{\inf }^1)}& & \tilde{D}(1,\dim (G))\\ & & \downarrow \downarrow & & \downarrow \downarrow & & \downarrow \downarrow \\ & \cdots & *& \overrightarrow{\overrightarrow{\to }}& *& \overrightarrow{\to }& *\end{array}$$\array{ && \vdots && \vdots && \vdots \\ &\cdots& \tilde D(3,dim(G))^{(\Delta^2_{inf})} && \tilde D(3,dim(G))^{(\Delta^1_{inf})} && \tilde D(3,dim(G)) \\ &&\downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow \\ &\cdots& \tilde D(1,dim(G))^{(\Delta^2_{inf})} && \tilde D(1,dim(G))^{(\Delta^1_{inf})} && \tilde D(1,dim(G)) \\ && \downarrow \downarrow && \downarrow \downarrow && \downarrow \downarrow \\ &\cdots& * &\stackrel{\to}{\stackrel{\to}{\to}}& * &\stackrel{\to}{\to}& * }

• the functions on infinitesimal simplices in an infinitesimal space that vanish on degenerate simplices are already isomorphic to covectors at the origin;

• the functions on infinitesimal $r$-simplices in a $\tilde{D}(k,\dim (G))$ for $r>k$ which vanish on degenerate simplices already vanish entirely.

Using this the total complex of $N{C}^{\mathrm{\infty }}(-)$ of this bisimplicial set is manifestly isomorphic to the Weil algebra.

…

Related concepts

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

see

• ∞-Lie algebroid valued differential forms

• ∞-Chern-Weil theory

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 dg-algebras or 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 topological spaces by dg-algebras. 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, rational homotopy theory in an (∞,1)-topos? and function algebras on ∞-stacks.

The explicit term $\mathrm{\infty }$-Lie algebroid / $L_{\mathrm{\infty }}$-algebroid as such appears in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential string- and fivebrane structures (ref)

The term also appears in

• Andrew James Bruce, From $L_{\mathrm{\infty }}$-algebroids to higher Schouten/Poisson structures (arXiv:1007.1389)