The Milnor-Moore theorem for $L_\infty$ algebras in rational homotopy theory

We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing a new $A_\infty$ model for simply connected rational homotopy types, and uncovering a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.


Introduction
The main goal of this paper is to construct a universal enveloping A ∞ algebra for a given L ∞ algebra, alternative to the already existing versions [15,3], and to study the consequences of such an structure in rational homotopy theory.
Let L be an L ∞ algebra. In Def. 2.4, we introduce the universal enveloping A ∞ algebra U t (L). It is isomorphic to the free symmetric algebra ΛL on L as a graded vector space, and arises from a transfer process. For dg Lie algebras, U t (L) coincides with the classical dg associative envelope U L. To motivate the definition of U t , we first prove the following result (Thm. 2.1(i )). That is, for every x i ∈ H , ıℓ n (x 1 , ..., x n ) = σ∈S n χ(σ) m n x σ(1) ⊗ · · · ⊗ x σ(n) = m L n (x 1 , ..., x n ).
The result above covers the case in which L is minimal, since any such can be obtained as a contraction of the dg Lie algebra L C (L). In general, U t (L) is defined as ΛL together with an A ∞ structure inherited from a contraction from ΩC (L) onto ΛL. Here, C are the Quillen chains, Ω the cobar construction, and L Quillen's Lie functor. See Section 2 for details.
The original motivation for introducing the envelope we present was for extending the classical Milnor-Moore theorem ( [24]) to L ∞ algebras in the rational setting. This is Thm. 4.1.

Background and notation
In this paper, graded objects are always taken over Z, with homological grading (differentials lower the degree by 1). The degree of an element x is denoted by |x|, and all algebraic structures are considered over a characteristic zero field.
An A ∞ algebra is a graded vector space A = {A n } n∈Z together with linear maps m k : A ⊗k → A of degree k − 2, for k ≥ 1, satisfying the Stasheff identities for every i ≥ 1: A differential graded algebra (DGA), is an A ∞ algebra for which m k = 0 for k ≥ 3. An A ∞ algebra is minimal if m 1 = 0. An A ∞ morphism f : A → B is a family of linear maps f k : A ⊗k → B of degree k − 1 such that the following equation holds for every i ≥ 1: i=r +s+t s≥1 r,t ≥0 (−1) r +st f r +1+t id ⊗r ⊗m s ⊗ id ⊗t = 1≤r ≤i i=i 1 +···+i r (−1) s m r f i 1 ⊗ · · · ⊗ f i r being s = r −1 ℓ=1 ℓ(i r −ℓ − 1). Such an f is an A ∞ quasi-isomorphism if f 1 : (A, m 1 ) → (A ′ , m ′ 1 ) is a quasi-isomorphism of complexes. The bar construction B A of an A ∞ algebra A is the differential graded coalgebra (DGC, henceforth) where T (s A) is the tensor coalgebra on the suspension s A of A (i.e., (s A) p = A p−1 ), and δ = k≥1 δ k is the codifferential such that The bar construction turns A ∞ morphisms A → C into DGC morphisms B A → BC , and preserves quasi-isomorphisms ( [16]). The cobar construction ΩC of a coaugmented DGC C is the augmented DGA where T s −1 C is the tensor algebra on the desuspension s −1 C of the cokernel C = coKer (K → C ) of the coaugmentation K → C (i.e., (s −1 C ) p = C p+1 ), and d = d 1 +d 2 is the differential determined by where δ is the codifferential of C and i x i ⊗ y i = ∆(x) − (1 ⊗ x + x ⊗ 1) is the reduced comultiplication of x. The cobar construction extends to A ∞ coalgebras, but we are not in the need of such a generality in this paper. An L ∞ algebra is a graded vector space L = {L n } n∈Z together with skew-symmetric linear maps ℓ k : L ⊗k → L of degree k − 2, for k ≥ 1, satisfying the generalized Jacobi identities for every n ≥ 1: Here, S(i , n −i ) are the (i , n −i ) shuffles, given by those permutations σ of n elements such that σ(1) < · · · < σ(i ) and σ(i + 1) < · · · < σ(n); and ε(σ), sgn(σ) stand for the Koszul sign and the signature associated to σ, respectively. A differential graded Lie algebra (DGL) is an L ∞ algebra L for which ℓ k = 0 for k ≥ 3.
An L ∞ algebra is minimal if ℓ 1 = 0. An L ∞ morphism f : L → L ′ is a family of skew-symmetric linear maps f n : L ⊗n → L ′ of degree n − 1 such that the following equation is satisfied for every n ≥ 1: is a quasi-isomorphism of complexes. The Quillen chains C (L) of an L ∞ algebra is the equivalent cocommutative DGC (CDGC, henceforth) where ΛsL is the cofree conilpotent cocommutative graded coalgebra on the suspension sL of L, and δ = k≥1 δ k is the codifferential whose correstrictions are determined by the L ∞ structure maps, i.e., The sign ε is determined by the Koszul sign rule.
The Quillen functor L (C ) on a coaugmented CDGC C is the DGL where L s −1 C is the free graded Lie algebra on the desuspension s −1 C of the cokernel of the coaugmentation, C = coKer (K → C ), and ∂ = ∂ 1 + ∂ 2 is the differential determined by where δ is the codifferential of C and i x i ⊗ y i is again the reduced comultiplication of x.
There is an antisymmetrization functor (−) L from the category of A ∞ algebras to that of L ∞ algebras which preserves quasi-isomorphisms ( [15]). For a given A ∞ algebra (A, {m n }), its antisymmetrization A L has the same underlying graded vector space and higher brackets ℓ n given by Here, S n is the symmetric group on n letters, and we shorten the notation by χ(σ) = ε(σ) sgn(σ) for σ ∈ S n . We will usually denote the higher brackets ℓ n of A L by m L n .
A contraction of M onto N is a diagram of the form  That is, p f = f q and f i = j f .
We will be concerned with the following particular instance of the homotopy transfer theorem (see [13,23,18,12,16,5]). The maps involved in the higher structure of Theorem 1.1 can be described in several ways. For the purposes of this paper, we will describe the maps using recursive algebraic formulas. We will consistently use the following convention for the rest of the paper: contractions of an L ∞ algebra will be denoted by (i , q, K ), whereas contractions of an A ∞ algebra will be denoted by ( j , p,G). The capital letters I ,Q or J , P will stand for the corresponding induced infinity quasi-isomorphisms.
The higher multiplications {m n } on N and the terms {J n } of the A ∞ quasi-isomorphism J are recursively given as follows. Formally, set Gλ 1 = − j , and define λ n : H ⊗n → A for n ≥ 2 recursively by Here, α(i 1 , ..., i k ) = j <k i j (i k − 1), see [5, §12]. Then, m n = p • λ n and J n = G • λ n for all n ≥ 2.
Similarly, the higher brackets {ℓ n } and the Taylor series {I n } of the L ∞ quasi-isomorphism I are recursively given as follows. Formally, set K θ 1 = −i , and define θ n : H ⊗n → L for n ≥ 2 recursively by In the equation above, S(i 1 , ..., i k ) are the (i 1 , ..., i k )-shuffle permutations of the symmetric group S n , whose elements are those σ ∈ S n such that σ(1) = 1, and The sign ε σ is determined by the Koszul convention. Then, ℓ n = q • θ n and I n = K • θ n for all n ≥ 2.

The universal enveloping A ∞ algebra as a transfer
We produce the universal enveloping A ∞ algebra of a given L ∞ algebra via a transfer process. To do so, we start by showing (Thm. 2.1) that the classical adjoint pair commutes with the transfer of higher structure. See [9, Chap. 21] for a careful exposition of the adjoint pair above. After the proof of Thm. 2.1, we explain how to produce such a universal envelope, which turns out to coincide with Baranovsky's construction [3] up to homotopy.  Baranovsky's enveloping construction on (H , {ℓ n }). The map ı : H → ΛH above is an L ∞ version of a PBW map L → U L. The proof of Thm. 2.1 relies in the following lemma, which is elementary but interesting in itself. It will be relevant for the enveloping A ∞ algebra as a transferred structure (Def. 2.4).
Therefore, the higher brackets are the antisymmetrization of the higher multiplications: the terms of the induced L ∞ quasi-isomorphisms I : M L → L are the antisymmetrization of the terms of the A ∞ quasi-isomorphism J : M A → A: . The general case follows exactly the same proof, but with more involved formulas that do not give any more insight. The multiplication map of A will be denoted by m. We prove equation (3) by induction on n, and deduce at each inductive step the corresponding equation for (4) and for (5). Let n = 2. Use, in the order given, the definition of θ 2 , that f is a Lie map for the brackets involved, that f i = j f , and recognize the recursive formula for λ 2 : Equation (3) is therefore proven. Using that f is a morphism of contractions, and the proof of the case n = 2 above, we can easily prove equations (4) and (5): Assume next that for every p ≤ n−1, equation (3) holds. Then, (4) and (5) also hold for p ≤ n−1, which follows from a manipulation identical to the one done for the case n = 2. Let us prove that equation (3) holds for p = n, and then also equations (4) and (5) for p = n are straightforward consequence of f being a morphism of contractions and the just proven case n of equation 3. To lighten notation, we write χ(σ) := ε(σ) sgn(σ) for any given permutation σ.
Use, in the order given: the definition of θ n , that f is a Lie map for the brackets involved, the identity f i = j f and the induction hypothesis, and rearrange the permutations accordingly, to end up with the recursive formula of λ n evaluated at the desired elements:   , ( j , p,G) is a contraction of U L onto U H which is a morphism of retracts for the inclusion To prove (i i ), denote by {µ n } the A ∞ algebra structure on U H induced by Baranovsky's construction, and by {m n } the induced by the contraction ( j , p,G). where {m n } is any A ∞ algebra structure arising by exhibiting ΛL as a contraction of ΩC (L). In particular, if L is minimal, then the A ∞ structure on ΛL is the one given in Theorem 2.1.
The definition given is basically equivalent to Baranovsky's. The difference is that we explicitly use Thm. 2.1 for constructing it, hence avoiding the use of Baranovsky's chain homotopy K [3, Thm. 1], and with explicit, more transparent formulas whenever L is minimal. A different way of reading Def. 2.4 is as follows. For an arbitrary L ∞ algebra L, the A ∞ structure {m n } on ΛL arises by forming the diagram: is made a morphism of contractions, where we contract ΩC (L) onto its homology H * (ΩC (L)), which is isomorphic as a graded vector space to ΛL (this isomorphism follows, for example, from [3, Thm. 1]). Given f : L 1 → L 2 an L ∞ morphism, and once chosen contractions enjoying properties similar to Baranovsky's definition on morphisms (see [3,Thm. 3]).

Homotopical properties and comparison with other envelopes
We collect the main properties regarding the homotopy type of the several universal enveloping constructions in Proposition 3.1.
Let L be an L ∞ algebra. Denote by U B (L) and U t (L) the construction of Baranovsky and the given in Def. 2.4, respectively. We will consider a third universal A ∞ envelope U d (L), see discussion after Conjecture 3.3. At this point, it suffices to know that U d (L) is isomorphic to ΛL as a graded vector space, and carries an A ∞ structure for which there is a DGC quasi-isomorphism The universal envelopes U B ,U t and U d are homotopy equivalent (Prop. 3.1 (i )). Quillen's foundation of rational homotopy theory, as well as other deep results (see for example [1,10,17]), heavily rely on the now classical fact that homology commutes with the classical universal enveloping algebra functor over characteristic zero fields,  [25]) asserts that for a given DGL L with universal enveloping DGA U L, there is a natural DGC quasi-isomorphism For L ∞ algebras, although C (L), BU t (L) and BU B (L) are DGC's, there is usually no direct DGC quasi-isomorphism as in (7). However, these DGC's are always weakly equivalent, which is the lift of the quasi-isomorphism (7) when dealing with infinity structures (Prop. 3.1 (i i )).
Proposition 3.1. Let L be an L ∞ algebra. Then, The three constructions are then the same up to homotopy, and U t (L) where U is any of the envelopes U t ,U B or U d , which is not generally a DGC map for U B or U t .
(iii) Assume that H * (L) carries an L ∞ structure induced by a contraction from L onto it. Then, there are A ∞ quasi-isomorphisms where U is any of the envelopes U t ,U B ,U d or U .
Proof. (i ) Theorem 2.1(i i ) asserts that U B (L) ≃ U t (L). It suffices to show that U B (L) ≃ U d (L). Indeed, the construction of U d is based on the existence of a differential d on T sΛ + L = BU d (L) so that there is a DGC quasi-isomorphism C (L) ≃ − → BU d (L). By [3,Thm 4 (ii)], there is a DGA quasiisomorphism ΩC (L) → ΩBU B (L). Since the bar construction preserves quasi-isomorphisms, and given that the unit of the bar-cobar adjunction is a quasi-isomorphism for conilpotent coalgebras, there is the following zig-zag of DGC quasi-isomorphisms, from which the result follows: Here, G * (L) is the associated graded A ∞ algebra for the ascending filtration of U L given by F 0 = Q, F 1 = Q ⊕ L, and for p ≥ 2 : F p L = Span Q m n (x 1 , ., , , .x n ) | n ≥ 2, x j ∈ F p j L, p 1 + · · · + p n ≤ p , and S * (L) = F (L, ℓ 1 ) /J is the quotient of the free A ∞ algebra on the chain complex (L, ℓ 1 ) by the ideal generated by imposing the vanishing on L of the antisymmetrization of the A ∞ structure µ n of F (L, ℓ 1 ) for n ≥ 2. That is, µ L n (x 1 , ..., x n ) = 0 for all n ≥ 2, x i ∈ L. Basically, S * is the "free A ∞ algebra symmetrized on L" (not to be confused with a C ∞ algebra, whose structural maps vanish on the image of the shuffle products). Denote by P the dg operad whose free algebras are given by S * (an explicit description in terms of planar trees is given in [19,Prop. 4.6]). Summarizing, for any L ∞ algebra L, there is an isomorphism of A ∞ algebras where S * (L) = P (L) is the free P -algebra for a certain dg operad P . Thus, after a possible change of homotopy in the contraction from L onto H , Berglund's generalization of the tensor trick to algebras over operads ([5, Thm. 1.2]) applies to the contraction (9). That is, there is a contraction To finish, choose any A ∞ quasi-isomorphism U L ≃ H * (U L), for instance by using Thm. 1.1.
Then, there are A ∞ quasi-isomorphisms  1 (iii). Indeed, any "natural" map U (H L) → H U L passes through a previous choice of infinity structures, thus one cannot expect an isomorphism. It gets even worst than that: no choice will ever be an isomorphism, except for the trivial case, given that by definition U H L carries a non-trivial differential, whereas H * (U L) does not.
For P a dg operad, recall that a P -algebra is formal if there exists a zig-zag of P -algebra quasiisomorphisms connecting it to its homology ( [16]). In presence of a contraction, Lemma 2.2 gives a straightforward proof of the fact that L is formal as a DGL if, and only if, U L is formal as a DGA. This result ( [27]), however, has been superseded by [8,Thm. B].
We conclude this section with a conjecture. If Conjecture 3.3 is true, then the universal enveloping U d studied in this section enjoys the homotopical properties of U . This justifies the study of U d . To finish the homotopical study of U , it suffices to prove the weaker version of Conjecture 3.3 relaxing the DGC quasi-isomorphism to a zig-zag of quasi-isomorphisms.

The Milnor-Moore infinity theorem and a new rational model
The algebraic formalism of Section 2 has interesting applications to rational homotopy theory. The monograph [9] is an excellent resource on rational homotopy theory. In this section, all L ∞ algebras are concentrated in non-negative degrees.

The Milnor-Moore infinity theorem
Let X be a simply connected complex. The classical Milnor-Moore theorem ( [24]) asserts that the rational homotopy Lie algebra L X = π * (ΩX ) ⊗ Q embeds as the primitive elements of the rational loop space Hopf algebra H * (ΩX ; Q). Furthermore, the latter Hopf algebra is precisely the universal enveloping algebra of L X , and the inclusion is given by the rationalization of the Hurewicz morphism, h : π * (ΩX ) ⊗ Q → H * (ΩX ; Q) = U (π * (ΩX ) ⊗ Q) .
If only the rational homotopy Lie algebra π * (ΩX ) ⊗ Q is taken into account, then non-equivalent rational spaces may share this invariant. For instance, the rationalization of CP 2 and of K (Z, 2) × K (Z, 5) are not equivalent, yet both have abelian two dimensional isomorphic rational homotopy Lie algebras. However, endowing an L ∞ structure to π * (ΩX ) ⊗ Q determines a unique rational homotopy type, even if we include the class of nilpotent finite type complexes. In this latter case, we need to restrict to finite type pronilpotent L ∞ algebras. The rational homotopy type encoded by such an L ∞ algebra L is determined by the DGL L C (L) in case L = L ≥1 , and by the Sullivan algebra C * (L) in case L = L ≥0 is finite type pronilpotent. Here, C * = ∨ • C is the linear dual ∨ of the Quillen chains C . See [6, Thm. 2.3] for details. By a beautiful result of Majewski, whenever X is simply connected of finite type, these two models are homotopy equivalent ( [17]).
Denote U = U t . The next result lifts the morphism (10) to the context of infinity algebras. is a strict L ∞ embedding. Therefore, the L ∞ structure on the rational homotopy Lie algebra is the antisymmetrized of the A ∞ structure on H * (ΩX ; Q): Proof. Assume that the rational homotopy Lie algebra π * (ΩX )⊗Q carries a minimal L ∞ structure {ℓ n } governing the rational homotopy type of X for which ℓ 2 is the Samelson bracket. For instance, from a CW-decomposition build the Quillen minimal model L = (L(V ), ∂) of X , satisfying as graded Lie algebras. The choice of a contraction from L onto π * (ΩX ) ⊗ Q gives an L ∞ structure as in the statement. The rational Hurewicz homomorphism of equation (10) is, after the choice of an ordered basis of L, the PBW map from L into U L. Therefore, h can be chosen to be h = ı = p ıi in the following diagram, which is under the hypotheses of Theorem 2.2: An application of Theorem 2.2 finishes the proof. Detecting when a given cocommutative Hopf algebra is the universal envelope of its primitives is a difficult problem. This has been studied, among others, by Anick, Cartier, Halperin, Kostant, Milnor and Moore. See for example [10]. The classical name of this sort of result is the Cartier-Milnor-Moore theorem. Does a similar statement hold in the infinity setting? Conjecture 4.3. Let A be an A ∞ algebra over a characteristic zero field such that there is a cocommutative, conilpotent coproduct ∆ on A which is a strict A ∞ morphism A → A ⊗2 . Then, the primitives for the coproduct L = Ker(∆) = P * (A) form an L ∞ algebra, and the inclusion L → A extends to an isomorphism of A ∞ algebras which respects the Hopf structure.
In the conjecture above, we expect U to be Lada and Markl's envelope, and maybe the diagonal ∆ needs to come from a "Hopf algebra up to homotopy", so that the isomorphism might be not only of A ∞ algebras, but of homotopy Hopf algebras. If X is a simply connected complex, and H * (ΩX ; Q) carries a universal enveloping A ∞ structure, then H * (ΩX ; Q) is a rational model for X . Indeed, by Remark 4.2, P * (H * (ΩX ; Q)) = π * (ΩX ) ⊗ Q is a fully-fledged L ∞ algebra capturing the rational homotopy type of X .
1. The simply connected sphere S n .
• For odd n, it is Λx with |x| = n − 1, with trivial differential and trivial higher multiplications of all orders. • For even n, it is Λ(x, y) with |x| = n − 1, |y| = 2n − 2, with a unique non-trivial multiplication map given by m 2 (x, x) = 1 2 y. 2. A finite product of simply-connected Eilenberg-Mac Lane spaces k i=1 K (Q, n i ). It is given by (Λx 1 , ..., x k ), where each |x i | = n i − 1, with trivial differential and higher multiplications of all orders.
3. The complex projective spaces CP k , for k ≥ 1.

Coformal spaces.
The universal enveloping A ∞ algebra model of any coformal space can be chosen to be the classical universal enveloping algebra of it. Indeed, if X is coformal, then L = π * (ΩX ) ⊗ Q together with ℓ 2 given by the Samelson product is an L ∞ model of X . Since L is a DGL with trivial differential, the universal enveloping A ∞ algebra of it coincides with the classical envelope, having the latter trivial differential as well. This includes examples 1 and 2.
The quadratic part ∂ 2 of the differential is the standard induced by the reduced coproduct of C (L) (see formula (2)), and ∂ 1 is explicitly given on generators by  Here, ε is the parity of n−1 j =1 |x j |(k − j ). If moreover the involved higher products are all uniquely defined, then the above containment is an equality of elements.
If moreover the triple products are all uniquely defined, then the above containment is an equality of elements.