nLab L-infinity-algebra

Idea

$L_\infty$-algebras (or strong homotopy Lie algebras) are a higher generalization (a “vertical categorification”) of Lie algebras: in an $L_\infty$-algebra the Jacobi identity is allowed to hold (only) up to higher coherent homotopy.

An $L_\infty$-algebra that is concentrated in lowest degree is an ordinary Lie algebra. If it is concentrated in the lowest two degrees is is a Lie 2-algebra, etc.

From another perspective: an $L_\infty$-algebra is a Lie ∞-algebroid with a single object.

$L_\infty$-algebras are infinitesimal approximations of smooth ∞-groups in analogy to how an ordinary Lie algebra is an infinitesimal approximation of a Lie group. Under Lie integration every $L_\infty$-algebra $\mathfrak{g}$ “exponentiates” to a smooth ∞-group $\Omega \exp(\mathfrak{g})$.

History

Remark

(history of the concept of (super-)$L_\infty$ algebras)

The identification of the concept of (super-)$L_\infty$-algebras has a non-linear history:

L-∞ algebras in the incarnation of higher brackets satisfying a higher Jacobi identity (def. ) were introduced in Stasheff 92, Lada-Stasheff 92, based on the example of such a structure on the BRST complex of the bosonic string that was reported in the construction of closed string field theory in Zwiebach 92.

Lada-Stasheff 92 credit Schlessinger-Stasheff 85 with the introduction of the concept, but while that article considers many closely related structures, it does not consider $L_\infty$-algebras as such. Lada-Markl 94 credit other work by Schlessinger-Stasheff as the origin, but that work appeared much later as Schlessinger-Stasheff 12.

According to Stasheff 16, slide 25, Zwiebach had this structure already in 1989, when Stasheff recognized it in a talk by Zwiebach at a GUT conference in Chapel-Hill. Zwiebach, in turn, was following the BV-formalism of Batalin-Vilkovisky 83, Batakin-Fradkin 83, whose relation to $L_\infty$-algebras was later amplified in Stasheff 96, Stasheff 97.

The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras is due to Lada-Markl 94. See Sati-Schreiber-Stasheff 08, around def. 13.

But in this dual incarnation, L-∞ algebras and more generally super L-∞ algebras (of finite type) had secretly been introduced, independently of the BV-formalism of Batalin-Vilkovisky 83, Batakin-Fradkin 83, within the supergravity literature already in D’Auria-Fré-Regge 80 and explicitly in van Nieuwenhuizen 82. The concept was picked up in the D'Auria-Fré formulation of supergravity (D’Auria-Fré 82) and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the supergravity literature (but beware that these dg-algebras are in general free only as graded-supercommutative superalgebras, not as differential algebras) The relation between super $L_\infty$-algebras and the “FDA”s of the supergravity literature is made explicit in (FSS 13).

higher Lie theory	supergravity
$\,$ super Lie n-algebra $\mathfrak{g}$ $\,$	$\,$ “FDA” $CE(\mathfrak{g})$ $\,$

The construction in van Nieuwenhuizen 82 in turn was motivated by the Sullivan algebras in rational homotopy theory (Sullivan 77). Indeed, their dual incarnations in rational homotopy theory are dg-Lie algebras (Quillen 69), hence a special case of $L_\infty$-algebras.

This close relation between rational homotopy theory and higher Lie theory might have remained less of a secret had it not been for the focus of Sullivan minimal models on the non-simply connected case, which excludes the ordinary Lie algebras from the picture. But the Quillen model of rational homotopy theory effectively says that for $X$ a rational topological space then its loop space ∞-group $\Omega X$ is reflected, infinitesimally, by an L-∞ algebra. This perspective began to receive more attention when the Sullivan construction in rational homotopy theory was concretely identified as higher Lie integration in Henriques 08. A modern review that makes this L-∞ algebra-theoretic nature of rational homotopy theory manifest is in Buijs-Félix-Murillo 12, section 2.

Definition

In terms of algebras over an operad

An $L_\infty$-algebra is an algebra over an operad in the category of chain complexes over the L-∞ operad.

In the following we spell out in detail what this means in components.

In terms of higher brackets

We now state the definition of $L_\infty$-algebras that is most directly related to the traditional definition of ordinary Lie algebras, namely as $\mathbb{Z}$-graded vector space $\mathfrak{g}$ equipped with $n$-ary multilinear and graded-skew symmetric maps $[-,\cdots,-]$ – the “brackets” – that satisfy a generalization of the Jacobi identity.

To that end, we here choose grading conventions such that the following definition of $L_\infty$-algebras reduces to that of ordinary Lie algebras when $\mathfrak{g}$ is concentrated in degree zero. Moreover we take the differential of the underlying chain complex of the $L_\infty$-algebra to have degree $-1$ (“homological grading”). Together this means in particular that $\mathfrak{g}$ is a Lie n-algebra for $n \in \mathbb{N}$, $n \geq 1$, if it is concentrated in degrees 0 to $n-1$.

Beware that there are also other conventions possible, and there are other conventions in use, for both these choices, leading to different signs in the following formulas.

Definition

(graded signature of a permuation)

Let $V$ be a $\mathbb{Z}$-graded vector space, and for $n \in \mathbb{N}$ let

$$\mathbf{v} = (v_1, v_2, \cdots, v_n)$$

be an n-tuple of elements of $V$ of homogeneous degree $\vert v_i \vert \in \mathbb{Z}$, i.e. such that $v_i \in V_{\vert v_i\vert}$.

For $\sigma$ a permutation of $n$ elements, write $(-1)^{\vert \sigma \vert}$ for the signature of the permutation, which is by definition equal to $(-1)^k$ if $\sigma$ is the composite of $k \in \mathbb{N}$ permutations that each exchange precisely one pair of neighboring elements.

We say that the $\mathbf{v}$-graded signature of $\sigma$

$$\chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\}$$

is the product of the signature of the permutation $(-1)^{\vert \sigma \vert}$ with a factor of $(-1)^{\vert v_i \vert \vert v_j \vert}$ for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the decomposition of the permuation as a sequence of swapping neighbour pairs.

Definition

An $L_\infty$-algebra is

1. a $\mathbb{Z}$-graded vector space $\mathfrak{g}$;

2. for each $n \in \mathbb{N}$, $n \geq 1$ a multilinear map, called the $n$-ary bracket, of the form

   $$l_n(\cdots) \;\coloneqq\; [-,-, \cdots, -]_n \;\colon\; \underset{n \; \text{copies}}{\underbrace{\mathfrak{g} \otimes \cdots \otimes \mathfrak{g}}} \longrightarrow \mathfrak{g}$$

   and of degree $n-2$

   (if one includes here $n = 0$ then one speaks of a curved L-infinity algebra)

such that the following conditions hold:

1. (graded skew symmetry) each $l_n$ is graded antisymmetric, in that for every permutation $\sigma$ of $n$ elements and for every n-tuple $(v_1, \cdots, v_n)$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ then

   $$l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)}) = \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n)$$

   where $\chi(\sigma,v_1,\cdots, v_n)$ is the $(v_1,\cdots,v_n)$-graded signature of the permuation $\sigma$, according to def. ;

2. (strong homotopy Jacobi identity) for all $n \in \mathbb{N}$, and for all $n$-tuples $(v_1, \cdots, v_n)$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ the following equation holds

   (1)$$\sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}} \sum_{\sigma \in UnShuff(i,j-1)} \chi(\sigma,v_1, \cdots, v_{n}) (-1)^{i(j-1)} l_{j} \left( l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right), v_{\sigma(i+1)} , \cdots , v_{\sigma(n)} \right) = 0 \,,$$

   where the inner sum runs over all $(i,j-1)$-unshuffles $\sigma$ and where $\chi$ is the graded signature sign from def. .

Example

In lowest degrees the generalized Jacobi identity says that

1. for $n = 1$: the unary map $\partial \coloneqq l_1$ squares to 0:

   $$\partial (\partial(v_1)) = 0$$

2. for $n = 2$: the unary map $\partial$ is a graded derivation of the binary map

   $$- [\partial v_1, v_2] - (-1)^{\vert v_1 \vert \vert v_2 \vert} [\partial v_2, v_1] + \partial [v_1, v_2] = 0$$

   hence

   $$\partial [v_1, v_2] = [\partial v_1, v_2] + (-1)^{\vert v_1 \vert} [v_1, \partial v_2] \,.$$

Example

When all higher brackets vanish, $l_{k \gt 2}= 0$ then for $n = 3$:

$$[[v_1,v_2],v_3] + (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)} [[v_2,v_3],v_1] + (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)} [[v_1,v_3],v_2] = 0$$

this is the graded Jacobi identity. So in this case the $L_\infty$-algebra is equivalently a dg-Lie algebra.

Example

When $l_3$ is possibly non-vanishing, then on elements $x_i$ on which $\partial = l_1$ vanishes, the generalized Jacobi identity for $n = 3$ gives

$$[[v_1,v_2],v_3] + (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)} [[v_2,v_3],v_1] + (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)} [[v_1,v_3],v_2] = - \partial [v_1, v_2, v_3] \,.$$

This shows that the Jacobi identity holds up to an “exact” term, hence up to homotopy.

In terms of semifree differential coalgebra

In (Lada-Stasheff 92) it was pointed out that the higher brackets of an $L_\infty$-algebra (def. ) induce on the graded-co-commutative cofree coalgebra $\vee^\bullet \mathfrak{g}$ over the underlying graded vector space $\mathfrak{g}$ the structure of a differential graded coalgebra, with differential $D = [-] + [-,-] + [-,-,-] + \cdots$ the sum of the higher brackets, extended as graded coderivations. The higher Jacobi identity is equivalently the condition that $D^2 = 0$. In (Lada-Markl 94) it was observed that conversely, such “semifree” differential graded coalgebras are an equivalent incarnation of $L_\infty$-algebras.

(If one uses unital dg-co-algebras then the $L_\infty$-algbras encoded with way are generally curved L-infinity algebras. To restrict to the non-curved one one either considers co-augmented unital dg-co-algebras or non-unital coalgebras.)

Notice that this immediately imples that if $\mathfrak{g}$ is degreewise finite dimensional, then passing to dual vector spaces turns semifree differential graded coalgebra into semifree differential graded algebras, which hence are opposite-equivalent to $L_\infty$-algebras of finite type. For $\mathfrak{g}$ an ordinary finite dimensional Lie algebra, then this dg-algebras is its Chevalley-Eilenberg algebra, hence we may generally speak of Chevalley-Eilenberg algebras of $L_\infty$-algebras of finite type (and also more generally, if one invokes pro-objects, see at model structure for L-infinity algebras – Use of pro-dg-algebras ).

In term of the operadic definition of $L_\infty$-algebras above this equivalence is an incarnation of the Koszul duality between the Lie operad and the commutative operad.

We now spell out this dg-coalgebraic incarnation of $L_\infty$-algebras.

A (connected) $L_\infty$-algebra is

• an $\mathbb{N}_+$-graded vector space $\mathfrak{g}$;

• equipped with a differential $D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g}$ of degree $-1$ on the free graded co-commutative coalgebra over $\mathfrak{g}$ that squares to 0

$$D^2 = 0 \,.$$

Here the free graded co-commutative co-algebra $\vee^\bullet \mathfrak{g}$ is, as a vector space, the same as the graded Grassmann algebra $\wedge^\bullet \mathfrak{g}$ whose elements we write as

$$3 t_1 \vee t_2 + t_3 + t_3 \vee t_4 \vee t_5$$

etc (where the $\vee$ is just a funny way to write the wedge $\wedge$, in order to remind us that:…)

but thought of as equipped with the standard coproduct

$$\Delta (v_1 \vee v_2 \cdots \vee v_n) \propto \sum_i \pm (v_1 \vee \cdots \vee v_i) \otimes (v_{i+1} \vee \cdots \vee v_n)$$

(work out or see the references for the signs and prefacors).

Since this is a free graded co-commutative coalgebra, one can see that any differential

$$D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g}$$

on it is fixed by its value “on cogenerators” (a statement that is maybe unfamiliar, but simply the straightforward dual of the more familar statement to which we come below, that differentials on free graded algebras are fixed by their action on generators) which means that we can decompose $D$ as

$$D = D_1 + D_2 + D_3 + \cdots \,,$$

where each $D_i$ acts as $l_i$ when evaluated on a homogeneous element of the form $t_1 \vee \cdots \vee t_n$ and is then uniquely extended to all of $\vee^\bullet \mathfrak{g}$ by extending it as a coderivation on a coalgebra.

For instance $D_2$ acts on homogeneous elements of word lenght 3 as

$$D_2(t_1 \vee t_2 \vee t_3) = D_2(t_1, t_2)\vee t_3 \pm permutations \,.$$

exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below.

Using this, one checks that the simple condition that $D$ squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:

$$(D^2 = 0) \Leftrightarrow \left( \forall n : \sum_{i+j = n} \sum_{shuffles \sigma} \pm l_i (l_j (v_{\sigma(1)}, \cdots, v_{\sigma(j)} ) , v_{\sigma(j+1)} , \cdots , v_{\sigma(n)} ) = 0 \right) \,.$$

So in conclusion we have:

An $L_\infty$-algebra is a dg-coalgebra whose underlying coalgebra is cofree and concentrated in negative degree.

In terms of semifree differential algebra

The reformulation of an $L_\infty$-algebra as simply a semi-co-free graded-co-commutative coalgebra $(\vee^\bullet \mathfrak{g}, D)$ is a useful repackaging of the original definition, but the coalgebraic aspect tends to be not only unfamiliar, but also a bit inconvenient. At least when the graded vector space $\mathfrak{g}$ is degreewise finite dimensional, we may simply pass to its degreewise dual graded vector space $\mathfrak{g}^*$.

(Fully generally the following works when using not just dg-algebras but pro-objects in dg-algebras, see at model structure for L-infinity algebras – Use of pro-dg-algebras).

Its Grassmann algebra $\wedge^\bullet \mathfrak{g}^*$ is then naturally equipped with an ordinary differential $d = D^*$ which acts on $\omega \in \wedge^\bullet \mathfrak{g}^*$ as

$$(d \omega) (t_1 \vee \cdots \vee t_n) = \pm \omega(D(t_1 \vee \cdots \vee t_n)) \,.$$

When the grading-dust has settled one finds that with

$$\wedge^\bullet \mathfrak{g}^* = k \oplus \mathfrak{g}^*_1 \oplus (\mathfrak{g}^*_1 \wedge \mathfrak{g}^*_1 \oplus \mathfrak{g}^*_2) \oplus \cdots$$

with the ground field in degree 0, the degree 1-elements of $\mathfrak{g}^*$ in degree 1, etc, that $d$ is of degree +1 and of course squares to 0

$$d^2 = 0 \,.$$

This means that we have a semifree dga

$$CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d) \,.$$

In the case that $\mathfrak{g}$ happens to be an ordinary Lie algebra, this is the ordinary Chevalley-Eilenberg algebra of this Lie algebra. Hence we should generally call $CE(\mathfrak{g})$ the Chevalley-Eilenberg algebra of the $L_\infty$-algebra $\mathfrak{g}$.

One observes that this construction is bijective: every (degreewise finite dimensional) cochain semifree dga generated in positive degree comes from a (degreewise finite dimensional) $L_\infty$-algebra this way.

This means that we may just as well define a (degreewise finite dimensional) $L_\infty$-algebra as an object in the opposite category of (degreewise finite dimensional) commutative dg-algebras that are semifree dgas and generated in positive degree.

(In general this corresponds to curved L-infinity algebra. The flat $L_\infty$-algebras $\mathfrak{g}$ dually correspond to the dg-algebras which are augmented over $\mathbb{R}$, i.e for which the canonical projection $CE(\mathfrak{g}) \longrightarrow \mathbb{R}$ is a homomorphism of dg-algebras.)

And this turns out to be one of the most useful perspectives on $L_\infty$-algebras.

In particular, if we simply drop the condition that the dg-algebra be generated in positive degree and allow it to be generated in non-negative degree over the algebra in degree 0, then we have the notion of the (Chevalley-Eilenberg algebra of) an L-infinity-algebroid.

Details

We discuss in explit detail the computation that shows that an $L_\infty$-algebra structure on $\mathfrak{g}$ is equivalently a dg-algebra-structure on $\wedge^\bullet \mathfrak{g}^*$.

Let $\mathfrak{g}$ be a degreewise finite-dimensional $\mathbb{N}_+$graded vector space equipped with multilinear graded-symmetric maps

$$[-,\cdots,-]_k : Sym^k \mathfrak{g} \to \mathfrak{g}$$

of degree -1, for each $k \in \mathbb{N}_+$.

Let $\{t_a\}$ be a basis of $\mathfrak{g}$ and $\{t^a\}$ a dual basis of the degreewise dual $\mathfrak{g}^*$. Equip the Grassmann algebra $Sym^\bullet \mathfrak{g}^*$ with a derivation

$$d : Sym^\bullet \mathfrak{g}^* \to Sym^\bullet \mathfrak{g}^*$$

defined on generators by

$$d : t^a \mapsto - \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, \cdots, t_{a_k}]^a_k \, t^{a_1} \wedge \cdots \wedge t^{a_k} \,.$$

Here we take $t^a$ to be of the same degree as $t_a$. Therefore this derivation has degree +1.

We compute the square $d^2 = d \circ d$:

$$\begin{aligned} d d t^a &= d (-1)\sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, \cdots, t_{a_k}]^a_k \, t^{a_1} \wedge \cdots \wedge t^{a_k} \\ & = \sum_{k,l = 1}^\infty \frac{1}{(k-1)! l!} [[t_{b_1}, \cdots, t_{b_l}], t_{a_2}, \cdots, t_{a_k}]^a \, t^{b_1} \wedge \cdots \wedge t^{b_l} \wedge t^{a_2} \wedge \cdots \wedge t^{a_{k}} \end{aligned} \,.$$

Here the wedge product on the right projects the nested bracket onto its graded-symmetric components. This is produced by summing over all permutations $\sigma \in \Sigma_{k+l-1}$ weighted by the Koszul-signature of the permutation:

$$\cdots = \sum_{k,l = 1}^\infty \frac{1}{(k+l-1)!} \sum_{\sigma \in \Sigma_{k+l-1}} (-1)^{sgn(\sigma)} \frac{1}{(k-1)! l!} [[t_{b_1}, \cdots, t_{b_l}], t_{a_2}, \cdots, t_{a_k}]^a \, t^{b_1} \wedge \cdots \wedge t^{b_l} \wedge t^{a_2} \wedge \cdots \wedge t^{a_{k}} \,.$$

The sum over all permutations decomposes into a sum over the $(l,k-1)$-unshuffles and a sum over permutations that act inside the first $l$ and the last $(k-1)$ indices. By the graded-symmetry of the bracket, the latter do not change the value of the nested bracket. Since there are $(k-1)! l!$ many of them, we get

$$\cdots = \sum_{k,l = 1}^\infty \frac{1}{(k+l-1)!} \sum_{\sigma \in Unsh(l,k-1)} (-1)^{sgn(\sigma)} \left[\left[t_{a_1}, \cdots, t_{a_l}\right], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}\right] \, t^{a_1} \wedge \cdots \wedge t^{a_{k+l-1}} \,.$$

Therefore the condition $d^2 = 0$ is equivalent to the condition

$$\sum_{k+l = n-1} \sum_{\sigma \in Unsh(l,k-1)} (-1)^{sgn(\sigma)} \left[\left[t_{a_1}, \cdots, t_{a_l}\right], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}\right] = 0$$

for all $n \in \mathbb{N}$ and all $\{t_{a_i} \in \mathfrak{g}\}$. This is equation (1) which says that $\{\mathfrak{g}, \{[-,\dots,-]_k\}\}$ is an $L_\infty$-algebra.

In terms of algebras over an operad

$L_\infty$-algebras are precisely the algebras over an operad of the cofibrant resolution of the Lie operad.

Examples

Special cases

• An $L_\infty$-algebra for which $V$ is concentrated in the first $n$ degree is a Lie $n$-algebra (sometimes also: “$L_n$-algebra”).

• An $L_\infty$-algebra for which only the unary operation and the binary bracket are non-trivial is a dg-Lie algebra: a Lie algebra internal to the category of dg-algebras. From the point of view of higher Lie theory this is a strict $L_\infty$-algebra: one for which the Jacobi identity does happen to hold “on the nose”, not just up to nontrivial coherent isomorphisms.

• So in particular

  • an $L_\infty$-algebra generated just in degree 1 is an ordinary Lie algebra ;

  • an $L_\infty$-algebra generated just in degree 1 and 2 is a Lie 2-algebra ;

    • an $L_\infty$-algebra generated just in degree 1 and 2 and with at most binary brackets is a strict Lie 2-algebra , equivalently encoded in a differential crossed module.

  • an $L_\infty$-algebra generated just in degree 1, 2 and 3 is a Lie 3-algebra ;

• if $\mathfrak{g}$ is a Lie algebra over $\mathbf{K}$, and $b^{k-1}\mathbb{K}$ is the complex consisting of the field $\mathbb{K}$ in degree $1-k$, then an $L_\infty$-algebra morphism from $\mathfrak{g}$ to $b^{k-1}\mathbb{K}$ is precisely a degree $k$ Lie algebra cocycle.

• The skew-symmetry of the Lie bracket is retained strictly in $L_\infty$-algebras. It is expected that weakening this, too, yields a more general vertical categorification of Lie algebras. For $n=2$ this has been worked out by Dmitry Roytenberg: On weak Lie 2-algebras.

• The horizontal categorification of $L_\infty$-algebras are $L_\infty$-algebroids.

• An $L_\infty$-algebra with only $D_n$ non-vanishing is called an n-Lie algebra – to be distinguished from a Lie $n$-algebra ! However, in large parts of the literature $n$-Lie algebras are considered for which $D_n$ is not of the required homogeneous degree in the grading, or in which no grading is considered in the first place. Such $n$-Lie algebras are not special examples of $L_\infty$-algebras, then. For more see n-Lie algebra.

• An $L_\infty$-algebra internal to super vector spaces is a super L-∞ algebra.

Classes of examples

• automorphism Lie 2-algebra?

• inner derivation Lie 2-algebra

• automorphism ∞-Lie algebra

• For every $\infty$-Lie algebra $\mathfrak{g}$ there is its automorphism ∞-Lie algebra. In terms of rational homotopy theory this models the rational automorphism group of the rational space corresponding to $\mathfrak{g}$.

• Poisson-bracket Lie n-algebra

• Heisenberg Lie n-algebra of an n-plectic manifold or more generally of an n-plectic smooth infinity-groupoid

• some classes of W-algebras are claimed to induce $L_\infty$-algebras in Blumenhagen-Fuchs-Traube 17

Specific examples

• line Lie n-algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

Properties

Ind-Conilpotency

Remark

(Pridham 10, remark 3.15, remark 3.13)

For $\mathfrak{g}$ an $L_\infty$-algebra, then its CE chain dgc-coalgebra $CE_\bullet(\mathfrak{g})$ (above) is ind-conilpotent.

This means that $CE_\bullet(\mathfrak{g})$ is a filtered colimit of sub-dg-coalgebras which are conilpotent, in that for each of them there is $n \in \mathbb{N}$ such that their $n$-fold coproduct vanishes. As such these are like “co-local Artin algebras”.

Moreover, since every dg-coalgebra is the union of its finite-dimensonal subalgebras (see at dg-coalgebra the section As filtered colimits of finite-dimensional pieces), this means that $CE_\bullet(\mathfrak{g})$ is a filtered colimit of finite dimensional conilpotent coalgebras.

This implies that the dual Chevalley-Eilenberg cochain algebra $CE^\bullet(\mathfrak{g})$ is a filtered limit of finite-dimensional nilpotent dgc-algebras (actual local Artin algebras).

Model category structure

See model structure for L-∞ algebras.

Relation to dg-Lie algebras

Every dg-Lie algebra is in an evident way an $L_\infty$-algebra. Dg-Lie algebras are precisely those $L_\infty$-algebras for which all $n$-ary brackets for $n \gt 2$ are trivial. These may be thought of as the strict $L_\infty$-algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.

Theorem

Let $k$ be a field of characteristic 0 and write $L_\infty Alg_k$ for the category of $L\infty$-algebras over $k$.

Then every object of $L_\infty Alg_k$ is quasi-isomorphic to a dg-Lie algebra.

Moreover, one can find a functorial replacement: there is a functor

$$W : L_\infty Alg_k \to L_\infty Alg_k$$

such that for each $\mathfrak{g} \in L_\infty Alg_k$

1. $W(\mathfrak{k})$ is a dg-Lie algebra;

2. there is a quasi-isomorphism

   $$\mathfrak{g} \stackrel{\simeq}{\to} W(\mathfrak{g}) \,.$$

This appears for instance as (Kriz & May 1995, Cor. 1.6).

For more see at relation between L-∞ algebras and dg-Lie algebras.

Relation to $\infty$-Lie groupoids

In generalization to how a Lie algebra integrates to a Lie group, $L_\infty$-algebras integrate to ∞-Lie groups.

See

Lie integration

and

Lie integrated ∞-Lie groupoids.

Related concepts

• A-∞ algebra

• E-∞ algebra

• $L_\infty$-algebra

  • homomorphism of L-∞ algebras

  • infinity-Lie algebra cohomology

  • universal enveloping E-n algebra

  • nilpotent L-∞ algebra

  • L-∞ algebroid

  • derived L-∞ algebroid

  • sheaf of L-∞ algebras

  • quantum L-∞ algebra

• relation between L-∞ algebras and dg-Lie algebras

• EL-∞ algebra

References

General

The concept of L-∞ algebras as graded vector spaces equipped with $n$-ary brackets satisfying a generalized Jacobi identity is due to:

• Jim Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, in Quantum groups Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992 (doi:10.1007/BFb0101184).

• Tom Lada, Jim Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993) 1087-1103 [doi:10.1007/BF00671791, arXiv:hep-th/9209099]

• Tom Lada, Martin Markl, Strongly homotopy Lie algebras, Communications in Algebra 23 6 (1995) [doi:10.1080/00927879508825335, arXiv:hep-th/9406095]

• Maxim Kontsevich, Section 4.3 of: Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003) 157-216 (arXiv:q-alg/9709040, doi:10.1023/B:MATH.0000027508.00421.bf)

At least Stasheff 92 was following Zwiebach 92, who had observed that the n-point functions in closed string field theory equip the BRST complex of the closed bosonic string with $L_\infty$-algebra structure (see further reference there). Zwiebach, in turn, was following the BV-formalism due to Batalin-Vilkovisky 83, Batakin-Fradkin 83.

See also at L-infinity algebra – History.

Discussion in terms of cofibrant resolutions of the Lie operad:

• Igor Kriz, Peter May, p. 28 of: Operads, algebras, modules and motives, Astérisque 233, Société Mathématique de France (1995) (pdf, numdam:AST_1995__233__1_0)

• Jean-Louis Loday, Bruno Vallette, Sec. 3.2.12 and onwards in: Algebraic Operads, Grundlehren der mathematischen Wissenschaften 346, Springer 2012 (ISBN 978-3-642-30362-3, pdf)

A historical survey is

• Jim Stasheff, Higher homotopy structures, then and now, talk at Opening workshop of Higher Structures in Geometry and Physics, MPI Bonn 2016 (pdf, arXiv:1809.02526)

See also

• Marilyn Daily, $L_\infty$-structures, PhD thesis, 2004 (web)

• Klaus Bering, Tom Lada, Examples of Homotopy Lie Algebras Archivum Mathematicum (arXiv:0903.5433)

Comprehensive survey with emphasis on $L_\infty$-algebra cohomology:

• Ben Reinhold, $L_\infty$-algebras and their cohomology, Emergent Scientist 3 4 (2019) [doi:10.1051/emsci/2019003]

Review for the special case of Lie 2-algebras with emphasis on the perspective of categorification:

• John Baez, Alissa Crans, Higher-dimensional algebra VI: Lie 2-algebras, TAC 12, (2004), 492–528. (arXiv:math/0307263)

As models for rational homotopy types

That $L_\infty$-algebras are models for rational homotopy theory is implicit in Quillen 69 (via their equivalence with dg-Lie algebras) and was made explicit in Hinich 98. Exposition is in

• Urtzi Buijs, Yves Félix, Aniceto Murillo, section 2 of $L_\infty$-rational homotopy of mapping spaces (arXiv:1209.4756), published as $L_\infty$-models of based mapping spaces, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524.

and genralization to non-connected rational spaces is discussed in

• Urtzi Buijs, Aniceto Murillo, Algebraic models of non-connected spaces and homotopy theory of $L_\infty$-algebras, Advances in Mathematics 236 (2013): 60-91. (arXiv:1204.4999)

$L_\infty$-algebras in physics

The following lists, mainly in chronological order of their discovery, L-∞ algebra structures appearing in physics, notably in supergravity, BV-BRST formalism, deformation quantization, string theory, higher Chern-Simons theory/AKSZ sigma-models and local field theory.

For more see also at higher category theory and physics.

In supergravity

Implicitly, in their equivalent formal dual guise of Chevalley-Eilenberg algebras (see above), $L_\infty$-algebras of finite type – in fact super L-∞ algebras – play a pivotal role in the D'Auria-Fré formulation of supergravity at least since

• Peter van Nieuwenhuizen, Free Graded Differential Superalgebras, in: Group Theoretical Methods in Physics, Lecture Notes in Physics 180, Springer (1983) 228–247 [doi:10.1007/3-540-12291-5_29, spire:182644]

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B 201 (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5, errata]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Ch III.6 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, ch III.6: pdf]

• Pietro Fré, §6.3 in: Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer (2013) [doi:10.1007/978-94-007-5443-0]

• Leonardo Castellani, §6 in: Supergravity in the group-geometric framework: a primer, Fortschr. Phys. 66 4 (2018) [doi:10.1002/prop.201800014, arXiv:1802.03407]

where they are called “free differential algebras” (“FDA”s, apparently following can Nieuwenhuizen 1982), which is a misnomer for what in mathematics are called semifree dgas (since it is only the underlying graded-commutative algebra that is required to be free, the differential is crucially not free in general, otherwise one has just a Weil algebra).

The translation of D'Auria-Fré formalism (“FDA”s) to explicit (super) $L_\infty$-algebra language was made in:

• Urs Schreiber, SuGra 3-Connection Reloaded (2006)

• Hisham Sati, Urs Schreiber, Jim Stasheff, example 5 in section 6.5.1, p. 54 of: $L_\infty$ algebra connections and applications to String- and Chern-Simons n-transport, in: Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, Int. J. of Geometric Methods in Modern Physics 12 02 (2015) 1550018 [arXiv:1308.5264, doi:10.1142/S0219887815500188]

connecting them to the higher WZW terms of the Green-Schwarz sigma models of fundamental super p-branes (The brane bouquet).

See also at supergravity Lie 3-algebra, and supergravity Lie 6-algebra.

Further exposition and review of the (dual) identification of supergravity “FDAs” with super $L_\infty$-algebras:

• Urs Schreiber, Homotopy Lie n-algebras in Supergravity, PhysicsForums-Insights (2015)

• Branislav Jurčo, Christian Saemann, Urs Schreiber, Martin Wolf: Higher Structures in M-Theory, Introduction to Higher Structures in M-Theory 2018, Fortsch. d. Phys. 67 8-9 (2019) 1910001 [arXiv:1903.02807, doi:10.1002/prop.201910001]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: The rational higher structure of M-theory, in Higher Structures in M-Theory 2018, Fortschr. der Physik 67 8-9 (2019) 1910017 [arXiv:1903.02834, doi:10.1002/prop.201910017]

Notice that there is a different concept of “Filipov n-Lie algebra” suggested by Bagger& Lambert 2006 to play a role in the description of the conformal field theory in the near horizon limit of black p-branes, notably the BLG model for the conformal worldvolume theory on the M2-brane .

A realization of these “Filippov $3$-Lie algebras” as 2-term $L_\infty$-algebras (Lie 2-algebras) equipped with a binary invariant polynomial (“metric Lie 2-algebras”) is in:

• Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)

• Patricia Ritter, Christian Saemann, section 2.5 of Lie 2-algebra models, JHEP 04 (2014) 066 (arXiv:1308.4892)

based on

• Paul de Medeiros, José Figueroa-O'Farrill, Elena Méndez-Escobar, Patricia Ritter, On the Lie-algebraic origin of metric 3-algebras, Commun.Math.Phys.290:871-902,2009 (arXiv:0809.1086)

See also

• José Figueroa-O'Farrill, section Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf)

Supergravity C-Field gauge algebra

Identifying the super-graded gauge algebra of the C-field in D=11 supergravity (with non-trivial super Lie bracket $[v_3, v_3] = -v_6$):

• Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Equation (2.6) of Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl.Phys. B 535 (1998) 242-292 [doi:10.1016/S0550-3213(98)00552-5, arXiv:hep-th/9806106]

• I. V. Lavrinenko, H. Lu, Christopher N. Pope, Kellogg S. Stelle, (3.4) in: Superdualities, Brane Tensions and Massive IIA/IIB Duality, Nucl. Phys. B 555 (1999) 201-227 [doi:10.1016/S0550-3213(99)00307-7, arXiv:hep-th/9903057]

• Jussi Kalkkinen, Kellogg S. Stelle, (75) of: Large Gauge Transformations in M-theory, J. Geom. Phys. 48 (2003) 100-132 [doi:10.1016/S0393-0440(03)00027-5, arXiv:hep-th/0212081]

• Igor A. Bandos, Alexei J. Nurmagambetov, Dmitri P. Sorokin, (86) in: Various Faces of Type IIA Supergravity, Nucl.Phys. B 676 (2004) 189-228 [doi:10.1016/j.nuclphysb.2003.10.036, arXiv:hep-th/0307153]

Identification as an $L_\infty$-algebra (a dg-Lie algebra, in this case):

• Hisham Sati, (4.9) in: Geometric and topological structures related to M-branes, in Superstrings, Geometry, Topology, and $C^\ast$-algebras, Proc. Symp. Pure Math. 81 (2010) 181-236 [ams:pspum/081, arXiv:1001.5020]

and identificatoin with the rational Whitehead $L_\infty$-algebra (the rational Quillen model) of the 4-sphere (cf. Hypothesis H):

• Hisham Sati, Alexander Voronov, (13) in: Mysterious Triality and M-Theory [arXiv:2212.13968]

• Hisham Sati, Urs Schreiber, (22) in: Flux Quantization on Phase Space [arXiv:2312.12517]

In BV-BRST formalism

The introduction of BV-BRST complexes as a model for the derived critical locus of the action functionals of gauge theories is due to

• Igor Batalin, Grigori Vilkovisky, Gauge Algebra and Quantization, Phys. Lett. B 102 (1981) 27–31. doi:10.1016/0370-2693(81)90205-7

• Igor Batalin, Grigori Vilkovisky, Feynman rules for reducible gauge theories, Phys. Lett. B 120 (1983) 166-170.

  doi:10.1016/0370-2693(83)90645-7

• Igor Batalin, Efim Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B122 (1983) 157-164.

• Igor Batalin, Grigori Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev. D 28 (10): 2567–258 (1983) doi:10.1103/PhysRevD.28.2567. Erratum-ibid. 30 (1984) 508 doi:10.1103/PhysRevD.30.508

as reviewed in

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press 1992. xxviii+520 pp.

• Joaquim Gomis, J. Paris, S. Samuel, Antibrackets, Antifields and Gauge Theory Quantization (arXiv:hep-th/9412228)

The understanding that these BV-BRST complexes mathematically are the formal dual Chevalley-Eilenberg algebra of a derived L-∞ algebroid originates around

• Jim Stasheff, Homological Reduction of Constrained Poisson Algebras, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 (arXiv:q-alg/9603021, Euclid)

• Jim Stasheff, The (secret?) homological algebra of the Batalin-Vilkovisky approach (arXiv:hep-th/9712157)

Discussion in terms of homotopy Lie-Rinehart pairs is due to

• Lars Kjeseth, Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. (Euclid)

The L-∞ algebroid-structure is also made explicit in (def. 4.1 of v1) of (Sati-Schreiber-Stasheff 09).

The extraction of $L_\infty$-algebras from the formal neighbourhood of a derived critical locus is maybe first made explicit in:

• Maxim Grigoriev, Dmitry Rudinsky, Notes on the $L_\infty$-approach to local gauge field theories $[$arXiv2303.08990$]$

In string field theory

The first explicit appearance of $L_\infty$-algebras in theoretical physics is the $L_\infty$-algebra structure on the BRST complex of the closed bosonic string found in the context of closed bosonic string field theory in

• Barton Zwiebach, Closed string field theory: Quantum action and the B-V master equation , Nucl.Phys. B390 (1993) 33 (arXiv:hep-th/9206084)

• Jim Stasheff, Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space Talk given at the Conference on Topics in Geometry and Physics (1992) (arXiv:hep-th/9304061)

Generalization to open-closed bosonic string field theory yields L-∞ algebra interacting with A-∞ algebra:

• Hiroshige Kajiura, Homotopy Algebra Morphism and Geometry of Classical String Field Theory (2001) (arXiv:hep-th/0112228)

• Hiroshige Kajiura, Jim Stasheff, Homotopy algebras inspired by classical open-closed string field theory, Comm. Math. Phys. 263 (2006) 553–581 (2004) (arXiv:math/0410291)

• Martin Markl, Loop Homotopy Algebras in Closed String Field Theory (1997) (arXiv:hep-th/9711045)

See also

• Jim Stasheff, Higher homotopy algebras: String field theory and Drinfeld’s quasiHopf algebras, proceedings of International Conference on Differential Geometric Methods in Theoretical Physics, 1991 (spire)

For more see at string field theory – References – Relation to A-infinity and L-infinity algebras.

In deformation quantization

The general solution of the deformation quantization problem of Poisson manifolds due to

• Kontsevich 97

makes crucial use of L-∞ algebra. Later it was understood that indeed L-∞ algebras are equivalently the universal model for infinitesimal deformation theory (of anything), also called formal moduli problems:

• Vladimir Hinich, DG coalgebras as formal stacks (arXiv:9812034)

• Jacob Lurie, Formal Moduli Problems

• Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)

In heterotic string theory

Next it was again $L_\infty$-algebras of finite type that drew attention. It was eventually understood that the string structures which embody a refinement of the Green-Schwarz anomaly cancellation mechanism in heterotic string theory have a further smooth refinement as G-structures for the string 2-group, which is the Lie integration of a Lie 2-algebra called the string Lie 2-algebra. This is due to

• John Baez, Alissa Crans, Urs Schreiber, Danny Stevenson, From loop groups to 2-groups, Homotopy, Homology and Applications 9 (2007), 101-135. (arXiv:math.QA/0504123)

• André Henriques, Integrating $L_\infty$ algebras, Compos. Math. 144 (2008), no. 4, 1017–1045 (doi,math.AT/0603563)

and the relation to the Green-Schwarz mechanism is made explicit in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics, 2012, Volume 315, Issue 1, pp 169-213 (arXiv:0910.4001)

This article also observes that an analogous situation appears in dual heterotic string theory with the fivebrane Lie 6-algebra in place of the string Lie 2-algebra.

Higher Chern-Simons field theory and AKSZ sigma-models

Ordinary Chern-Simons theory for a simple gauge group is all controled by a Lie algebra 3-cocycle. The generalization of Chern-Simons theory to AKSZ-sigma models was understood to be encoded by symplectic Lie n-algebroids (later re-popularized as “shifted symplectic structures”) in

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv:9910078)

• Pavol Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, based on a talk at “Poisson 2000”, CIRM Marseille, June 2000; (arXiv:0105080)

• Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids in Quantization, Poisson Brackets and Beyond , Theodore Voronov (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (arXiv)

• Dmitry Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories Lett.Math.Phys.79:143-159,2007 (arXiv:hep-th/0608150).

The globally defined AKSZ action functionals obtained this way were shown in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (arXiv:1108.4378)

to be a special case of the higher Lie integration process of

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735)

Further exmaples of non-symplectic $L_\infty$-Chern-Simons theory obtained this way include 7-dimensional Chern-Simons theory on string 2-connections:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229–321

In local prequantum field theory

Infinite-dimensional $L_\infty$-algebras that behaved similar to Poisson bracket Lie algebras – Poisson bracket Lie n-algebras – were noticed

• Chris Rogers, $L_\infty$ algebras from multisymplectic geometry , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

In

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)

these were shown to be the infinitesimal version of the symmetries of prequantum n-bundles as they appear in local prequantum field theory, in higher generalization of how the Poisson bracket is the Lie algebra of the quantomorphism group.

These also encode a homotopy refinement of the Dickey bracket on Noether conserved currents which for Green-Schwarz sigma models reduces to Lie $n$-algebras of BPS charges which refine super Lie algebras such as the M-theory super Lie algebra:

• Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges (arXiv:1507.08692)

• Igor Khavkine, Urs Schreiber, Lie n-algebras of higher Noether currents

This makes concrete the suggestion that there should be $L_\infty$-algebra refinements of the Dickey bracket of conserved currents in local field theory that was made in

• Glenn Barnich, Ronald Fulp, Tom Lada, Jim Stasheff, The sh Lie structure of Poisson brackets in field theory (arXiv:hep-th/9702176)

Comprehesive survey and exposition of this situation is in

• Urs Schreiber, Higher Prequantum Geometry, in Gabriel Catren, Mathieu Anel (eds.), New Spaces for Mathematics and Physics, 2016

In perturbative quantum field theory

Further identification of L-∞ algebras-structure in the Feynman amplitudes/S-matrix of Lagrangian perturbative quantum field theory:

• Markus Fröb, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories Communications in Mathematical Physics 2019 (online first) (arXiv:1803.10235)

• Alex Arvanitakis, The $L_\infty$-algebra of the S-matrix (arXiv:1903.05643)

In double field theory

In double field theory:

• Andreas Deser, Jim Stasheff, Even symplectic supermanifolds and double field theory, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 (arXiv:1406.3601)

• Olaf Hohm, Barton Zwiebach, $L_\infty$ Algebras and Field Theory (arXiv:1701.08824)

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