Homotopical rigidity of the pre-Lie operad
aa r X i v : . [ m a t h . K T ] F e b HOMOTOPICAL RIGIDITY OF THE PRE-LIE OPERAD
VLADIMIR DOTSENKO AND ANTON KHOROSHKIN
This paper is dedicated to Martin Markl on the occasion of his sixtieth birthday A BSTRACT . We show that the celebrated operad of pre-Lie algebras is very rigid: it has no “non-obvious” degreesof freedom from either of the three points of view: deformations of maps to and from the “three graces of operadtheory”, homotopy automorphisms, and operadic twisting. I NTRODUCTION
Recall that a pre-Lie algebra is a vector space equipped with a bilinear operation a , b a ⊳ b satisfying theidentity ( a ⊳ a ) ⊳ a − a ⊳ ( a ⊳ a ) = ( a ⊳ a ) ⊳ a − a ⊳ ( a ⊳ a ),known as the pre-Lie identity or the right-symmetric identity. Pre-Lie algebras appear in a wide range of con-texts across pure and applied mathematics, from algebra and combinatorics to differential geometry, numer-ical methods, and theory of renormalisation. In this paper, we study the operad of pre-Lie algebras from thehomotopy theory viewpoint.It is almost immediate from the definition that each pre-Lie algebra may be regarded as a Lie algebra withthe bracket [ a , b ] : = a ⊳ b − b ⊳ a , each associative algebra may be regarded as a pre-Lie algebra, and, of course,each commutative associative algebra may be regarded as a pre-Lie algebra. These facts correspond to state-ments on the level of operads: namely, there are maps of the corresponding operads which induce these func-tors between categories of algebras. The first goal of this paper is to show that deformation complexes of thesemaps to and from the operad PreLie are acyclic. Moreover, the deformation complex of the identity map ofthe operad
PreLie is also acyclic, implying that the only homotopy automorphisms of the operad
PreLie arethe intrinsic ones given by re-scaling. Our second goal is to study the result of applying to the operad
PreLie the general construction of operadic twisting due to Thomas Willwacher [Wil15]. We compute the homologyof the operad Tw(
PreLie ), showing that it coincides with the operad
Lie . Our main motivation to study the op-erad Tw(
PreLie ) was coming from search of operations that are naturally defined on deformation complexes ofmaps of operads. Our theorem shows that the only homotopy invariant structure one may define functoriallystarting from the convolution pre-Lie algebra structure is that of a dg Lie algebra.Both results of this paper are strongly connected to Martin Markl’s work on deformation theory. Deforma-tion theory of maps of operads was developed by van der Laan [vdL04] drawing as an inspiration from Markl’swork on models for operads and the cotangent cohomology [Mar96a, Mar96b]. Moreover, it appears that thefirst time the deformation complex of the identity map of a Koszul operad received substantial attention wasin paper [Mar07a] where Markl proposed a general framework for studying natural operations on homologyof deformation complexes; in that paper, the deformation complex of the identity map was christened the“soul of the cohomology of P -algebras”. That paper also features Markls’s version of the Deligne conjecturefor the operad
Lie [Mar07a, Conj. 7] stating that the shifted Lie algebra structure is the only natural homo-topy invariant algebraic structure defined on cotangent complexes of Lie algebras; this led to a sequel paper[Mar07b] where Markl gave a beautiful but mysterious cohomological description of Lie elements in free pre-Lie algebras. In fact, we found a way to de-mystify that description: the cryptic construction of the operad rPL in [Mar07b] is best understood in the context of the operadic twisting, as a sort of a “soul” of the operadTw(
PreLie ). As a consequence, an earlier related result of Willwacher [Wil17, Th. 3.6] does in fact settle Markl’sversion of the Deligne conjecture for deformations of Lie algebras. In view of all these connections, it is onlynatural that we wish to dedicate our work to Martin for his birthday, wishing him many happy returns of theday. A perhaps unfortunate consequence of this terminology is that cohomology of algebras over many important operads ends up beingsoulless. ur arguments use filtration arguments that appear quite often when working with graph complexes [Wil15,Wil17], so in addition to furnishing the proofs of several new results, this paper hopefully might serve a ped-agogical purpose, giving the reader an insight into several useful tricks that are normally hidden deep insidevery condensed papers.For all the relevant definition from the operad theory we refer the reader to the monograph [LV12]. All vectorspaces in this paper are defined over a field k of characteristic zero, and all chain complexes are homological(with the differential of degree − s to handle suspensions, and the symbol S for operadicsuspensions. When writing down elements of operads, we use small latin letters as placeholders; when work-ing with algebras over operads that carry nontrivial homological degrees, there are extra signs which arise fromapplying operations to arguments via the usual Koszul sign rule.1. C OMBINATORICS RELATED TO THE PRE -L IE OPERAD
Most of the arguments of this paper rely on the description of the operad
PreLie controlling pre-Lie algebrasin terms of labelled rooted trees due to Chapoton and Livernet [CL01] (which was perhaps known to Cayley[Cay57]). In modern terms, the underlying S -module of the operad PreLie is the linearisation of the species RT of labelled rooted trees, and the operadic insertion of trees can be described combinatorially as follows.For a labelled rooted tree S ∈ RT ( J ) and a labelled rooted tree T ∈ RT ( I ), the composition T ◦ i S is equal to thesum X f : in T ( i ) → J T ◦ fi S ,where the sum is over all functions f from the set of incoming edges of the vertex labelled i to the set J of allvertices of S : the labelled rooted tree T ◦ fi S is obtained by grafting the tree S in the place of the vertex i , andgrafting the subtrees growing from the vertex i in T at the vertices of S according to the function f , so that theset of incoming edges of each vertex j becomes in S ( j ) ⊔ f − ( j ). For example, we have '&%$ !" ❂❂❂ '&%$ !" ✁✁✁ '&%$ !" ◦ '&%$ !" a '&%$ !" c = '&%$ !" ❄❄❄ '&%$ !" a '&%$ !" ⑧⑧⑧ '&%$ !" c + '&%$ !" ❄❄❄ '&%$ !" ⑧⑧⑧ '&%$ !" a '&%$ !" c + '&%$ !" '&%$ !" ❄❄❄ '&%$ !" a '&%$ !" c + '&%$ !" '&%$ !" a '&%$ !" ⑧⑧⑧ '&%$ !" c .The Koszul dual operad PreLie ! is also very easy to describe combinatoriallly. That operad is usually denoted Perm , and the corresponding algebras are called permutative algebras; a permutative algebra is an associativealgebra that additionally satisfies the identity a a a = a a a . It is clear that the underlying species of theoperad Perm is the species of sets with one marked element (the first element of the n -fold product), so thecorresponding representation of the symmetric group is the standard permutation representation. While per-mutative algebras themselves do not often arise naturally, the operad Perm is used in an important generalconstruction of the operad theory going back to work of Chapoton [Cha01]: for any operad P , one may definethe operad of P -dialgebras as the Hadamard product P ⊗ H Perm . Intuitively, one should think of P -dialgebrasas of P -algebras with one element underlined. In some proofs of this paper, we shall encounter associativedialgebras [Lod01] and pre-Lie dialgebras [Fel11, Kol08].2. D EFORMATION THEORY FOR MAPS OF K OSZUL OPERADS
In this section, we assume each operad P reduced ( P (0) = P (1) ∼= k ), and with finite-dimensional components. We begin with briefly recalling a particular case of the general results of [MV09,vdL04] that highlight the role of pre-Lie algebras in deformation theory; these instances of pre-Lie algebrasserve as a motivation for Section 5 of this paper. Let P be a Koszul operad, and let f : P → Q be a map from P to a dg operad Q . In this case, the general recipe for computing the deformation complex of the map f pro-duces a small and tractable chain complex in several easy steps. First, one should consider the convolution op-erad between the Koszul dual cooperad of P and the dg operad Q ; its underlying dg S -module is Hom( P ¡ , Q ),and the operad composition maps ◦ i are computed using the general philosophy behind convolution prod-ucts: to evaluate the operadic composition φ ◦ i ψ on α ∈ P ¡ , one applies the cooperad decomposition map ∆ i of the cooperad P ¡ to α , computes the tensor product of maps φ ⊗ ψ on the result, and computes the compo-sition ◦ i of the operad Q . As any operad, the convolution operad can be made into a pre-Lie algebra using theformula φ ⊳ ψ = X i φ ◦ i ψ . ne can check that invariants of symmetric groups are closed under this convolution product, so one mayconsider the following convolution pre-Lie algebra Hom S ( P ¡ , Q ) : = Y n ≥ Hom( P ¡ ( n ), Q ( n )) S n .Finally, using the Lie bracket [ a , b ] : = a ⊳ b − b ⊳ a mentioned in the introduction, one may consider that spaceas a Lie algebra called the convolution Lie algebra ; it is a dg Lie algebra (with zero differential if the operad Q has zero differential). Recall that the Maurer–Cartan equation in a dg Lie algebra is the equation d ( α ) +
12 [ α , α ] = α of degree − P ∞ = Ω ( P ¡ ) to Q . In general, a Maurer–Cartan element in a Lie algebra can be used to twist the differential,letting d α = d + [ α , − ].We shall twist the differential in our dg Lie algebra using a particular Maurer–Cartan element α correspondingto the map from P ∞ to Q that is obtained from f by the pre-composition with the projection P ∞ ։ P . Bydefinition, the deformation complex of map f is the dg Lie algebraDef( f : P → Q ) : = ¡ Hom S ( P ¡ , Q ), d α ¢ .This complex controls the deformation theory of the map f in the following sense: the Maurer–Cartan ele-ments of that differential graded Lie algebra, that is elements λ of degree − d α ( λ ) +
12 [ λ , λ ] = f ; gauge equivalence of Maurer–Cartan elementscorresponds to equivalence of deformations. In fact, one may replace the deformation complex by its homol-ogy with the transferred L ∞ -algebra structure: that L ∞ -algebra is filtered, so one may work with its Maurer–Cartan elements instead, not losing any information [Ber15, DR15].We remark that it is common to remove the counit from the cooperad P ¡ and define the deformation com-plex as ³ Hom S ( P ¡ , Q ), d α ´ .Since our operads are assumed to be connected, the difference between the two complexes is a one-dimensionalspace, so it accounts just for one extra homology class. (If Q = P , the corresponding homology class accountsfor the inner derivation of P given by the commutator with the operad unit, or, after exponentiation, to re-scaling operations [KS07, Rem. 6.3.3].) We prefer to work with the bigger complex Hom S ( P ¡ , Q ), since it allowsfor more elegant results; however, it is important to remember that the abovementioned extra homology classalways exists.Let us record here a homology computation (due to Markl) for deformation complex of the identity map foreach of the “three graces of operad theory” which is one of the earliest such computations in the literature ;particular cases of these results (for the homology in degrees 1 and 2) have also recently been re-proved in[BWX + Theorem 2.1 ([Mar07a, Th. 13 & Ex. 14]) . The following complexes are acyclic:(1) the deformation complex of the identity map of the Lie operad,(2) the deformation complex of the identity map of the commutative operad,(3) the deformation complex of the identity map of the associative operad.
Suppose that P = T ( X )/( R ) and Q = T ( Y )/( S ) are two Koszul operads, and let us assume for simplicitythat these operads are generated by binary operations (the reader is invited to adapt the proof for the generalcase). Suppose that f : P → Q is a map of operads induced by a map of quadratic data ( X , R ) → ( Y , S ). Itfollows that we also have a map of Koszul dual cooperads f ¡ : P ¡ → Q ¡ . Moreover, under finite-dimensionalityassumptions, one may dualise and take the Hadamard tensor product of the map f ∗◦ : Y ∗ → X ∗ with s − S − to obtain a well defined map of operads f ! : Q ! → P ! . We remark that unlike in the case of linear duality, it isnot possible to predict what properties the Koszul dual map of a map of operads would have: It would be fair to note that the argument of [Mar07a, Ex. 14] needs a minor correction: for n ≥ S n -module Lie ( n ) does notcontain the sign representation [Kly74], so the deformation complexes for the identity maps of operads Com and
Lie almost completelycollapse even before passing to the homology. the Koszul dual of the surjection PreLie → Com is the map
Lie → Perm which is neither surjective norinjective (in fact, one can prove that the image of that map is isomorphic to the operad of Lie algebrassatisfying the identity [[ a , a ],[ a , a ]] = • the Koszul dual of the embedding Lie → PreLie is the surjection
Perm → Com , • the Koszul dual of the surjection PreLie → Ass is the surjection
Ass → Perm .However, deformation complexes behave well under Koszul duality, as we shall now show.
Theorem 2.2.
We have an isomorphism of differential graded Lie algebras
Def( f : P → Q ) ∼= Def( f ! : Q ! → P ! ). Proof.
Let us use the notation f ◦ : X → Y for the map of generators which induces the map f : P → Q . TheMaurer–Cartan element α corresponding to the map f in the convolution Lie algebra Hom S ( P ¡ , Q ) is equal to s − f ◦ , where s − f ◦ ∈ s − Hom S ( X , Y ) ∼= Hom S ( s X , Y ) ⊂ Hom S ( P ¡ , Q ).Let us examine the formula for the deformation complex a bit closer. Using [LV12, Sec. 7.2.3], we note that theunderlying S -module of the convolution operad Hom( P ¡ , Q ) is( P ¡ ) ∗ ⊗ H Q ∼= S ⊗ H P ! ⊗ H Q ,with the operad structure given by the factor-wise operad structure on the Hadamard tensor product, and thatthe S -module of generators of the Koszul dual operad P ! is s − S − ⊗ H X ∗ . In particular, the S -submodule s − X ∗ ⊗ H Q ∼= Hom( s X , Q ) ⊂ Hom( P ¡ , Q )consisting of maps supported at the cogenerators of P ¡ is identified with the submodule S ⊗ H µ s − S − ⊗ H X ∗ ¶ ⊗ H Q of the Hadamard product, and the space s − Hom S ( X , Y ) is identified with ¡ S (2) ⊗ ( s − S − (2) ⊗ X ∗ (2)) ⊗ Y (2) ¢ S .Let us denote by µ the basis element s − ∈ k s − ∼= S (2) and by ν the basis element s ∈ k s ∼= S − (2). If we denoteby { x i } a basis of X (2) and by { x ∨ i = s − ν ⊗ x ∗ i } the corresponding basis of X ! (2), the element in the Hadamardproduct corresponding to the Maurer–Cartan element α used to twist the differential is X i µ ⊗ x ∨ i ⊗ f ( x i ).Since the operad Q is Koszul, we may use the same techniques for the map f ! . The corresponding convolutionoperad is S ⊗ H P ! ⊗ H Q ∼= S ⊗ H Q ⊗ H P ! ∼= S ⊗ H ( Q ! ) ! ⊗ H P ! .We note that this operad is isomorphic to the convolution operad corresponding to the morphism f . Conse-quently, the convolution Lie algebra on Hom S ( P ¡ , Q ) is isomorphic to that on Hom S (( Q ! ) ¡ , P ! ). The canonicalisomorphism Hom S ( X , Y ) ∼= Hom S ( Y ∗ , X ∗ )sends f ◦ to f ∗◦ , which easily implies that under our identifications the Maurer–Cartan elements correspondingto f and f ! are identified. This completes the proof. (cid:3) In a particular case where P = Q and the map f is the identity map, this theorem states that the deformationtheory of the identity map is the same for a Koszul operad and its dual. In the case of associative algebras (i.e.operads concentrated in arity one), the deformation complex of the identity map of a Koszul algebra A definedabove has the same homotopy type as the Hochschild cohomology complex C −• ( A , A ); for the case of operadsit is an appropriate generalisation of the Hochschild complex. The fact that for a Koszul associative algebra thehomotopy type of that differential graded Lie algebra is invariant under Koszul duality is due to Keller [Kel03]. . D EFORMATION COMPLEXES OF MAP BETWEEN THE PRE -L IE OPERAD AND THE “ THREE GRACES ”In this section, we show that the maps between the operad
PreLie and the operads
Com , Lie and
Ass , chris-tened by Jean–Louis Loday the “three graces of operad theory”, are homotopically rigid. It is easy to show thatthere are, up to re-scaling, just three such maps mentioned in the introduction: the projection from
PreLie to Com sending the pre-Lie product to the product in the commutative operad, the map from
Lie to PreLie send-ing [ a , b ] to a ⊳ b − b ⊳ a , and the projection from PreLie to Ass sending the pre-Lie product to the associativeproduct.3.1.
The map to the commutative operad.
Our first vanishing theorem uses a simple representation-theoreticargument.
Theorem 3.1.
The deformation complex of the projection
PreLie → Com is acyclic.Proof.
In this case, the corresponding convolution dg Lie algebra is Y n ≥ Hom S n ( PreLie ¡ ( n ), Com ( n ))equipped with the differential [ α , − ]; here α is the map that sends s ( a ⊳ a ) to a · a . The S n -module isomor-phism PreLie ¡ ( n ) ∼= s n − sgn n ⊗ Perm ( n ) implies that Y n ≥ Hom S n ( PreLie ¡ ( n ), Com ( n )) ∼= Y n ≥ s − n Hom S n ( Perm ( n ),sgn n ).Since the S n -module Perm ( n ) is isomorphic to the standard permutation representation, it does not containthe sign representation for n ≥
3. It follows that the convolution Lie algebra is concentrated in degrees zeroand −
1; the differential of the class of degree 0 kills the class of degree − (cid:3) Using Theorem 2.2, we obtain the Koszul dual result.
Corollary 3.2.
The deformation complex of the map
Lie → Perm is acyclic.
The map from the Lie operad.
Our result of this section is the first slightly non-trivial rigidity theoreminvolving the pre-Lie operad. A similar but much simpler argument shows that the deformation complex ofthe analogous map from the Lie operad to the associative operad is also acyclic.
Theorem 3.3.
The deformation complex of the inclusion
Lie → PreLie is acyclic.Proof.
The corresponding convolution Lie algebra is Y n ≥ Hom S n ( Lie ¡ ( n ), PreLie ( n ))equipped with the differential [ α , − ]; here α is the map that sends s [ a , a ] to a ◦ a − a ◦ a . Because of theisomorphisms Y n ≥ Hom S n ( Lie ¡ ( n ), PreLie ( n )) ∼= Y n ≥ Hom S n ( s n − sgn n , PreLie ( n )) ∼= s Y n ≥ PreLie ( n ) ⊗ S n ( k s − ) ⊗ n ,it is obvious that if we shift the homological degrees by one, that algebra may be identified with the underlyingspace of the free pre-Lie algebra generated by the element s − . Elements of that space are combinations ofunlabelled rooted trees (or, rather, rooted trees whose vertices are all labelled s − ). The Maurer–Cartan element α in this case is the tree . The differential d α is the “usual” graph complex differential [Kon93]: the image ofeach tree T is obtained by adding • the sum over all vertices of T of all possible ways to split that vertex into two, and to connect theincoming edges of that vertex to one of the two vertices thus obtained, taken with the plus sign (corre-sponding to operadic insertions of the Maurer–Cartan element at vertices of T ): ❍❍❍ ... ... ✈✈✈♦♦♦♦ X ❄❄ ... ✈✈✈❈❈❈ ... ☎☎ • grafting the tree T at the new root, taken with the minus sign, and the sum of all possible ways to createone extra black leaf, taken with the plus sign (corresponding to operadic insertions of T at vertices ofthe Maurer–Cartan element). et us consider the filtration of this chain complex defined as follows. We define the frame of a tree T ofa rooted tree as the longest path starting from the root and consisting of vertices that have exactly one child(the last point of the frame is the first vertex with at least two children or a leaf). Consider the filtration bythe number of vertices in a tree complementary to the frame. The differential of the associated graded chaincomplex increases the length of the frame and therefore has a much simpler differential: for a tree T with theframe of even length, the differential just grafts T at the new root, and for a tree with the frame of odd length,the differential is zero. This complex is manifestly acyclic. (cid:3) Using Theorem 2.2, we obtain the Koszul dual result.
Corollary 3.4.
The deformation complex of the quotient map
Perm → Com is acyclic.
The map to the associative operad.
The next result can be viewed as a toy model of a deeper result provedin Section 4.
Theorem 3.5.
The deformation complex of the quotient map
PreLie → Ass is acyclic.Proof.
The corresponding convolution Lie algebra is Y n ≥ Hom S n ( PreLie ¡ ( n ), Ass ( n ))equipped with the differential [ α , − ]; here α is the map that sends s ( a ⊳ a ) to a · a . The S n -module isomor-phism PreLie ¡ ( n ) ∼= s n − sgn n ⊗ Perm ( n ) implies that Y n ≥ Hom S n ( PreLie ¡ ( n ), Ass ( n )) ∼= s Y n ≥ ( Ass ( n ) ⊗ Perm ( n )) ⊗ S n ( k s − ) ⊗ n ,so if we shift the homological degrees by one, this convolution Lie algebra may be identified with the underly-ing space of the free associative dialgebra generated by the element s − . The Maurer–Cartan element α in thiscase is binary product a a with the first element underlined. If we denote by e in the product of n copies of s − where the i -th factor is underlined, we have d ( e in ) = e in + + ( − n − e n + − i − X k = ( − k − e i + n + − n X k = i ( − k − e in + = e in + + ( − n − e n + − + ( − i e i + n + + ( − i + ( − n − i e in + ,or in other words, d ( e in ) = e n + , i odd , n odd , e in + − e n + , i odd , n even , e in + + e n + − e i + n + , i even , n odd , − e i + n + − e n + , i even , n even .We note that the span of all e n is an acyclic subcomplex (it is in fact isomorphic to the deformation complex ofthe map Lie → Ass ), and the quotient by that subcomplex is also acyclic. (cid:3)
Using Theorem 2.2, we obtain the Koszul dual result.
Corollary 3.6.
The deformation complex of the quotient map
Ass → Perm is acyclic.
4. D
EFORMATION COMPLEX OF THE PRE -L IE OPERAD
In this section, we use intuition coming from examples of Section 3 to prove the first “serious” rigidity the-orem: the acyclicity of the deformation complex of the operad
PreLie . Theorem 4.1.
The deformation complex of the identity map
PreLie → PreLie is acyclic.
As we mentioned in the introduction, this result implies that the group of homotopy automorphisms ofthe operad
PreLie is the one-dimensional group of intrinsic automorphisms that multiply each element of thecomponent
PreLie ( n ) by λ n − for some scalar λ . Proof.
In this case, the corresponding convolution Lie algebra is Y n ≥ Hom S n ( PreLie ¡ ( n ), PreLie ( n )) quipped with the differential [ α , − ]; here α is the map that sends s ( a ◦ a ) to a ◦ a . Arguing as in the proofof Theorem 3.5, we see that Y n ≥ Hom S n ( PreLie ¡ ( n ), PreLie ( n )) ∼= s Y n ≥ ( PreLie ( n ) ⊗ Perm ( n )) ⊗ S n ( k s − ) ⊗ n ,so if we shift the homological degrees by one, this convolution Lie algebra may be identified with the underly-ing space of the free pre-Lie dialgebra generated by the element s − . For the reader who prefers a more com-binatorial viewpoint, we would like to indicate that on the level of species, the set RT ( n ) ⊗ Perm ( n ) representsJoyal’s vertebrates on n labelled vertices [Joy81]. Elements of that space are combinations of unlabelled rootedtrees (or, rather, rooted trees whose vertices are all labelled s − ) where one of the vertices (root or non-root) isdistinguished; we call that vertex “special” and other vertices “normal”. The Maurer–Cartan element α in thiscase is the tree (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) , where the black vertex is normal, and the white vertex is special: the distinguished vertex ofthe tree encoding the identity map of the operad PreLie is the root vertex. The differential d α is similar to theusual graph complex differential: the image of each tree T is obtained by adding • the sum over all normal vertices of T of all possible ways to split that vertex into two normal ones,and to connect the incoming edges of that vertex to one of the two vertices thus obtained, taken withthe plus sign (corresponding to operadic insertions of the Maurer–Cartan element at normal verticesof T ): ❍❍❍ ... ... ✈✈✈♦♦♦♦ X ❄❄ ... ✈✈✈❈❈❈ ... ☎☎ • the sum over all possible ways to split the special vertex into two, and to connect the incoming edgesof that vertex to one of the two vertices thus obtained, so that the one of the two vertices that is closerto root becomes the special vertex of the new tree (corresponding to the operadic insertion of theMaurer–Cartan element at the special vertex): ❍❍❍ ... ... ✈✈✈♦♦♦♦ /.-,()*+ X ❄❄ ... ✈✈✈❈❈❈ ... ☎☎ /.-,()*+ • grafting the tree T at the new root that becomes the special vertex in the new tree, taken with theminus sign, and the sum of all possible ways to create one extra normal leaf, taken with the plus sign(corresponding to operadic insertions of T at vertices of the Maurer–Cartan element).To compute the homology of this complex, we shall argue in two steps. First, let us consider the spacespanned by all trees whose special vertex is the root vertex (the degenerate vertebrates of [Joy81]). It is clear thatthis space is a subcomplex, moreover, if forgetting about the speciality of the root proves that this subcomplexis isomorphic to the standard graph complex discussed in the proof of Theorem 3.3. Thus, it is acyclic. Thequotient by this subcomplex is spanned by the trees whose root vertex is normal. Let us consider the spine ofsuch a tree defined as the path connecting the root to the special vertex. Clearly, the differential is made ofterms that preserve the length of the spine and terms that increase it. We may consider the filtration for whichthe associated graded differential preserves the length of the spine. That associated graded complex splits asa sum of complexes with the spine of given length m , and such summand is the m -fold tensor product ofcomplexes corresponding to the individual trees attached along the vertices of the spine (this is a homologicalversion of the relationship about vertebrates and rooted trees [Joy81, Ex. 9]). Each factor attached at a normalvertex is the standard graph complex discussed in the proof of Theorem 3.3, and since the root vertex is normal,there is at least one such factor. It remains to use the Künneth formula to complete the proof. (cid:3) Using Theorem 2.2, we obtain the Koszul dual result.
Corollary 4.2.
The deformation complex of the identity map
Perm → Perm is acyclic.
5. T
WISTING OF THE PRE -L IE OPERAD
General operadic twisting was defined by Willwacher [Wil15, Appendix I] to work with Kontsevich’s graphcomplexes; it is an endofunctor of the category of differential graded operads equipped with a morphism fromthe shifted operad L ∞ . There exists a counterpart of that endofunctor for operads equipped with a morphismfrom the operad L ∞ , see [DW15, Sec. 3.5]. In particular, the general definition can be applied to a dg operad( P , d P ) equipped with a map of dg operads f : ( Lie ,0) → ( P , d P ) that sends the generator of Lie to a certainbinary operation of P that we denote [ − , − ]. In this case, the operad Tw( P ), the result of applying the twisting rocedure to the operad P , is a differential graded operad that can be defined as follows [DSV18]. Denote by α a new operation of arity 0 and degree −
1. The underlying non-differential operad of Tw( P ) is the completedcoproduct à P ∨ k α . To define the differential, one performs two steps. First, one considers the operadMC( P ) = ³ à P ∨ k α , d P + d MC ´ ,where differential d P is the differential of P , and the differential d MC vanishes on P and satisfies d MC ( α ) =− [ α , α ]. An algebra over that operad is a dg P -algebra with a Maurer–Cartan element. The element ℓ α ∈ MC( P )(1) defined by the formula ℓ α ( a ) = [ α , a ] is an operadic Maurer–Cartan element of MC( P ) which wecan use to twist the differential of that operad. One putsTw( P ) = ³ à P ∨ k α , d Tw = d P + d MC + [ ℓ α , − ] ´ .Note that the differential of α now has a different sign: d ( α ) + ℓ α ( α ) = −
12 [ α , α ] + [ α , α ] =
12 [ α , α ].Also, the operation [ − , − ] inside P ⊂ Tw( P ) is a cycle: d Tw ([ − , − ]) = − , − ] is annihilatedby d P and satisfies the Jacobi identity. This means that there is a map of dg operads from ( Lie ,0) to Tw( P ).The reason to be interested in the operad Tw( P ) is the following. Suppose that A is a dg P -algebra, andsuppose that α is a Maurer–Cartan element of the algebra A viewed as a Lie algebra. As discussed in Section 2,one can twist the differential of A ; the twisted dg Lie algebra ( A , d α ) can in fact be extended to a Tw( P )-algebrastructure.Let us remark that for each operad P with zero differential, all operations in the image of d Tw contain at leastone occurrence of α . Thus, the image of Lie in Tw( P ) on the level of cohomology is isomorphic to the imageof the map f : Lie → P . In case of the operad PreLie , the map
Lie → PreLie is injective (even the composite
Lie → PreLie → Ass is injective) so the inclusion of dg operads
Lie → Tw(
PreLie ) is injective on the level ofhomology. We shall now prove that the homology of the operad Tw(
PreLie ) is exhausted by the image of thatinclusion. This result is close to [Wil17, Th. 3.6]; its proof, like the one in loc. cit. , mimics [LV14, Lemma 8.5].
Theorem 5.1.
The inclusion of dg operads ( Lie ,0) → Tw(
PreLie ) induces a homology isomorphism.Proof. We shall examine the differential more carefully and then argue by induction on arity. The arity n com-ponent Tw( PreLie )( n ) is spanned by rooted trees with “normal” vertices labelled 1, ..., n and a certain numberof “special” vertices labelled α . The differential in Tw( PreLie ) is similar to the usual graph complex differential:the image of each tree T is obtained by adding • the sum over all possible ways to split a normal vertex into a normal one retaining the label and aspecial one, and to connect the incoming edges of that vertex to one of the two vertices thus obtained,so that the term where the vertex further from the root retains the label is taken with the plus sign, andthe other term is taken with the minus sign (corresponding to the operadic insertions of ℓ α at labelledvertices): ❉❉❉ ... ... ③③③rrrr ?>=<89:; s X ❀❀ ... ①①① ?>=<89:; s ❄❄❄ ... ✟✟✟ ?>=<89:; α − ❃❃ ... ①①① ?>=<89:; α ❄❄❄ ... ✡✡✡ ?>=<89:; s • the sum over all special vertices of T of all possible ways to split that vertex into two special ones, and toconnect the incoming edges of that vertex to one of the two vertices thus obtained, taken with the plussign (corresponding to computing the differential of the Maurer–Cartan element; here it is reasonableto note that in our case [ α , α ] = α ⊳ α ): ❉❉❉ ... ... ③③③rrrr ?>=<89:; α X ❃❃ ... ✈✈✈ ?>=<89:; α ❅❅❅ ... ✟✟✟ ?>=<89:; α • grafting the tree T at the new special root, taken with the minus sign, and the sum of all possible waysto create one extra special leaf, taken with the plus sign (corresponding to operadic insertions of thetree T at the only vertex of ℓ α ). n particular, we note that Tw( PreLie )(0) is, as a complex, isomorphic to the deformation complex of the inclu-sion
Lie → PreLie from Theorem 3.3, and as such is acyclic. Consider some arity n >
0, and the decompositionTw(
PreLie )( n ) = V ( n ) ⊕ W ( n ), where V ( n ) is spanned by the trees where the normal vertex with label 1 has lessthan two incident edges, and W ( n ) is spanned by the trees where the normal vertex with label 1 has at least twoincident edges. The differential has components mapping V ( n ) to V ( n ), mapping W ( n ) to V ( n ), and mapping W ( n ) to W ( n ). We consider the filtration F • Tw(
PreLie )( n ) for which F p V ( n ) is spanned by trees from V ( n )with at least p edges and F p W ( n ) is spanned by trees from W ( n ) with at least p + W ( n ) to V ( n ); ittakes the normal vertex labelled 1 in T , makes this vertex special, and creates a new normal univalent vertexlabelled 1 that is connected to v : ✿✿ ... ☎☎ ?>=<89:; ?>=<89:; ●●●● ✺✺✺ ... ✠✠✠ ?>=<89:; α .This map is clearly injective. For n = Lie (1) → Tw(
PreLie )(1) is a quasi-isomorphism. For n >
1, the cokernel is spannedby the trees T for which the normal vertex labelled 1 is univalent and connected to another normal one. It canbe thus split into a direct sum of subcomplexes according to the number k of that latter normal vertex; thenumber of such subcomplexes in arity n is equal to n −
1. We may proceed by induction by erasing the normalvertex labelled 1: each of these subcomplexes is assumed to have homology
Lie ( n −
1) of dimension ( n − PreLie ) in arity n is ( n − Lie ( n ),implying that the inclusion of the operad Lie to the homology of the operad Tw(
PreLie ) is an isomorphism. (cid:3)
Our main motivation to study the operad Tw(
PreLie ) was coming from search of operations that are nat-urally defined on deformation complexes. In the cases discussed in Section 2, those complexes arise frompre-Lie algebras, so for each of them the dg Lie algebra structure is in fact a part of a Tw(
PreLie )-algebra struc-ture. Our theorem shows that the the only homotopy invariant structure one may define functorially startingfrom the original pre-Lie algebra is that of a dg Lie algebra.6. A
PPLICATIONS AND FURTHER RESULTS
Lie elements in pre-Lie algebras.
We shall now recall Markl’s criterion for Lie elements in the free pre-Liealgebra
PreLie ( V ) generated by a vector space V [Mar07b], and discuss its relationship to Theorem 5.1. To statethat criterion, one builds the pre-Lie algebra rPL ( V ) defined by the formula rPL ( V ) : = PreLie ( V ⊕ k ◦ )( ◦ ⊳ ◦ ) ,where ◦ is an additional generator of degree −
1. It is possible to show that there exists a well defined map d : rPL ( V ) → rPL ( V ) of degree − d ( a ⊳ b ) = d ( a ) ⊳ b + ( − | a | a ⊳ d ( b ) + Q ( a , b ),where Q ( a , b ) = ( ◦ ⊳ a ) ⊳ b −◦ ⊳ ( a ⊳ b ), and that d =
0, so the pre-Lie algebra rPL ( V ) becomes a chain complex.In [Mar07b], Markl proved the following beautiful result. Theorem 6.1 ([Mar07b, Th. 3.3]) . The subspace of Lie elements in
PreLie ( V ) equals the kernel of the differen-tial d on the space of degree elements rPL ( V ) ∼= PreLie ( V ) : Lie ( V ) ∼= ker( d : rPL ( V ) → rPL ( V ) ).In fact, as explained in loc. cit. , one can define a differential graded operad rPL and view the chain complex( rPL ( V ), d ) as the result of evaluating the Schur functor corresponding to differential graded S -module rPL onthe vector space V .To obtain a different interpretation of the rPL construction, we recall that d ( a ⊳ b ) − d ( a ) ⊳ b − ( − | a | a ⊳ d ( b )is simply the operadic differential ∂ ( − ⊳ − ) = [ d , − ⊳ − ] evaluated on the product a ⊗ b , so the differential of thepre-Lie product in the dg operad rPL is equal to Q ( − , − ). We now note that in terms of rooted trees one has Q ( a , b ) = ( ◦ ⊳ a ) ⊳ b − ◦ ⊳ ( a ⊳ b ) = '&%$ !" a ✺✺✺✺ /.-,()*+ b ✠✠✠✠ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) . his brings us very close to the key revelation: in the operad Tw( PreLie ), we have d Tw à '&%$ !" '&%$ !" ! = '&%$ !" ✽✽✽✽ '&%$ !" ✝✝✝✝ '&%$ !" α ,so one must think of the element ◦ as a shadow of the element α ∈ Tw(
PreLie ). Now we are ready to make theconnection precise. We note that the pre-Lie ideal of Tw(
PreLie ) generated by α ⊳ α is closed under differential,and so one may consider the filtration by powers of that ideal and the associated graded chain complex. Anargument identical to that in the proof of Theorem 5.1 can be used to compute the homology of the associatedgraded complex is already the operad Lie . In particular, one can prove the following result, confirming theexpectation of [Mar07b, Problem 7.6].
Proposition 6.2.
The quotient dg operad
Tw(
PreLie )/( α ⊳ α ) is isomorphic to the dg operad rPL , and we haveH • ( rPL , d rPL ) ∼= Lie .We also remark that one can consider [Wil17, Sec. 3.2] a version of the operad Tw(
PreLie ) where one onlyallows trees whose special vertices have at least two children. That operad is very close to Markl’s operad B Lie ∞ of natural operations on deformation complexes of Lie ∞ -algebras, and to the Lie ∞ -version of the minimaloperad of Kontsevich and Soibelman [KS00]. Thus, the fact that the homology of that operad is also isomorphicto the operad Lie [Wil17, Th. 3.6] essentially settles Markl’s version of the Deligne conjecture for deformationcomplexes of Lie algebras [Mar07a, Conj. 7].6.2. Deformation of maps into the brace operad.
A natural companion of the pre-Lie operad from the combi-natorial viewpoint is the operad
Brace of brace algebras discovered independently by Getzler [Get93], Kadeishvili[Kad88], and Ronco [Ron00]. Algebraically, it is an operad generated by infinitely many operations{ a ; a ,... , a n }, n ≥ planar rooted trees with operadic insertions described by a planar analogueof the Chapoton–Livernet formula, see [Cha02, Foi02]. Let us record here two analogues of our results that canbe proved by filtration arguments for graph complexes made of planar trees. Theorem 6.3.
The following complexes are acyclic:(1) the deformation complex of the map
Lie → Brace sending [ a , a ] to { a ; a } − { a ; a } ,(2) the deformation complex of the map PreLie → Brace sending a ⊳ a to { a ; a } . We note that deformaton theory of maps from the brace operad is much harder to study since that operadis not quadratic and therefore not Koszul.6.3.
Twisting of analogues of the pre-Lie operad.
It is also possible to apply operadic twisting to the braceoperad; the result is much more complicated than that for the pre-Lie operad. In fact, according to [DW15,Th. 9.3], the differential graded operad Tw(
Brace ) is quasi-isomorphic to the differential graded brace operadthat prominently features in various proofs of the Deligne conjecture [GV95, KS00, MS02, VG95], which allowsto compute the homology of the operad Tw(
Brace ). Theorem 6.4 ([DW15]) . We have the operad isomorphism H • (Tw( Brace )) ∼= S Gerst . In particular, one has H (Tw( Brace )) ∼= Lie ,and, as indicated in [Mar07b, Sec. 1.4], one can use it to describe Lie elements in free brace algebras. It isnatural to ask for which Hopf cooperads C we have H (Tw( PreLie C )) ∼= Lie ,where
PreLie C is the operad of C -enriched rooted trees [CW15, DF20] which coincides with PreLie for C = uCom ∗ and with Brace for C = uAss ∗ . This question should be compared to a much more general conjectureof Markl [Mar07a, Conjecture 22]. CKNOWLEDGEMENTS
We thank Martin Markl for comments on a draft version of this paper. Research of the second author wascarried out within the HSE University Basic Research Program and supported in part by the Russian AcademicExcellence Project ’5-100’ and in part by the Simons Foundation. This work started during the second author’svisit to Trinity College Dublin which became possible because of the financial support of Visiting Professor-ships and Fellowships Benefaction Fund. Results of Section 2 (in particular, Theorem 2.2) have been obtainedunder support of the RSF grant No.19-11-00275. R
EFERENCES[Ber15] Alexander Berglund. Rational homotopy theory of mapping spaces via Lie theory for L ∞ -algebras. Homology Homotopy Appl. ,17(2):343–369, 2015.[BWX +
20] Yan-Hong Bao, Yan-Hua Wang, Xiao-Wei Xu, Yu Ye, James J. Zhang, and Zhi-Bing Zhao. Cohomological invariants of algebraicoperads, I. Available from the webpage https://arxiv.org/abs/2001.05098 , 2020.[Cay57] A. Cayley Esq. XXVIII. On the theory of the analytical forms called trees.
The London, Edinburgh, and Dublin PhilosophicalMagazine and Journal of Science , 13(85):172–176, 1857.[Cha01] Frédéric Chapoton. Un endofoncteur de la catégorie des opérades. In
Dialgebras and related operads , volume 1763 of
LectureNotes in Math. , pages 105–110. Springer, Berlin, 2001.[Cha02] Frédéric Chapoton. Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces.
J.Pure Appl. Algebra , 168(1):1–18, 2002.[CL01] Frédéric Chapoton and Muriel Livernet. Pre-Lie algebras and the rooted trees operad.
Internat. Math. Res. Notices , (8):395–408,2001.[CW15] Damien Calaque and Thomas Willwacher. Triviality of the higher formality theorem.
Proc. Amer. Math. Soc. , 143(12):5181–5193, L ∞ -algebras. J. Algebra ,430:260–302, 2015.[DSV18] Vladimir Dotsenko, Sergey Shadrin, and Bruno Vallette. The twisting procedure. Available from the webpage https://arxiv.org/abs/1810.02941 , 2018.[DW15] Vasily Dolgushev and Thomas Willwacher. Operadic twisting—with an application to Deligne’s conjecture.
J. Pure Appl. Alge-bra , 219(5):1349–1428, 2015.[Fel11] Raúl Felipe. A brief foundation of the left-symmetric dialgebras. Comunicación del CIMAT No I-11-02 available on the webpage http://cimat.repositorioinstitucional.mx/jspui/handle/1008/595 , 2011.[Foi02] L. Foissy. Les algèbres de Hopf des arbres enracinés décorés. II.
Bull. Sci. Math. , 126(4):249–288, 2002.[Get93] Ezra Getzler. Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology. In
Quantum deformations ofalgebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) , volume 7 of
Israel Math. Conf. Proc. , pages65–78. Bar-Ilan Univ., Ramat Gan, 1993.[GV95] Murray Gerstenhaber and Alexander A. Voronov. Homotopy G -algebras and moduli space operad. Internat. Math. Res. Notices ,(3):141–153, 1995.[Joy81] André Joyal. Une théorie combinatoire des séries formelles.
Adv. in Math. , 42(1):1–82, 1981.[Kad88] T. V. Kadeishvili. The structure of the A ( ∞ )-algebra, and the Hochschild and Harrison cohomologies. Trudy Tbiliss. Mat. Inst.Razmadze Akad. Nauk Gruzin. SSR , 91:19–27, 1988.[Kel03] Bernhard M. Keller. Derived invariance of higher structures on the Hochschild complex. Preprint from the author’s web page https://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf , 2003.[Kly74] Aleksandr A. Klyachko. Lie elements in a tensor algebra.
Sibirsk. Mat. Ž. , 15:1296–1304, 1430, 1974.[Kol08] P. S. Kolesnikov. Varieties of dialgebras, and conformal algebras.
Sibirsk. Mat. Zh. , 49(2):322–339, 2008.[Kon93] Maxim Kontsevich. Formal (non)commutative symplectic geometry. In
The Gel’fand Mathematical Seminars, 1990–1992 , pages173–187. Birkhäuser Boston, Boston, MA, 1993.[KS00] Maxim Kontsevich and Yan Soibelman. Deformations of algebras over operads and the Deligne conjecture. In
Conférence
Moshé Flato 1999, Vol. I (Dijon) , volume 21 of
Math. Phys. Stud. , pages 255–307. Kluwer Acad. Publ., Dordrecht, 2000.[KS07] Maxim Kontsevich and Yan Soibelman. Deformation theory I. Book draft available via , 2007.[Lod01] Jean-Louis Loday. Dialgebras. In
Dialgebras and related operads , volume 1763 of
Lecture Notes in Math. , pages 7–66. Springer,Berlin, 2001.[LV12] Jean-Louis Loday and Bruno Vallette.
Algebraic operads , volume 346 of
Grundlehren der Mathematischen Wissenschaften [Fun-damental Principles of Mathematical Sciences] . Springer, Heidelberg, 2012.[LV14] Pascal Lambrechts and Ismar Voli´c. Formality of the little N -disks operad. Mem. Amer. Math. Soc. , 230(1079):viii+116, 2014.[Mar96a] Martin Markl. Cotangent cohomology of a category and deformations.
J. Pure Appl. Algebra , 113(2):195–218, 1996.[Mar96b] Martin Markl. Models for operads.
Comm. Algebra , 24(4):1471–1500, 1996.[Mar07a] Martin Markl. Cohomology operations and the Deligne conjecture.
Czechoslovak Math. J. , 57(132)(1):473–503, 2007.[Mar07b] Martin Markl. Lie elements in pre-Lie algebras, trees and cohomology operations.
J. Lie Theory , 17(2):241–261, 2007.[MS02] James E. McClure and Jeffrey H. Smith. A solution of Deligne’s Hochschild cohomology conjecture. In
Recent progress in homo-topy theory (Baltimore, MD, 2000) , volume 293 of
Contemp. Math. , pages 153–193. Amer. Math. Soc., Providence, RI, 2002. [MV09] Sergei Merkulov and Bruno Vallette. Deformation theory of representations of prop(erad)s. II.
J. Reine Angew. Math. , 636:123–174, 2009.[Ron00] María Ronco. Primitive elements in a free dendriform algebra. In
New trends in Hopf algebra theory (La Falda, 1999) , volume267 of
Contemp. Math. , pages 245–263. Amer. Math. Soc., Providence, RI, 2000. vdL04] P. P. I. van der Laan. Operads : Hopf algebras and coloured Koszul duality. PhD thesis, Utrecht University, available on thewebpage https://dspace.library.uu.nl/handle/1874/31825 , 2004.[VG95] A. A. Voronov and M. Gerstenkhaber. Higher-order operations on the Hochschild complex. Funktsional. Anal. i Prilozhen. ,29(1):1–6, 96, 1995.[Wil15] Thomas Willwacher. M. Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra.
Invent. Math. , 200(3):671–760, 2015.[Wil17] Thomas Willwacher. Pre-Lie pairs and triviality of the Lie bracket on the twisted hairy graph complexes. Available from thewebpage https://arxiv.org/abs/1702.04504 , 2017.I
NSTITUT DE R ECHERCHE M ATHÉMATIQUE A VANCÉE , UMR 7501, U
NIVERSITÉ DE S TRASBOURG ET
CNRS, 7
RUE R ENÉ -D ESCARTES ,67000 S
TRASBOURG
CEDEX, F
RANCE
E-mail address : [email protected] N ATIONAL R ESEARCH U NIVERSITY H IGHER S CHOOL OF E CONOMICS , 20 M
YASNITSKAYA STREET , M
OSCOW
USSIA & I
NSTITUTEFOR T HEORETICAL AND E XPERIMENTAL P HYSICS , M
OSCOW
USSIA
E-mail address : [email protected]@hse.ru