\infty-Lie theory

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Rational homotopy theory



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

An L 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 L_\infty-algebra is a Lie ∞-algebroid with a single object.

L 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 L_\infty-algebra 𝔤\mathfrak{g} “exponentiates” to a smooth ∞-group Ωexp(𝔤)\Omega \exp(\mathfrak{g}).



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

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

L-∞ algebras in the incarnation of higher brackets satisfying a higher Jacobi identity (def. 2) 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 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 had been following the BV-formalism of Batalin-Vilkovisky 83, Batakin-Fradkin 83, whose relation to L 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 L_\infty-algebras and the “FDA”s of the supergravity literature is made explicit in (FSS 13).

higher Lie theorysupergravity
\, super Lie n-algebra 𝔤\mathfrak{g} \,\, “FDA” CE(𝔤)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 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 XX a rational topological space then its loop space ∞-group ΩX\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.


In terms of algebras over an operad

An L 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 L_\infty-algebras that is most directly related to the traditional definition of ordinary Lie algebras, namely as \mathbb{Z}-graded vector space 𝔤\mathfrak{g} equipped with nn-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 L_\infty-algebras reduces to that of ordinary Lie algebras when 𝔤\mathfrak{g} is concentrated in degree zero. Moreover we take the differential of the underlying chain complex of the L L_\infty-algebra to have degree 1-1 (“homological grading”). Together this means in particular that 𝔤\mathfrak{g} is a Lie n-algebra for nn \in \mathbb{N}, n1n \geq 1, if it is concentrated in degrees 0 to n1n-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.


(graded signature of a permuation)

Let VV be a \mathbb{Z}-graded vector space, and for nn \in \mathbb{N} let

v=(v 1,v 2,,v n) \mathbf{v} = (v_1, v_2, \cdots, v_n)

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

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

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

χ(σ,v 1,,v n){1,+1} \chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\}

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


An L L_\infty-algebra is

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

  2. for each nn \in \mathbb{N}, n1n \geq 1 a multilinear map, called the nn-ary bracket, of the form

    l n()[,,,] n:𝔤𝔤ncopies𝔤 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 n2n-2

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

such that the following conditions hold:

  1. (graded skew symmetry) each l nl_n is graded antisymmetric, in that for every permutation σ\sigma of nn elements and for every n-tuple (v 1,,v n)(v_1, \cdots, v_n) of homogeneously graded elements v i𝔤 |v i|v_i \in \mathfrak{g}_{\vert v_i \vert} then

    l n(v σ(1),v σ(2),,v σ(n))=χ(σ,v 1,,v n)l n(v 1,v 2,v n) 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 χ(σ,v 1,,v n)\chi(\sigma,v_1,\cdots, v_n) is the (v 1,,v n)(v_1,\cdots,v_n)-graded signature of the permuation σ\sigma, according to def. 1;

  2. (strong homotopy Jacobi identity) for all nn \in \mathbb{N}, and for all (n+1)-tuples (v 1,,v n+1)(v_1, \cdots, v_{n+1}) of homogeneously graded elements v i𝔤 |v i|v_i \in \mathfrak{g}_{\vert v_i \vert} the followig equation holds

    (1) i,ji+j=n+1 σUnShuff(i,j)χ(σ,v 1,,v n)(1) i(j1)l j(l i(v σ(1),,v σ(i)),v σ(i+1),,v σ(n))=0, \sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}} \sum_{\sigma \in UnShuff(i,j)} \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)(i,j)-unshuffles σ\sigma and where χ\chi is the graded signature sign from def. 1.


In lowest degrees the generalized Jacobi identity says that

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

    ((v 1))=0 \partial (\partial(v_1)) = 0
  2. for n=2n = 2: the unary map \partial is a graded derivation of the binary map

    [v 1,v 2](1) |v 1||v 2|[v 2,v 1]+[v 1,v 2]=0 - [\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


    [v 1,v 2]=[v 1,v 2]+(1) |v 1|[v 1,v 2]. \partial [v_1, v_2] = [\partial v_1, v_2] + (-1)^{\vert v_1 \vert} [v_1, \partial v_2] \,.

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

[[v 1,v 2],v 3]+(1) |v 1|(|v 2|+|v 3|)[[v 2,v 3],v 1]+(1) |v 2|(|v 1|+|v 3|)[[v 1,v 3],v 2]=0 [[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 L_\infty-algebra is equivalently a dg-Lie algebra.


When l 3l_3 is possibly non-vanishing, then on elements x ix_i on which =l 1\partial = l_1 vanishes then the generalized Jacobi identity for n=3n = 3 gives

[[v 1,v 2],v 3]+(1) |v 1|(|v 2|+|v 3|)[[v 2,v 3],v 1]+(1) |v 2|(|v 1|+|v 3|)[[v 1,v 3],v 2]=[v 1,v 2,v 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] = - \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 L_\infty-algebra (def. 2) induce on the graded-co-commutative cofree coalgebra 𝔤\vee^\bullet \mathfrak{g} over the underlying graded vector space 𝔤\mathfrak{g} the structure of a differential graded coalgebra, with differential D=[]+[,]+[,,]+D = [-] + [-,-] + [-,-,-] + \cdots the sum of the higher brackets, extended as graded coderivations. The higher Jacobi identity is equivalently the condition that D 2=0D^2 = 0. In (Lada-Markl 94) it was observed that conversely, such “semifree” differential graded coalgebras are an equivalent incarnation of L L_\infty-algebras.

(If one uses unital dg-co-algebras then the L 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 𝔤\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 L_\infty-algebras of finite type. For 𝔤\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 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 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 L_\infty-algebras.

A (connected) L L_\infty-algebra is

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

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

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

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

3t 1t 2+t 3+t 3t 4t 5 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 throught of as equipped with the standard coproduct

Δ(v 1v 2v n) i±(v 1v i)(v i+1v n) \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: 𝔤 𝔤 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 DD as

D=D 1+D 2+D 3+, D = D_1 + D_2 + D_3 + \cdots \,,

where each D iD_i acts as l il_i when evaluated on a homogeneous element of the form t 1t nt_1 \vee \cdots \vee t_n and is then uniquely extended to all of 𝔤\vee^\bullet \mathfrak{g} by extending it as a coderivation on a coalgebra.

For instance D 2D_2 acts on homogeneous elements of word lenght 3 as

D 2(t 1t 2t 3)=D 2(t 1,t 2)t 3±permutations. 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 DD squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:

(D 2=0)(n: i+j=n shufflesσ±l i(l j(v σ(1),,v σ(j)),v σ(j+1),,v σ(n))=0). (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 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 L_\infty-algebra as simply a semi-co-free graded-co-commutative coalgebra ( 𝔤,D)(\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 𝔤\mathfrak{g} is degreewise finite dimensional, we may simply pass to its degreewise dual graded vector space 𝔤 *\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 𝔤 *\wedge^\bullet \mathfrak{g}^* is then naturally equipped with an ordinary differential d=D *d = D^* which acts on ω 𝔤 *\omega \in \wedge^\bullet \mathfrak{g}^* as

(dω)(t 1t n)=±ω(D(t 1t n)). (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

𝔤 *=k𝔤 1 *(𝔤 1 *𝔤 1 *𝔤 2 *) \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 𝔤 *\mathfrak{g}^* in degree 1, etc, that dd is of degree +1 and of course squares to 0

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

This means that we have a semifree dga

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

In the case that 𝔤\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(𝔤)CE(\mathfrak{g}) the Chevalley-Eilenberg algebra of the L L_\infty-algebra 𝔤\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 L_\infty-algebra this way.

This means that we may just as well define a (degreewise finite dimensional) L 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 L_\infty-algebras 𝔤\mathfrak{g} dually correspond to the dg-algebras which are augmented over \mathbb{R}, i.e for which the canonical projection CE(𝔤)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 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.


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

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

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

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

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

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

defined on generators by

d:t a k=1 1k![t a 1,,t a k] k at a 1t a k. 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 at^a to be of the same degree as t at_a. Therefore this derivation has degree +1.

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

ddt a =d(1) k=1 1k![t a 1,,t a k] k at a 1t a k = k,l=1 1(k1)!l![[t b 1,,t b l],t a 2,,t a k] at b 1t b lt a 2t a k. \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 σΣ k+l1\sigma \in \Sigma_{k+l-1} weighted by the Koszul-signature of the permutation:

= k,l=1 1(k+l1)! σΣ k+l1(1) sgn(σ)1(k1)!l![[t b 1,,t b l],t a 2,,t a k] at b 1t b lt a 2t a k. \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,k1)(l,k-1)-unshuffles and a sum over permutations that act inside the first ll and the last (k1)(k-1) indices. By the graded-symmetry of the bracket, the latter do not change the value of the nested bracket. Since there are (k1)!l!(k-1)! l! many of them, we get

= k,l=1 1(k+l1)! σUnsh(l,k1)(1) sgn(σ)[[t a 1,,t a l],t a l+1,,t a k+l1]t a 1t a k+l1. \cdots = \sum_{k,l = 1}^\infty \frac{1}{(k+l-1)!} \sum_{\sigma \in Unsh(l,k-1)} (-1)^{sgn(\sigma)} [[t_{a_1}, \cdots, t_{a_l}], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}] \, t^{a_1} \wedge \cdots \wedge t^{a_{k+l-1}} \,.

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

k+l=n1 σUnsh(l,k1)(1) sgn(σ)[[t a 1,,t a l],t a l+1,,t a k+l1]=0 \sum_{k+l = n-1} \sum_{\sigma \in Unsh(l,k-1)} (-1)^{sgn(\sigma)} [[t_{a_1}, \cdots, t_{a_l}], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}] = 0

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

In terms of algebras over an operad

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


Special cases

  • An L L_\infty-algebra for which VV is concentrated in the first nn degree is a Lie nn-algebra (sometimes also: “L nL_n-algebra”).

  • An L 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 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

  • if 𝔤\mathfrak{g} is a Lie algebra over K\mathbf{K}, and b k1𝕂b^{k-1}\mathbb{K} is the complex consisting of the field 𝕂\mathbb{K} in degree 1k1-k, then an L L_\infty-algebra morphism from 𝔤\mathfrak{g} to b k1𝕂b^{k-1}\mathbb{K} is precisely a degree kk Lie algebra cocycle.

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

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

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

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

Classes of examples

Specific examples


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 L_\infty-algebra. Dg-Lie algebras are precisely those L L_\infty-algebras for which all nn-ary brackets for n>2n \gt 2 are trivial. These may be thought of as the strict L L_\infty-algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.


Let kk be a field of characteristic 0 and write L Alg kL_\infty Alg_k for the category of LL\infty-algebras over kk.

Then every object of L Alg kL_\infty Alg_k is quasi-isomorphic to a dg-Lie algebra.

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

W:L Alg kL Alg k W : L_\infty Alg_k \to L_\infty Alg_k

such that for each 𝔤L Alg k\mathfrak{g} \in L_\infty Alg_k

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

  2. there is a quasi-isomorphism

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

This appears for instance as (KrizMay, cor. 1.6).

Relation to \infty-Lie groupoids

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


Lie integration


Lie integrated ∞-Lie groupoids.



The concept of L L_\infty-algebras as graded vector spaces equipped with nn-ary brackets satisfying a generalized Jacobi identity was introduced in this generality in

These authors were following Zwiebach 92, who had found in his work on closed string field theory that the n-point functions equip the BRST complex of the closed bosonic string with such a structure. Zwiebach in turn was following the BV-formalism due to Batalin-Vilkovisky 83, Batakin-Fradkin 83

See also at L-infinity algebra – History.

A discussion in terms of resolutions of the Lie operad is for instance in

A historical survey is

See also

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

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

A detailed reference for Lie 2-algebras is:

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

L 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

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

In the supergravity literature these CE-algebras are referred to as “FDA”s. This is short for “free differential algebra”, which is a slight misnomer for what in mathematics are called semifree dgas (or sometimes “quasi-free” dga-s).

The translation of D'Auria-Fré formalism to explicit (super) L L_\infty-algebra language is made in

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

Further exposition of this includes

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

Notice that there is a different concept of “Filipov n-Lie algebra” suggested in (Bagger-Lambert 06) 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 thse “Filippov 33-Lie algebras” as 2-term L L_\infty-algebras (Lie 2-algebras) equipped with a binary invariant polynomial (“metric Lie 2-algebras”) is in

based on

See also

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

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

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

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).

In string field theory

The first explicit appearance of L L_\infty-algebras in theoretical physics is the L 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:

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

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:

In heterotic string theory

Next it was again L 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

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

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

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

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

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

In local prequantum field theory

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


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 nn-algebras of BPS charges which refine super Lie algebras such as the M-theory super Lie algebra:

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

Comprehesive survey and exposition of this situation is in

In double field theory

In double field theory:

Revised on February 16, 2017 02:52:40 by Urs Schreiber (