M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra

Abstract

We show that the zeroth cohomology of M. Kontsevich’s graph complex is isomorphic to the Grothendieck–Teichmüller Lie algebra $\mathfrak {{grt}}_1$. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by $\mathfrak {{grt}}_1$, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) $E_n$ operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for $\mathfrak {{grt}}_1$-elements implies the hexagon equation.

Appendices

Appendix A: Harrison complex of the cofree coalgebra

Let us recall some general (and well known) facts from homological algebra. Let

$$\begin{aligned} \mathbb {F}_{ Com ^*}(X_1,\dots , X_k) \end{aligned}$$

be the free cocommutative coalgebra cogenerated by symbols $X_1,\dots , X_k$. Since this coalgebra is cofree, its reduced Harrison complex $Harr (\mathbb {F}_{ Com ^*}(X_1,$ $\dots , X_k))$ has cohomology

$$\begin{aligned} H^j( Harr (\mathbb {F}_{ Com ^*}(X_1,\dots , X_k))) \cong {\left\{ \begin{array}{ll} \mathbb {K}X_1\oplus \dots \oplus \mathbb {K}X_k &{} \text {for j=1} \\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

Here $\mathbb {K}X_1\oplus \dots \oplus \mathbb {K}X_k\cong \mathbb {K}^k$ shall denote the $k$-dimensional vector space generated by the set $X_1,\dots , X_k$. Since the differential on $Harr (\cdots )$ cannot create or annihilate any of the formal variables, the complex

$$\begin{aligned} Harr (\mathbb {F}_{ Com ^*}(X_1,\dots , X_k)) \end{aligned}$$

inherits a $\mathbb {Z}^k$ grading. Of particular interest to us is the subcomplex of degree $(1,1,\dots , 1)$,

$$\begin{aligned} Harr ^{(1,\dots , 1)}(\mathbb {F}_{ Com ^*}(X_1,\dots , X_k)). \end{aligned}$$

It immediately follows from the formula for the cohomology of $Harr (\cdots )$ above that

$$\begin{aligned} H^j( Harr ^{(1,\dots , 1)}(\mathbb {F}_{ Com ^*}(X_1,\dots , X_k))) \cong {\left\{ \begin{array}{ll} \mathbb {K}&{} \text {for j=1 and k=1} \\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

Let us denote the $p$-fold symmetric tensor product of a dg vector space $V$ by $S^p V$, i.e.,

$$\begin{aligned} S^pV=(V^{\otimes p})^{S_p}. \end{aligned}$$

$S^p Harr (\mathbb {F}_{ Com ^*}(X_1,\dots , X_k))$ inherits a $\mathbb {Z}^k$ grading. The subcomplex of degree $(1,\dots , 1)$ is of special importance to us and we will abbreviate

$$\begin{aligned} V_{p,k,n} := (S^p( Harr (\mathbb {F}_{ Com ^*}(X_1,\dots , X_k))[1-n]))^{(1,\dots , 1)}. \end{aligned}$$

As taking invariants with respect to finite group actions commutes with taking cohomology,

$$\begin{aligned} H^j(V_{p,k,n}) \cong {\left\{ \begin{array}{ll} \mathbb {K}&{} \text {for j=kn and p=k} \\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

Note that $V_{p,k,n}$ carries a natural action of the symmetric group $S_k$ by permuting the indices of the variables $X_j$. Again because taking invariants with respect to finite group actions commutes with taking cohomology, the following Lemma is evident.

Lemma A.1

Let $p, k,n\in \mathbb {N}$, let $G\subset S_k$ be a subgroup, and let $M$ be some $G$-module. Then

$$\begin{aligned} H^j((V_{p,k,n}\otimes M)^{G}) \cong {\left\{ \begin{array}{ll} (M\otimes sgn ^{\otimes n})^{G} &{} \text {for j=kn and p=k} \\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

Here $V_{p,k,n}$ is considered a $G$-module, and the $G$-action on the tensor product is the diagonal action.

Appendix B: The deformation complex of $n$-algebras

1.1 B.1. The (co)operads $e_n$ and $e_n^\vee$

The operad $e_n$ is the operad governing $n$-algebras. An $n$-algebra is a (graded) vector space $V$ with binary operations $\wedge$ of degree 0 and $\left[ {\cdot },{\cdot }\right]$ of degree $1-n$ satisfying the following relations:

• (1) $(V,\wedge )$ is a graded commutative algebra.

• (2) $(V[n-1],\left[ {\cdot },{\cdot }\right] )$ is a graded Lie algebra.

• (3) For all (homogeneous) $v\in V$, the unary operation $\left[ {v},{\cdot }\right]$ is a derivation of degree $|v|+1-n$ on $(V,\wedge )$.

Elements of $e_n(N)$ can be written as linear combinations of expressions of the form

$$\begin{aligned} L_1(X_{1},\ldots ,X_{N})\wedge \ldots \wedge L_k(X_{1},\ldots ,X_{N}) \end{aligned}$$

(38)

where $X_1,\dots ,X_N$ are formal variables, $L_j$ are $Lie \{n-1\}$ words and each $X_i$ occurs exactly once in the expression. The action of the symmetric group $S_N$ on $e_n(N)$ is given by permuting the labels on the $X_1,\dots ,X_N$. From the length of the individual Lie words one can derive various filtrations on $e_n$. We denote by by

$$\begin{aligned} e_n(N)_{k_1,k_2,\dots } \end{aligned}$$

the subspace spanned by elements of the form (38) with $k_j$ the number of Lie words of length $j$, $j=1,2\dots$. Clearly $\sum _j jk_j =N$ and furthermore the cohomological degree is $N-\sum _j k_j$.

Example B.1

For example, in the expression

$$\begin{aligned} X_1\wedge \left[ {X_4},{X_3}\right] \wedge X_2 \end{aligned}$$

the number of Lie words of length 1 is $k_1=2$.

The cooperad $e_n^\vee$ is the Koszul dual cooperad to $e_n$. One can show that $e_n^\vee =e_n^*\{n\}$. Dualizing the direct sum decomposition of $e_n(N)$ into subspaces $e_n(N)_{k_1,k_2,\dots }$ we may write

$$\begin{aligned} e_n^\vee (N) = \bigoplus _{k_1,k_2,\dots } e_n^\vee (N)_{k_1,k_2,\dots } \end{aligned}$$

(39)

where $e_n^\vee (N)_{k_1,k_2,\dots }$ is dual to $(e_n\{n\})(N)_{k_1,k_2,\dots }$.

Remark B.2

One may represent $e_n^\vee$ using graphs. Then $e_n^\vee (n)_{k_1,k_2,\dots }$ is the space of graphs with $k_1$ isolated vertices, $k_2$ connected components of size 2, $k_3$ connected components of size 3 etc.

1.2 B.2. $hoe _n$ and a(nother) filtration on the deformation complex

Recall that $hoe _n=\Omega (e_n^\vee )$ is the operadic cobar construction of the cooperad $e_n^\vee$. Inserting the direct sum decomposition above into (3) we see that

$$\begin{aligned} \mathsf {Def}( hoe _n\rightarrow e_n) = \prod _{k_1,k_2,\dots } \mathrm {Hom}_{S_N}(e_n^\vee (N)_{k_1,k_2,\dots }, e_n(N)) \end{aligned}$$

where we abbreviate $N=\sum _j jk_j$ within the product on the right hand side. The differential on $\mathsf {Def}( hoe _n\rightarrow e_n)$ contains one part, say $d_+$, that raises $k_1$ by one, and one part that leaves $k_1$ constant. Let us put a filtration on $\mathsf {Def}( hoe _n\rightarrow e_n)$ so that $d_+$ is the differential on the associated graded. Concretely,

$$\begin{aligned} {\mathcal {F}}^p \mathsf {Def}( hoe _n\rightarrow e_n) = \prod _{k_1,k_2,\dots } \mathrm {Hom}_{S_N}(e_n^\vee (N)_{k_1,k_2,\dots }, e_n(N))_{ \ge k_1-p } \end{aligned}$$

where the final subscript shall indicate that only the subspace of cohomological degrees $\ge k_1-p$ is taken.

1.3 B.3. A more concrete description of the differential $d_+$

Let

$$\begin{aligned} e_n^\vee (N)' := \bigoplus _{k_2, k_3,\dots } e_n^\vee (N)_{0, k_2, k_3,\dots }. \end{aligned}$$

Then we may write

$$\begin{aligned} {\mathrm {gr}}\mathsf {Def}( hoe _n\rightarrow e_n)&\cong \prod _N \prod _{k_1\le N} \mathrm {Hom}_{S_{k_1}\times S_{N-k_1}}( sgn ^n_{k_1}\otimes e_n^\vee (N-k_1)',e_n(N)) \\&\cong \prod _N \prod _{k_1\le N} \mathrm {Hom}_{S_{N-k_1}}( e_n^\vee (N{-}k_1)',\mathrm {Hom}_{S_{k_1}}( sgn ^n_{k_1}, e_n(N))). \end{aligned}$$

The differential $d_+$ on this complex may be seen as induced by a differential on

$$\begin{aligned} \mathrm {Hom}_{S_{k_1}}( sgn ^n_{k_1}, e_n(N) ) \cong \left( sgn ^n_{k_1}\otimes e_n(N) \right) ^{S_{k_1}} \end{aligned}$$

which we abusively also denote by $d_+$. Elements of $\left( sgn ^n_{k_1}\otimes e_n(N) \right) ^{S_{k_1}}$ may be understood as expressions $P(X_1,\dots ,X_{k_1}, A_1,\dots ,A_m)$ in formal variables $X_1,\dots , X_{k_1}, A_1,\dots ,A_m$, with $m=N-k_1$, of the form (38), invariant under permutations (with signs) of $X_1,\dots , X_{k_1}$. Using this notation, the formula for the differential $d_+$ is:

$$\begin{aligned}&\pm (d_+ P)(X_1,\dots ,X_{k_1+1},A_1,\dots A_m) \\&\quad = \sum _{i=1}^{k_1+1}(-1)^{n (i+1)} \left[ {X_i},{P(X_1,\dots ,\hat{X}_i,\dots ,X_{k_1+1},A_1,\dots ,A_m)}\right] \\&\quad \quad - \sum _{1\le i<j\le k_1+1}(\pm )(-1)^{n (i+j+1)}\\&P\left( \left[ {X_i},{X_j}\right] ,X_1,\dots ,\hat{X}_i,\dots ,\hat{X}_j,\dots ,X_{k_1+1},A_1,\dots ,A_m\right) \\&\quad \quad - \sum _{i=1}^{k_1+1}\sum _{j=1}^m (\pm ) (-1)^{n (i+1)}\\&P(X_1,\dots ,\hat{X}_i,\dots ,X_{k_1+1},A_1,\dots ,\left[ {X_i},{A_j}\right] , \dots ,A_m)\\&\quad =\sum _{1\le i<j\le k_1+1}(\pm )(-1)^{n (i+j+1)}\\&P\left( \left[ {X_i},{X_j}\right] ,X_1,\dots ,\hat{X}_i,\dots ,\hat{X}_j,\dots ,X_{k_1+1},A_1,\dots ,A_m\right) . \end{aligned}$$

Here the signs $(\pm )$ occur only in the case of even $n$ and are a bit tricky, owed to the oddness of the Lie bracket.¹¹ For example, in the third line, one should write down a term of $P$, and then compute the number of brackets to the left of the first argument. This gives the sign. The last equality follows since $\left[ {X_i},{\cdot }\right]$ is a derivation with respect to the product and the Lie bracket.

Remark B.3

In particular, note that $d_+=0$ on the subspace with $k_1=0$.

1.4 B.4. The cohomology is concentrated in degree $k_1=0$

Let

$$\begin{aligned} \Xi = \prod _{k_2,k_3, \dots } \mathrm {Hom}_{S_N}(e_n^\vee (N)_{0,k_2,k_3,\dots }, e_n(N)) \subset {{\mathrm{Der}}}( hoe _n \rightarrow e_n) \end{aligned}$$

be the subspace of ${{\mathrm{Der}}}( hoe _n \rightarrow e_n)$ corresponding to $k_1=0$. The main result of this section is the following.

Proposition B.4

$\Xi \subset {{\mathrm{Der}}}( hoe _n \rightarrow e_n)$ is a subcomplex. The inclusion is a quasi-isomorphism.

The first statement follows directly from Remark B.3. For the second statement, let us compute the spectral sequence associated to the filtration by $k_1$ introduced above. Because $\Xi$ is a subcomplex, it will be sufficient to show that the cohomology of the associated graded ${\mathrm {gr}}\mathsf {Def}( hoe _n\rightarrow e_n)$ is $\Xi$. Given the description of the differential on the associated graded above, it is sufficient to show that the complexes

$$\begin{aligned} C_m := \left( \bigoplus _{k_1\ge 1} \left( sgn ^n_{k_1}\otimes e_n(m+k_1) \right) ^{S_{k_1}}, d_+ \right) \end{aligned}$$

are acyclic. Note that clearly $Lie \{ n-1 \} (m+k_1)\subset e_n(m+k_1)$. One can make a first reduction on the problem.

Lemma B.5

If the complexes

$$\begin{aligned} L_m := \left( \bigoplus _{k_1\ge 1} \left( sgn ^n_{k_1}\otimes Lie \{ n-1 \}(m+k_1) \right) ^{S_{k_1}}, d_+ \right) \end{aligned}$$

are acyclic, then so are the complexes $C_m$.

Proof

Consider a complex $\tilde{C}_m$ that is (like $C_m$) spanned by products

$$\begin{aligned} P(X_1,\dots ,X_{k_1}, A_1,\dots ,A_m) \end{aligned}$$

of Lie words, but with the $A_1,\dots ,A_m$ allowed to occur with repetitions. It splits as a sum of subcomplexes according to the number of occurences of $A_1, A_2$ etc. $C_m\subset \tilde{C}_m$ is simply the subcomplex in which each $A_1,A_2,\dots$ occurs exactly once. It is sufficient to prove that the cohomology of $\tilde{C}_m$ is precisely its $k_1=0$-part. However, $\tilde{C}_m$ is a cofree cocommutative coalgebra, the coproduct being the deconcatenation of Lie words. The cogenerators are those expressions $P(X_1,\dots ,X_{k_1}, A_1,\dots ,A_m)$ containing only a single Lie word. If $L_m$ is acyclic, the cohomology of the space of generators is precisely its $k_1=0$-part. $\square$

Hence we are left with showing that the complexes $L_m$ are acylic. For $L_0$ this is easy to see ($L_0$ is only two-dimensional by the Jacobi identity). Now suppose that $m\ge 1$. Then the space of Lie words in $X_1,\dots ,X_{k_1}, A_1,\dots ,A_m$, each symbol occuring exactly once, can be identified with the space of ordinary (associative) words in symbols $X_1,\dots ,X_{k_1}, A_2,\dots ,A_m$, each occuring exactly once. For example (for $m=3$, $k_1=2$)

$$\begin{aligned} \left[ {X_1},{\left[ {A_3},{\left[ {X_2},{\left[ {A_2},{A_1}\right] }\right] }\right] }\right] \leftrightarrow X_1A_3X_2A_2. \end{aligned}$$

A basis for the (anti-)symmetric part in the $X_j$ can be given by symbols like

$$\begin{aligned} XA_3XA_2.\leftrightarrow X_1A_3X_2A_2 \pm X_2A_3X_1A_2. \end{aligned}$$

In this basis, the differential is given by “doubling” $X$’s, i.e.,

$$\begin{aligned} d_+(XA_3XA_2) = XXA_3XA_2 \pm XA_3XXA_2. \end{aligned}$$

It is well known that this complex is acyclic. An explicit homotopy is “contracting” pairs of $X$’s. Hence the proposition is proven.

Appendix C: $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n)$ and $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n)$

At several points of this paper we use combinatorial arguments to compute the cohomology of $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n^\circlearrowleft )$ or some subcomplex or quotient. The combinatorics are much easier to understand and to describe in words if one identifies elements of $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n^\circlearrowleft )$ with certain linear combinations of graphs. Using the splitting (39) again, we may write

$$\begin{aligned} \mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n^\circlearrowleft ) = \prod _{k_1,k_2,\dots }\\ \mathrm {Hom}_{S_N}(e_n^\vee (N)_{k_1,k_2,\dots }, {\mathsf {Graphs}}_n^\circlearrowleft (N)) \end{aligned}$$

where again $N=\sum _j jk_j$ within the product.

The space $\mathrm {Hom}_{S_N}(e_n^\vee (N)_{k_1,k_2,\dots }, {\mathsf {Graphs}}_n^\circlearrowleft (N))$ may now (-up to a degree shift-) be understood as the subspace of elements $\Gamma \in {\mathsf {Graphs}}_n^\circlearrowleft (N)$ with the following symmetry properties:

• (1) Consider the $N$ external vertices to be organized into clusters, with $k_1$ clusters of $1$ vertices, followed by $k_2$ clusters of 2 vertices, etc. Then the (linear combination of) graphs $\Gamma$ must be invariant under interchange (with sign) of clusters of the same size. The sign is $(-1)^{j+n+1}$ for the interchange of clusters of size $j$.

• (2) $\Gamma$ must “vanish on shuffles” in any cluster. This means the following. Fix some cluster of length $j$, and fix $j_1,j_2\ge 1$ s.t. $j=j_1+j_2$. Let $USh _{j_1,j_2}$ be the set of $(j_1,j_2)$-unshuffle permutations, which we consider acting on ${\mathsf {Graphs}}_n^\circlearrowleft (N)$ by permuting the vertices in the cluster under consideration. Then we require

  $$\begin{aligned} \sum _{\sigma \in USh _{j_1,j_2}} \pm \, \sigma \cdot \Gamma = 0. \end{aligned}$$

The complex $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )$ has a similar interpretation in terms of graphs. Indeed, it is obtained from $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n^\circlearrowleft )$ by sending all graphs with internal vertices to zero. The complexes $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n)$ and $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n)$ can similarly be given a graphical description, by just dis-allowing tadpoles in the graphs.

Remark C.1

The degree of a graph can be computed as $n\cdot (\#vertices-1)-(n-1)\cdot (\#edges)$, when we count a cluster with $k$ vertices as $k$ vertices (of course) and $k-1$ edges.

Example C.2

Let us give one example to illustrate the concept. The following linear combination of graphs

represents an element in $\mathsf {Def}( hoe _2\rightarrow {\mathsf {Graphs}}_2)$ that sends the coproduct operation in $e_2^*(2)$ to the graph

and the cobracket operation in $e_2^*(2)$ to the graph

in ${\mathsf {Graphs}}_2(2)$. The first term in the sum above is of degree $-1$, while the second term is of degree $1$.

1.1 C.1. A Graphical description of the differential

Fig. 14

A schematic illustration of the various parts of the differential on the complex $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n)$. Using the splitting of the differential as in (7), the first line corresponds to $\delta$, the second to $d_{\left[ {},{}\right] }$ and the third to $d_\wedge$. For a detailed description, see the text below

Let us describe the differential on $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Graphs}}_n^\circlearrowleft )$ combinatorially, using the graphical language from the last subsection. The differential has four parts:

• (1) The first part splits an internal vertex into two internal vertices. It comes from the differential on ${\mathsf {Graphs}}_n^\circlearrowleft$.

• (2) The second part splits an external vertex into an external and an internal vertex. It also comes from the differential on ${\mathsf {Graphs}}_n^\circlearrowleft$.

• (3) The third part, $d_L$, splits a cluster of external vertices into two clusters, by splitting one external vertex in that cluster. It creates an edge between the two vertices that the original vertex was split up into (see Fig. 14). This part of the differential is denoted by $d_{\left[ {},{}\right] }$ in (6).

• (4) The fourth part, denoted $d_\wedge$ in (6), also creates an external vertex, but does not split the cluster and does not introduce a new edge. Hence it maps a cluster of length $j$ to one of length $j+1$. (For a picture, see again Fig. 14.)

In the complex $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n)$ the first two terms are absent.

Appendix D: Sketch of proof of Lemma 5.7

Our goal is to show that the double complex

$$\begin{aligned} 0\rightarrow \Xi _{ conn }\rightarrow \mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )_{ conn }\rightarrow \mathsf {Def}( hoLie _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )_{ conn }\rightarrow 0 \end{aligned}$$

is acyclic. Equivalently, we have to show that the inclusion

$$\begin{aligned} \Xi _{ conn }\rightarrow \mathsf {Def}'( hoe _n\rightarrow ({\mathsf {Gra}}_n^\circlearrowleft ))_{ conn }\end{aligned}$$

is a quasi-isomorphism, where the $'$ on the right hand side indicates that one restricts to those derivations with vanishing $hoLie _n$-part.

To go further, we need to use notation from Appendix C, in particular the graphical representation of $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )$. For each graph occurring in $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )_{ conn }$ one can associate a number $k_1$, the number of “clusters” (notation as in Appendix C) of length 1 in that graph. This yields a filtration on the graded vector space $\mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )_{ conn }$, which descends to a filtration on $\mathsf {Def}'( hoe _n\rightarrow ({\mathsf {Gra}}_n^\circlearrowleft ))_{ conn }$. One takes an appropriate spectral sequence such that the first differential increases $k_1$ by exactly one. More concretely, this differential, say $d_+$, acts by spliting off a length one cluster from any vertex. Let us introduce new terminology. Let us call vertices in clusters of length 1 “internal” and all others external. Then a computation very much similar to the computation of $H({\mathsf {Graphs}}_n^\circlearrowleft )\cong e_n$ shows that the cohomology of $\mathsf {Def}'( hoe _n\rightarrow ({\mathsf {Gra}}_n^\circlearrowleft ))_{ conn }$ is given by closed graphs without internal vertices. In other words, graphs with all clusters of length $\ge 2$, and which actually lie in $\mathsf {Def}( hoe _n\rightarrow e_n)_{ conn }\subset \mathsf {Def}( hoe _n\rightarrow {\mathsf {Gra}}_n^\circlearrowleft )_{ conn }$. But that space is the space we called $\Xi _{ conn }$.

Appendix E: (Re-)Derivation of Furusho’s result

In a remarkable paper [19], H. Furusho showed that the hexagon equation (25) is a consequence of the pentagon equation (24), if one requires that $\phi \in {\mathbb {F}_{ Lie }}(X,Y)$ (as in those equations) does not contain the term $\left[ {X},{Y}\right]$. Rephrasing this result in operadic language, it reads as follows.

Proposition E.1

$$\begin{aligned} H^1(\mathsf {Def}( Ass _\infty \rightarrow t[1])) \cong H^1(\mathsf {Def}( Com _\infty \rightarrow t[1])) \oplus \mathbb {K}[-1]. \end{aligned}$$

Here the $\mathbb {K}[-1]$ corresponds to the cohomology class represented by $\phi =\left[ {X},{Y}\right]$.

Proof

We will actually show that $H^1(\mathsf {Def}( Ass _\infty \rightarrow {\mathsf {ICG}}^\circlearrowleft [1])) \cong H^1(\mathsf {Def}$ $( Com _\infty \rightarrow {\mathsf {ICG}}^\circlearrowleft [1])) \oplus \mathbb {K}[-1]$. Take a spectral sequence as in the proof of Lemma 4.4. The difference to the situation there is that we now have to compute the Hochschild instead of the Harrison cohomology of a cofree cocommutative coalgebra. But it is well known that the Hochschild cohomology of such an algebra is the Koszul dual algebra, i. e., an (anti-)commutative free algebra. Translated into graphical language, it means that the cohomology is given by graphs, whose external vertices are connected by exactly one edge each, and which are antisymmetric under interchange of external vertices. The space of such graphs with $q$ external vertices forms a subcomplex $C_q\subset \mathsf {Def}( Ass _\infty \rightarrow {\mathsf {ICG}}^\circlearrowleft [1])$. In the language of [29] this subcomplex computes the part of the cohomology of Hodge degree $q$. Let us compute the degree 1 cohomology of $C_q$ for various $q$. We already saw that $H(C_1)\cong H^1(\mathsf {Def}( Com _\infty \rightarrow t[1]))\cong \mathfrak {{grt}}_1$.

Next, suppose we are given a closed linear combination of graphs $x_q \in C_q$ of cohomological degree 1. We split the differential on $\mathsf {Def}( Ass _\infty \rightarrow {\mathsf {ICG}}^\circlearrowleft [1])$ into a part $d_s$ creating an external vertex, and a part $\delta$ not creating one. In fact, $\delta$ is the part of the differential coming from the differential on ${\mathsf {ICG}}^\circlearrowleft$. If $q\ne 3$ we can write $x_q=-\delta y_q$ for some $y_q$ since $H({\mathsf {ICG}}^\circlearrowleft )\cong \mathfrak {{t}}$ is concentrated in degree 0. Hence our cohomology class is also represented by $d_s y_q=:x_{q+1}$. If $q\ge 4$ we can continue in this manner (i. e., $x_{q+1}=-\delta y_{q+1}$, etc...) indefinitely by degree reasons and see that the cohomology class represented by $x_q$ is trivial. Hence we have

$$\begin{aligned} H^1(\mathsf {Def}( Ass _\infty \rightarrow t[1])) \cong H^1(C_1)\oplus H^1(C_2) \oplus H^1(C_3). \end{aligned}$$

If $q=3$, then $x_q$ describes a non-trivial cohomology class only if it describes an element of $\mathfrak {{t}}_3$ which can be recovered by projecting to the internal-trivalent-tree part of the graphs. However, since the external vertices are of valence 1, the only possible tree with all external vertices of valence 1 is the graph, say $T_3$, with one internal vertex, see Fig. 15. This graph corresponds to $\left[ {t_{12}},{t_{23}}\right] \in \mathfrak {{t}}_3$. Hence $H^1(C_3)\cong \mathbb {K}$, the one cohomology class represented by $T_3$.

The statement of the proposition is thus reduced to showing that $H^1(C_2)=0$. So suppose $x_2\in {\mathbb {C}}_2$ is a degree 1 cocycle. We decompose $x_2=x_{2a}+x_{2b}$ where $x_{2a}$ contains those graphs for which both external vertices are connected to the same internal vertex and $x_{2b}$ the remainder. One can see that necessarily $x_{2a}=\delta y_{2a}+(\cdots )$ where $y_{2a}\in C_2$ and $(\cdots )$ is a linear combination of graphs with the two (univalent) external vertices connected to different internal vertices. Hence we may assume that in fact $x_{2a}=0$ from the start. In this case $x_2=x_{2b}=\delta y_2$ for a $y_2$ obtained by contracting one of the external edges. One checks that combinatorially $d_s y_2$ cannot contain any trivalent-tree-part: Such a tree must necessarily have (at least) two internal vertices connected to different external vertices. Since one external vertex of every graph in $y$ has valence one, it means that one of the internal vertices must have two edges connected to the same external vertex, which is a contradiction (or rather, the graph is 0). Hence $d_s y$ is $\delta$-exact again and hence by the same reasoning as in the $q>3$-case, the cohomology class of $x_q$ vanishes. Hence $H^1(C_2)=0$ and we are done. $\square$

Remark E.2

Note that the calculation of $H^1(\mathsf {Def}( Ass _\infty \rightarrow t[1]))$ above is similar to calculations of $H({\mathrm {gr}}^p \mathsf {Def}( hoe _2 \rightarrow e_2)_{ conn })$ performed in Sect. 7, up to some sign and degree differences. The Hodge degree $q$ corresponds to the parameter $p+1$ in Sect. 7.

Fig. 15

The graph corresponding to the $\left[ {t_{12}},{t_{23}}\right] \in \mathfrak {{t}}_3$

Appendix F: The one vertex irreducible part of $\mathsf {GC}_n$ is quasi-isomorphic to $\mathsf {GC}_n$

Let $\mathsf {GC}_{n}^{1vi}\subset \mathsf {GC}_n$ be the subspace of 1-vertex irreducible graphs, i.e., those graphs that remain connected after deleting one vertex.

Lemma F.1

$\mathsf {GC}_{1vi}\subset \mathsf {GC}$ is a sub-dg Lie algebra.

Proof

It is clear. $\square$

The following Proposition has been shown by Conant, Gerlits and Vogtman.

Proposition F.2

([10]) $\mathsf {GC}_n^{1vi}\hookrightarrow \mathsf {GC}$ is a quasi-isomorphism.

We nevertheless give a different short sketch of proof for completeness, following the idea of Lambrechts and Volic [30].

Proof ((very sketchy) Sketch of proof)

[(very sketchy) Sketch of proof] We need to show that $\mathsf {GC}_n/\mathsf {GC}_n^{1vi}$ is acyclic. Any non-1vi graph can be written using the following data: (i) a family of 1-vertex-irreducible graphs (“1vi components”) (ii) a tree (iii) for each vertex in the tree, a subset of vertices in the irrucible components. All vertices in that subset are glued together to form the graph. The differential can be decomposed into two parts: One part that changes one of the 1vi components, and one that does not. One can set up a spectral sequence, such that its first term is the latter part of the differential (leaving invariant the 1vi components).¹² The resulting complex splits into subcomplexes according to the family of 1vi components. Fix one such subcomplex, say $C$, and fix one vertex $v$ in one of the $1vi$ components, that belongs to the subset associated to vertex $t$ of the tree. There is a filtration $C\supset C_1$, where $C_1$ is the subspace containing graphs such that the number of vertices in the subset of $t$ is one and $t$ has only one incident edge. Take the spectral sequence. Its first term contains a differential mapping

$$\begin{aligned} C/C_1 \rightarrow C_1 \end{aligned}$$

which can easily be seen to be an isomorphism. $\square$

Appendix G: A note on the convergence of spectral sequences

In this paper, we use spectral sequences to compute the cohomology of graph complexes at various places, in particular for $\mathsf {GC}_n$ and $\mathsf {fGC}_n$. We claim that these spectral sequences indeed converge to the cohomology.

For $\mathsf {GC}_n$ this is very easy to see. $\mathsf {GC}_n$ splits into a direct sum of finite dimensional subcomplexes, for fixed values of the difference

$$\begin{aligned} \Delta =e-v:=\#\text {edges}-\#\text {vertices}. \end{aligned}$$

Indeed, since each vertex is at least trivalent, one has $e\ge \frac{3}{2} v$ and hence $v\le 2\Delta$ is bounded within each subcomplex. But there are only finitely many graphs with fixed $\Delta$ and bounded $v$, and hence each subcomplex is finite dimensional.

For $\mathsf {fGC}_n$ the argument is a bit more subtle. Let $\mathcal {F}$ be a filtration on $\mathsf {fGC}_n$, compatible with the differential. Let $\mathsf {fGC}_n'$ be the same complex as $\mathsf {fGC}_n$, but with the degrees shifted by $(n-\frac{1}{2})\Delta$, so that the new degree of a graph is

$$\begin{aligned} n(v-1)-(n-1)e +\left( n-\frac{1}{2}\right) \Delta =\frac{1}{2}(v+e)-n. \end{aligned}$$

It is clear that (i) each grading component of $\mathsf {fGC}_n'$ is finite dimensional, that (ii) the cohomology of $\mathsf {fGC}_n'$ is the same as that of $\mathsf {fGC}_n$ up to degree shifts and that (iii) one has a filtration $\mathcal {F}'$ on $\mathsf {fGC}_n'$. By (i) the filtration $\mathcal {F}'$ is bounded and hence the associated spectral sequence converges to the cohomology of $\mathsf {fGC}_n'$. But the spectral sequence associated to $\mathcal {F}$ is the same as that associated to $\mathcal {F}'$, up to some degree shifts. Hence it converges to the degree shifted cohomology of $\mathsf {fGC}_n'$, which by (ii) is the cohomology of $\mathsf {fGC}_n$.

Appendix H: $\mathfrak {{t}}$, $\mathfrak {{grt}}_1$, $\mathfrak {{sder}}$

Let $\mathfrak {{sder}}$ be the operad of Lie algebras of special derivations of free Lie algebras (see [3], or [16]). Elements of $\mathfrak {{sder}}$ can be seen as internal trivalent trees in ${\mathsf {ICG}}$, modulo the Jacobi identity. As noted in [44] there is a spectral sequence coming from the filtration on ${\mathsf {ICG}}$ by internal loops, whose first term contains $\mathfrak {{sder}}$.

In particular, $\mathfrak {{t}}$ is a sub-operad of Lie algebras of $\mathfrak {{sder}}$. The map $\mathfrak {{t}}\rightarrow \mathfrak {{sder}}$ sends a generator $t_{ij}$ to the graph with a single edge between external vertices $i$ and $j$.

Proposition H.1

Let $\phi \in \mathfrak {{t}}_3$ and $\Gamma$ its image in $\mathfrak {{sder}}(3)$. Then $\phi$ can be recovered from $\Gamma$ as follows:

• (1) Forget all graphs in $\Gamma$ that have more than one edge incident to vertex 2.

• (2) The coeffiecient $c$ of the (single possible) graph with no vertex at 2 yields the coefficient of $t_{13}$ in $\phi$.

• (3) Interpret each remaining tree as a Lie tree rooted at 2, corresponding to a Lie expression $\psi \in F_ Lie (X,Y)$.

• (4) Then $\phi = c\cdot t_{13} + \psi (t_{12},t_{13})$.

Proof

By induction on the degree. $\square$

When interpreting the tree as a Lie tree, we use the following sign convention. In the ordering of the edges, the root edge of the tree must come first, then all vertices of its left subtree, and then all vertices of the right subtree. For each subtree, we apply the same convention recursively. For example, consider the following tree with the indicated ordering of the edges.

This tree is to be interpreted as the Lie word

$$\begin{aligned}{}[X,[X,[X,Y]]]\,. \end{aligned}$$

Appendix I: Twisting of operads

1.1 I.1. A Foreword

Any operad describes a certain kind of algebraic object. Often the algebraic object (the representation of the operad) is easier to describe and comprehend than the operad itself. So let us describe first what algebraic situation we want to consider by defining a “twisted operad”.

Suppose we are given some operad $\mathcal P$ together with a $\mathcal P$-algebra $A$. Suppose further that we have a map $hoLie \rightarrow \mathcal P$. In particular, this means that $A$ is also a $hoLie$-algebra. Sweeping convergence issues under the rug, it makes sense to talk about Maurer–Cartan elements in $A$, which are simply Maurer–Cartan (MC) elements in the $hoLie$-algebra $A$. Fix such an $MC$ element $m$. Twisting the $hoLie$ structure on $A$ by $m$, one can in particular endow $A$ with a new differential. Furthermore one can construct new operations on $A$ by inserting $m$’s into the $\mathcal P$-operations on $A$. The twisted operad ${ Tw }\mathcal P$ is defined such that the algebra $A$ with twisted differential and with this extended set of operations is a ${ Tw }\mathcal P$-algebra.

The important claim for this paper is that there is an action of the deformation complex $\mathsf {Def}( hoLie \rightarrow \mathcal P)$ on the operad ${ Tw }\mathcal P$. Concretely, it is defined as follows: Suppose that we have a closed degree zero element $x\in \mathsf {Def}( hoLie \rightarrow \mathcal P)$. Such an element gives a $hoLie$ derivation (i.e., infinitesimal automorphism) of the $hoLie$-algebra $A$. Having an MC element $m\in A$, one can twist this derivation by $m$. One obtains in particular an (infinitesimally) different MC element $m'\in A$ and hence also a new ${ Tw }\mathcal P$-structure on $A$. The action of $x$ on the operad ${ Tw }\mathcal P$ is defined such that it induces that change of ${ Tw }\mathcal P$-structure.

Remark I.1

It is important that we define ${ Tw }\mathcal P$ such that $A$ as above with differential twisted by $m$ is a ${ Tw }\mathcal P$-algebra.

1.2 I.2. The construction

Let $\mathcal P$ be any (dg) operad. We will consider it here as a contravariant functor from the category of finite sets (with bijections as morphisms) to the category of dg vector spaces. For a finite set $S$, the space $\mathcal P(S)$ can be seen as the space of $\# S$-ary operations, with inputs labelled by elements of $S$. We will write for short $\mathcal P(n):= \mathcal P(\{1,\dots , n\})$. For some operation $a\in \mathcal P(n)$ and for some symbols $s_1,\dots ,s_n$ we will write

$$\begin{aligned} a(s_1,\dots ,s_n) \in \mathcal P(\{s_1,\dots ,s_n\}) \end{aligned}$$

for the image of $a$ under the map $\mathcal P(f)$, where $f: \{s_1,\dots ,s_n\} \rightarrow [n]$ is the bijection sending $s_j\mapsto j$. Similarly, for $a\in \mathcal P(n)$, $b\in \mathcal P(m)$, and symbols $s_1,\dots ,s_{m+n-1}$ we will write

$$\begin{aligned}&a(s_1,\dots , s_{j-1},b(s_{j},\dots ,s_{j+m-1}), s_{j+m},\dots ,s_{m+n-1})\nonumber \\&\quad =(a\circ _{j}b)(s_1,\dots ,s_{m+n-1}) \end{aligned}$$

for the operadic composition.

Let $hoLie _{k+1}:=( Lie \{k\})_\infty$ be the minimal resolution of the degree shifted Lie operad. Concretely, the Lie bracket here has degree $-k$. An operad map $hoLie _{k+1}\rightarrow \mathcal P$ is described by a Maurer–Cartan element $\mu$ in

$$\begin{aligned} \mathfrak {{g}} = \mathsf {Def}( hoLie _{k+1}\rightarrow \mathcal P)\cong \prod _j (\mathcal P(j)\otimes ({\mathbb {R}}[-k-1])^{\otimes j})^{S_j}[k+1]. \end{aligned}$$

On the right, the permutation group $S_j$ acts on $\mathcal P(j)$ as usual and on the tensor product of ${\mathbb {R}}[-k-1]$’s is by permutation, with appropriate signs. Up to degree shift, elements of $\mathfrak {{g}}$ are sums of symmetric or antisymmetric elements of $\mathcal P$.

Remark I.2

Fixing a Maurer–Cartan element $\mu \in \mathfrak {{g}}$ fixes the notion of “Maurer–Cartan element” in any (nilpotent) $\mathcal P$-algebra $A$. Concretely, the latter is just a Maurer–Cartan element in the $hoLie _{k+1}$-algebra $A$.

We next want to define an operad ${ Tw }\mathcal P$ that governs $\mathcal P$-algebras “twisted by” a Maurer–Cartan element in the sense of the Remark. The underlying functor is defined on a set $S$ as

$$\begin{aligned} { Tw }\mathcal P(S) = \prod _{j\ge 0} (\mathcal P(S\sqcup \{\bar{1},\dots \bar{j}\})\otimes ({\mathbb {R}}[k+1])^{\otimes j})^{S_j} \end{aligned}$$

where the symmetric group $S_j$ acts by permutation on the symbols $\bar{1},\dots , \bar{j}$ and be appropriate permutation (with signs) on the ${\mathbb {R}}[k+1]$. An element of the $j$-th factor in the product should be seen as an operation in $\mathcal P$ invariant under permutations of the last $j$ slots. For an element $a$ in the $j$-th factor of ${ Tw }\mathcal P(n)$ and some symbol set $S=\{s_1,\dots , s_{n+j}\}$ the expression

$$\begin{aligned} a(s_1,\dots ,s_n,s_{n+1},\dots s_{n+j}) \in \mathcal P(S)[j(k+1)] \end{aligned}$$

accordingly makes sense. The operadic compositions are defined for homogeneous (wrt. both the grading by $j$ and the degree) elements $a\in { Tw }\mathcal P(m), b\in { Tw }\mathcal P(n)$

$$\begin{aligned}&(a\circ _l b)(s_1,\dots , s_{m+n-1}, \bar{1},\dots \overline{j_1+j_2})= \\&\quad =\sum _{I\sqcup J=[j_1+j_2]} sgn (I,J)^{k+1}(-1)^{|b|j_1 (k+1)}\\&\qquad a(s_1,\dots ,s_{l-1},b(s_l,\dots ,s_{l+n-1},\bar{J}), \dots ,s_{m+n-1}, \bar{I})\\&\quad \in (\mathcal P(S\sqcup \{\bar{1},\dots \overline{j_1+j_2} \})\otimes ({\mathbb {R}}[k+1])^{\otimes (j_1+j_2)})^{S_{j_1+j_2}} \end{aligned}$$

where $\bar{I}$ is shorthand for $\bar{i}_1, \bar{i}_2,\dots$ with $i_1<i_2<\dots$ being the members of the set $I$, and similarly for $\bar{J}$. The sign is the sign of the shuffle permutation bringing $i_1, \dots ,j_1, \dots$ in the correct order.

Remark I.3

An element of ${ Tw }\mathcal P$ should be thought of as an operation in $\mathcal P$, with Maurer–Cartan elements inserted into some of its slots.

Next we want to define a differential on ${ Tw }\mathcal P$. First, denote by $\widetilde{{ Tw }}\mathcal P$ the operad ${ Tw }\mathcal P$ as defined so far, with the differential solely the one coming from the differential on $\mathcal P$. On $\widetilde{{ Tw }}\mathcal P$ we have a right action of the (dg) Lie algebra $\mathfrak {{g}}$. Concretely, for homogeneous $x\in \mathfrak {{g}}$, $a\in \widetilde{{ Tw }}\mathcal P(n)$ we have

$$\begin{aligned}&(a\cdot x)(1,\dots ,n,\bar{1},\dots \overline{j_1+j_2}) \nonumber \\&\quad =\sum _{I\sqcup J=[j_1+j_2]} sgn (I,J)^{k+1} a(1,\dots ,n, \bar{I}, x( \bar{J})). \end{aligned}$$

Lemma I.4

This formula describes a right action by operadic derivations.

Proof

For homogeneous elements we compute

$$\begin{aligned}&((a\circ _l b)\cdot x ) (s_1,\dots ,s_{m+n-1}, \bar{1},\dots , \overline{j_1+j_2+j_3})\\&=\sum _{I\sqcup J\sqcup K=[j_1+j_2+j_3]} sgn (I\cup J,K)^{k+1} sgn (I,J)^{k+1}(-1)^{|b|j_1 (k+1)}\\&\quad \quad \quad \quad \quad \quad a(s_1,\dots ,s_{l-1},b(s_l,\dots ,s_{l+n-1},\bar{J}), \dots ,s_{m+n-1}, \bar{I}, x(\bar{K}))\\&\quad \quad \quad \quad \quad \quad + sgn (I\cup J,K)^{k+1} sgn (I,J)^{k+1}(-1)^{(|b|+|x|)j_1 (k+1)}\\&\quad \quad \quad \quad \quad \quad a(s_1,\dots ,s_{l-1},b(s_l,\dots ,s_{l+n-1},\bar{J}, x(\bar{K})), \dots ,s_{m+n-1}, \bar{I})\\&=((\!-\!1)^{|b||x|}(a\cdot x)\circ _l b \!+\! a\circ _l (b\cdot x)) (s_1,\dots ,s_{m\!+\!n\!-\!1}, \bar{1},\dots , \overline{j_1\!+\!j_2\!+\!j_3}) \end{aligned}$$

For the last equality we used that

$$\begin{aligned}&sgn (I\cup J,K) sgn (I,J)= sgn (I, J\cup K) sgn (J,K)\nonumber \\&\quad = sgn (I\cup K, J) sgn (I,K)(-1)^{|J||K|}. \end{aligned}$$

$\square$

Of course, multiplying by a sign, one can make this right action into a left action.

For any operad $\mathcal Q$, the unary operations $\mathcal Q(1)$ form an algebra, hence in particular a Lie algebra, which acts on $\mathcal Q$ by operadic derivations. Concretely, for $q\in \mathcal Q(1), a\in \mathcal Q(n)$ the formula is

$$\begin{aligned} q\cdot a = q\circ _1 a -(-1)^{|a||c|}\sum _{j=1}^n a\circ _j q. \end{aligned}$$

(40)

Suppose that in addition some Lie algebra $\mathfrak {{h}}$ acts from the left on $\mathcal Q$ by operadic derivations. Then also the Lie algebra $\mathfrak {{h}}\ltimes \mathcal Q(1)$ acts on $\mathcal Q$ by operadic derivations. Applying this to our case, we see that the Lie algebra

$$\begin{aligned} \hat{\mathfrak {{g}}}=\mathfrak {{g}}\ltimes \widetilde{{ Tw }}\mathcal P(1) \end{aligned}$$

acts on $\widetilde{{ Tw }}\mathcal P(1)$ by operadic derivations. Given some (homogeneous) element $x \in \mathfrak {{g}}$, we construct an element $x_1\in \widetilde{{ Tw }}\mathcal P(1)$ by the formula

$$\begin{aligned} x_1(1,\bar{1},\dots , \bar{j}) = x(1,\bar{1},\dots , \bar{j}). \end{aligned}$$

(41)

Lemma I.5

If $\mu \in \mathfrak {{g}}$ is a Maurer–Cartan element, then $\hat{\mu }=\mu -\mu _1$ is a Maurer–Cartan element in $\hat{\mathfrak {{g}}}=\mathfrak {{g}}\ltimes \widetilde{{ Tw }}\mathcal P(1)$.

Proof

Let us compute this for homogeneous $\mu$, the general case is analogous.

$$\begin{aligned}&\frac{1}{2}\left[ {\mu _1},{\mu _1}\right] (s_1, \bar{1}, \dots , \overline{2m-2}) =\sum _{I\sqcup J = [2m-2]} sgn (I,J)^{k+1} \mu (\mu (s_1, \bar{I}), \bar{J})\\&\quad =-\sum _{I\sqcup J = [2m-2]} sgn (I,J)^{k+1} (-1)^{m(k+1)} \mu (\mu (\bar{I}), s_1, \bar{J})\\&\quad =- \sum _{I\sqcup J = [2m-2]} sgn (I,J)^{k+1} (-1)^{k+1} \mu (s_1, \bar{J}, \mu (\bar{I}))\\&\quad =-\sum _{I\sqcup J = [2m-2]} sgn (I,J)^{k+1} (-1)^{m(m-1)(k+1)} \mu (s_1, \bar{I}, \mu (\bar{J}))\\&\quad =-\mu _1\cdot \mu (s_1, \bar{1},\dots , \overline{2m-2}) =\mu \cdot \mu _1(s_1, \bar{1},\dots , \overline{2m-2}) \end{aligned}$$

$\square$

Given a Maurer–Cartan element in $\hat{\mathfrak {{g}}}$ we can twist the differential on $\widetilde{{ Tw }}\mathcal P$.

Definition I.6

Let $\mathcal P$ be any operad and $k\in \mathbb {Z}$ be an integer. Let $\mu \in \mathsf {Def}( Lie _\infty ^{(k)}\rightarrow \mathcal P)$ be a Maurer–Cartan element. Then we define the $\mu$-twisted operad ${ Tw }\mathcal P$ as the operad constructed above with differential

$$\begin{aligned} d_{\mathcal P} + \hat{\mu }\cdot . \end{aligned}$$

By construction, one has an action on ${ Tw }\mathcal P$ by the $\mu$-twisted version of the Lie algebra $\mathfrak {{g}}$, i.e., the Lie algebra $\mathfrak {{g}}$ with the term $\left[ {\mu },{\cdot }\right]$ added to the differential.

Remark I.7

Let $\mathcal P, k, \mu$ be as above and suppose that a $\mathcal P$-algebra $A$ is given. Let $\mathfrak {{n}}$ be a commutative nilpotent or pro-nilpotent algebra, e.g., $\mathfrak {{n}}=\epsilon {\mathbb {R}}[[\epsilon ]]$. Let $m\in A\otimes \mathfrak {{n}}$ be a Maurer–Cartan element, i.e., a Maurer–Catan element in the $hoLie _k$-algebra $A\otimes \mathfrak {{n}}$. Then $A\otimes \mathfrak {{n}}$ is an algebra over the operad ${ Tw }\mathcal P$ by the formula

$$\begin{aligned} p(x_1,\dots , x_n) = \frac{1}{j!} p(x_1,\dots ,x_n,m,\dots ,m) \end{aligned}$$

for $p\in { Tw }\mathcal P$ homogeneous wrt. the degree in $j$ and $a_1,\dots ,a_n\in A$.¹³

1.3 I.3. More explicit description of the action on the Graphs operad

Let us specialize the above constructions to the case $\mathcal P = {\mathsf {Gra}}_n^\circlearrowleft$. Then $\mathfrak {{g}}=\mathsf {Def}( hoLie _{n-1}\rightarrow \mathcal P)=\mathsf {fGC}_n^\circlearrowleft$ is the “full” graph complex, containing all possible graphs, possibly with multiple connected components, tadpoles or multiple edges. Elements of $\mathfrak {{g}}$ should be considered as (possibly infinite) linear combinations of graphs with numbered vertices, invariant under permutation of vertex labels. In pictures we draw a graph with black unlabelled vertices. This should be understood as the sum of all possible numberings of the vertices, divided by the order of the automorphism group. Note that the picture is still inaccurate since we do not specify the overall sign. An explicit Maurer–Cartan element $\mu \in \mathfrak {{g}}$ is given by the graph with two vertices and one edge.

By twisting one obtains the operad ${ Tw }\mathcal P={\mathsf {fGraphs}}_n^\circlearrowleft$. The $N$-ary operations of ${ Tw }\mathcal P$ are (possibly infinite) linear combinations of graphs, having two kinds of numbered vertices, “internal” and “external”. It is required that there are exactly $N$ external vertices and that the linear combination is invariant under interchange of the labels on the internal vertices. In pictures, we draw the internal vertices black without labels, with the convention that one should sum up over all possible numberings, and divide by the order of the symmetry group. From the Maurer–Cartan element $\mu$ one obtains the element $\mu _1\in { Tw }\mathcal P(1)$. It is given by the graph with one external and one internal vertex, and an edge between them. The differential on ${ Tw }\mathcal P$ has three terms: (i) There is a term coming from the action of $\mu$ as in Lemma I.4. Concretely, this amounts to splitting each internal vertex into two and reconnecting the incoming edges. (ii) There is a term $\sum _{j} (\cdot )\circ _j \mu _1$. This amounts to splitting off from each external vertex one internal vertex, and reconnecting the incoming edges. (iii) There is a term $\mu _1\circ _1 (\cdot )$. This term adds a new internal vertex and connects it to every other vertex (but one at a time). Note that if all internal vertices are at least bivalent, the terms (iii) precisely cancel those terms from (i) and (ii) that contain graphs with univalent internal vertices.

Next consider more generally the action of an arbitrary element $\gamma \in \mathfrak {{g}}$ on some $\Gamma \in { Tw }\mathcal P$. It again contains three terms: (i) There is a term coming from the action as in Lemma I.4. It amounts to inserting $\gamma$ at the internal vertices of $\Gamma$ and reconnecting the incoming edges. (ii) Build the element $\gamma _1\in { Tw }\mathcal P(1)$ by marking the vertex 1 in $\gamma$ as external. Then there is a term in the action stemming from $\sum _{j} \Gamma \circ _j \mu _1$. This amounts to inserting $\gamma _1$ at all external vertices. (iii) There is the term $\gamma _1\circ _1 \Gamma$. This amount to inserting $\Gamma$ at the external vertex of $\gamma _1$.

Remark I.8

A Hopf operad is an operad in the symmetric monoidal category of counital coalgebras (see [32, section 5.3.5]). The operad ${\mathsf {Graphs}}_n$ is a (cocommutative) Hopf operad. Concretely, ${\mathsf {Graphs}}_n(N)=S({\mathsf {ICG}}_n(N)[1])$ may be identified with a symmetric (co)product space (more precisely the the Chevalley–Eilenberg complex) of the $L_\infty$ algebra of internally connected graphs ${\mathsf {ICG}}_n(N)$, cf (10). The operad $e_n$ is a (quasi-isomorphic) sub-Hopf operad of ${\mathsf {Graphs}}_n$. Combinatorially, the action of $\mathsf {GC}_n$ on ${\mathsf {Graphs}}_n$ always merges zero or more internally connected components into one new internally connected component. Hence it is compatible with (i.e., a derivation with respect to) the coproduct. Furthermore it respects the counit. Thus $\mathsf {GC}_n$ acts on $e_n$ by Hopf operad derivations, up to homotopy.

Appendix J: The Tamarkin map $\mathfrak {{grt}}_1\rightarrow H^0(\mathsf {GC}_2)$ and Algorithm 2 of Sect. 9.

Part of the arguments in this subsection came out of a discussion with Pavol Ševera.

1.1 J.1. The map

Let us first describe the map $\mathfrak {{grt}}_1\rightarrow H^0(\mathsf {GC}_2)$, which is the universal version of the action of $\mathfrak {{grt}}_1$ on $T_\mathrm{poly}$ discussed in Sect. 6.3.2. As in Sect. 6.3.1, we have a chain of quasi-isomorphisms of operads

$$\begin{aligned} C{\mathcal {N}} PaCD \rightarrow BU\mathfrak {{t}} \leftarrow e_2 \leftarrow hoe _2. \end{aligned}$$

(42)

Fix a lift up to homotopy $F: hoe _2 \rightarrow C{\mathcal {N}} PaCD$ for now. It exists because $hoe _2$ is cofibrant.

The Lie algebra $\mathfrak {{grt}}_1$ acts on $PaCD$. Let some $\phi \in \mathfrak {{grt}}_1$ be given. Its action on $F$ determines an element $\phi '\in \mathsf {Def}( hoe _2 \rightarrow C{\mathcal {N}} PaCD )$. Since the map $F$ is a quasi-isomorphism, we may decompose $\phi '$ into a lift $l\in {{\mathrm{Der}}}( hoe _2)$ up to a homotopy $h\in \mathsf {Def}( hoe _2 \rightarrow C{\mathcal {N}} PaCD )$. The lift may furthermore be chosen so that it has trivial $hoLie _2$ part by degree reasons. So we have the following diagram.

The (infinitesimal) automorphism $l$ of $hoe _2$ describes the action of the $\mathfrak {{grt}}_1$-element $\phi$ on $hoe _2$. The composition of the homotopy $h$ with the maps $hoLie _2 \rightarrow hoe _2$ from the left and $C{\mathcal {N}} PaCD \rightarrow {\mathsf {Gra}}$ from the right yields a degree 0 cocycle

$$\begin{aligned} \xi \in \mathsf {Def}( hoLie _2 \rightarrow {\mathsf {Gra}}) \cong \mathsf {fGC}\end{aligned}$$

which represents a cohomology class in the (full) graph complex. Since $H^0(\mathsf {fGC})\cong H^0(\mathsf {GC})$ this class corresponds to a cohomology class of M. Kontsevich’s graph complex $\mathsf {GC}$. A representative in $\mathsf {GC}$ may be obtained by dropping from $\xi$ all graphs that are not connected or have a less then trivalent vertex. Composing $\xi$ with the map ${\mathsf {Gra}}\rightarrow {{\mathrm{End}}}( T_\mathrm{poly} )$ we recover D. Tamarkin’s action of $\phi$ on $T_\mathrm{poly}$.

Remark J.1

Concretely, the map $C{\mathcal {N}} PaCD \rightarrow {\mathsf {Gra}}$ occuring above is constructed as follows. First there is a map $C{\mathcal {N}} PaCD \rightarrow BU(\mathfrak {{t}})$ to the bar construction of the completed universal enveloping algebra of $\mathfrak {{t}}$, by forgetting the parenthesization. Next one projects to the abelianization, $BU(\mathfrak {{t}})\rightarrow BU(\mathfrak {{t}}/\left[ {\mathfrak {{t}}},{ \mathfrak {{t}}}\right] )$. For an abelian Lie algebra there is a canonical map from the bar construction of the universal enveloping algebra to the Chevalley complex, $BU(\mathfrak {{t}}/\left[ {\mathfrak {{t}}},{ \mathfrak {{t}}}\right] )\rightarrow C(\mathfrak {{t}}/\left[ {\mathfrak {{t}}},{ \mathfrak {{t}}}\right] )$. Finally one realizes that $C(\mathfrak {{t}}/\left[ {\mathfrak {{t}}},{ \mathfrak {{t}}}\right] ) \cong {\mathsf {Gra}}$. Concretely, the latter isomorphism is obtained by sending a product $t_{i_1j_1}\wedge \cdots \wedge t_{i_r j_r}\in C(\mathfrak {{t}}_n/\left[ {\mathfrak {{t}}_n},{ \mathfrak {{t}}_n}\right] )$ to a graph with $n$ vertices and edges between vertices $i_1$ and $j_1$, $i_2$ and $j_2$ etc.

Remark J.2

Note that the graph cohomology class $[\xi ]$ we associate to the element $l\in {{\mathrm{Der}}}( hoe _2)$ by the above recipe is the same as the one obtained using the map of Theorem 1.3: From $l$ we obtain an element in $\mathsf {Def}( hoe _2 \rightarrow e_2)$ by composition with $hoe _2\rightarrow e_2$ and degree 1 element $l'\in \mathsf {Def}( hoe _2 \rightarrow {\mathsf {Gra}})$ by further composition with $e_2\rightarrow {\mathsf {Gra}}$. Furthermore from the homotopy $h$ we obtain a degree zero element $h'\in \mathsf {Def}( hoe _2 \rightarrow {\mathsf {Gra}})$, whose coboundary is $l'$. Using these data to compute the connecting homomorphism in the long exact sequence from Proposition 5.5, we see that graph cocycle obtained is just the $hoLie _2$ part of $h'$, i. e., $\xi$. (Note that $l'$ is not guaranteed to be in the subcomplex $\Xi _{ conn }$, but one may remedy by adding some exact elements with trivial $hoLie _2$ part.)

1.2 J.2. The lift $hoe _2 \rightarrow C{\mathcal {N}} PaCD$

To obtain more information one has to consider the lift $F: hoe _2 \rightarrow C{\mathcal {N}} PaCD$ of $hoe _2 \mathop {\rightarrow }\limits ^{f}BU(\mathfrak {{t}}) \mathop {\twoheadleftarrow }\limits ^{g}C{\mathcal {N}} PaCD$. To determine $F$, it suffices to specify the image of the generators, i. e., of $e_2^\vee (N)[1]\cong e_2^*\{2\}(N)[1]$ for $N=2,3,\dots$ (respecting $S_N$ equivariance). These images are obtained by recursively (in $N$) solving the equations

$$\begin{aligned} dF(x)&= F(dx) \\ g(F(x))&= f(x) \end{aligned}$$

for $x\in e_2^\vee (N)[1]$, where we denote the differentials on $hoe _2$ and $C{\mathcal {N}} PaCD$ both by $d$, abusing notation. Solutions exists since the maps $f$ and $g$ are quasi-isomorphisms, but they are not unique.

Note furthermore that there are additional gradings on the objects involved: On $C{\mathcal {N}} PaCD$ and $BU(\mathfrak {{t}})$ there is the grading by $t$-degree, while on $hoe _2$ there is the grading of Sect. 4.1. The maps $f$, $g$ and the differential $d$ respect these gradings and hence we may choose $F$ in such a way that it respects the additional grading as well.

We will be interested in the restrictions to $Com _\infty \subset hoe _2$ and $hoLie _2\subset hoe _2$. Denote by $m_2, m_3,\dots$ generators of $Com _\infty$ and by $\mu _2,\mu _3,\dots$ the generators of $hoLie _2$. The gradings are as follows. $m_j$ has cohomological degree $2-j$, while the additional degree (of Sect. 4.1) is $0$. $\mu _j$ has cohomological degree $3-2n$ and additional degree $n-1$.

Example J.3

For $N=2$ we may pick

$$\begin{aligned} F(m_2)&= \frac{1}{2} (1_{12}+1_{21})&F(\mu _2)&= \frac{1}{2} \left( t_{12}\otimes { id }_{12} + t_{12}\otimes { id }_{21} \right) \end{aligned}$$

where $1_{12}$ and $1_{21}$ denote the two zero-chains “over” the two objects $(12)$ and and $(21)$ of $PaCD (2)$, and $t_{12}\otimes { id }_{12}$ and $t_{12}\otimes { id }_{21}$ are one-chains made out of the identity morphisms of $(12)$ and $(21)$ and $t_{12}\in \mathfrak {{t}}_2\subset U(\mathfrak {{t}}_2)$.

Note also that the $t$ degree of $F(m_2)$ is zero and that of $F(\mu _2)$ is $1$, so that $F$ respects also the additional degrees.

1.3 J.3. Sketch of proof of Lemma 6.10 and hence of Theorem 6.9

Defining $m_3$ appropriately (there is a choice) one finds that $F(m_3)$ has to satisfy the equation

$$\begin{aligned} dF(m_3)&= F(m_2)\circ _1 F(m_2) - F(m_2)\circ _2 F(m_2)\\&= \frac{1}{4} ( 1_{(12)3} +1_{(21)3} + 1_{3(12)} + 1_{3(21)}\\&- 1_{1(23)} - 1_{1(32)} - 1_{(23)1} - 1_{(32)1}). \end{aligned}$$

where we used the notation $1_{(12)3}$ for the zero chain above object $(12)3$ etc. As remarked before, we may pick all $F(m_j)$ to have $t$-degree 0.

Let us show Lemma 6.10. Pick some $\mathfrak {{grt}}_1$ element $\phi$. To obtain the corresponding element of $\mathsf {Def}( hoe _2\rightarrow BU(\mathfrak {{t}}) )$ we have to let it act on the lift map $F: hoe _2$ and $C{\mathcal {N}} PaCD$ and then compose with the projection $C{\mathcal {N}} PaCD \rightarrow BU(\mathfrak {{t}})$. Note that the action on $F(m_j)$ produces a $j-2$-chain, in which all of the $j-2$ involved morphisms except for one have $t$-degree 0. Since $BU(\mathfrak {{t}})$ is the normalized bar construction, the image in $BU(\mathfrak {{t}})$ hence vanishes unless $j=3$.

Remark J.4

Let $h$ be the unique morphism in $PaCD (3)$ from any of the four objects $(ij)k$, $k(ij)$, $(ji)k$ or $k(ji)$ to any of the objects $i(jk)$, $(jk)i$, $(kj)i$ or $i(kj)$, where $i,j,k$ is a permutation of $1,2,3$. Then acting with $\phi$ on the 1-chain described by $h$ and projecting the result to $BU(\mathfrak {{t}})$ yields the degree $-1$ element $\phi (t_{ij}, t_{jk})\in BU(\mathfrak {{t}})$.

From the remark and the hexagon identity it is not hard to see that acting with $\phi$ on $F(m_3)$ and projecting the result $BU(\mathfrak {{t}})$ we obtain the element $\phi (t_{12}, t_{23})$. Note that explicit knowledge of $F(m_3)$ is not required here, only knowledge of $dF(m_3)$ as above. But $\phi (t_{12}, t_{23})$ is the $\mathfrak {{grt}}_1$-element we started with, considering $\mathfrak {{grt}}_1$ as a subset of $\mathfrak {{t}}_3$. This shows Lemma 6.10 and hence D. Tamarkin’s Theorem 6.9.

1.4 J.4. Comparison to Algorithm 2 of Sect. 9

We will show that the cochain $\gamma '$ produced in Algorithm 2 of Sect. 9 agrees with $\xi$ as in Sect. 1 up to exact terms. Restrict the map $F: hoe _2 \rightarrow C{\mathcal {N}} PaCD$ we picked to $hoLie _2\subset hoe _2$ to obtain a map $hoLie _2\rightarrow C{\mathcal {N}} PaCD$. It is defined by specifying the images of the generators $\mu _2,\mu _3,\dots$. They have to satisfy equations of the form

$$\begin{aligned} d F(\mu _n) =(\text {linear combination of} \, \mu _j\circ \mu _{k} for j+k=n+1.). \end{aligned}$$

(43)

The cohomological degree of $F(\mu _n)$ must be $3-2n$. This means that $\mu _n$ is some linear combination of chains of morphisms of $PaCD$ of length $2n-3$. Furthermore, as before, we may pick $F$ such that $F(\mu _n)$ has $t$-degree $n-1$.

Remark J.5

In particular, this means that a chain of morphisms of $PaCD$ occuring in $\mu _n$ must contain at least $2n-3-(n-1)=n-2$ morphisms of $t$-degree zero. In particular, mapping $\mu _n$ along $C{\mathcal {N}} PaCD \rightarrow {\mathsf {Gra}}$ yields zero, except for $n=2$.

Next consider the action of $\phi \in \mathfrak {{grt}}_1$. It produces some

$$\begin{aligned} a \in \mathsf {Def}( hoe _2 \rightarrow C{\mathcal {N}} PaCD ) \end{aligned}$$

of degree 1. Precompose with $hoLie _2\rightarrow hoe _2$ and compose with $C{\mathcal {N}} PaCD \rightarrow BU\mathfrak {{t}}$ to obtain some element

$$\begin{aligned} a' \in \mathsf {Def}( hoLie _2 \rightarrow BU\mathfrak {{t}}). \end{aligned}$$

By the last Remark one sees that $a'$ is determined solely by the image of $\mu _3$ under the action of $x$. By an explicit calculation one can see that $a'$ in fact agrees with the element $T_3\in \mathsf {Def}( hoLie _2 \rightarrow C\mathfrak {{t}})\subset \mathsf {Def}( hoLie _2 \rightarrow BU\mathfrak {{t}})$ from the second algorithm in Sect. 6.3.1. Consider next the homotopy $h$. Precompose it with $hoLie _2\rightarrow hoe _2$ and compose with $C{\mathcal {N}} PaCD \rightarrow BU\mathfrak {{t}}$ so as to obtain some degree 0 element

$$\begin{aligned} h' \in \mathsf {Def}( hoLie _2 \rightarrow BU\mathfrak {{t}}). \end{aligned}$$

Because $h$ was the homotopy making the lower right cell in the commutative diagram above commute, one has

$$\begin{aligned} a'=dh' \end{aligned}$$

where $d$ is now the differential in $\mathsf {Def}( hoLie _2 \rightarrow BU\mathfrak {{t}})$. Since $a'=T_3$ actually lives in $\mathsf {Def}( hoLie _2 \rightarrow C\mathfrak {{t}})$ and $C\mathfrak {{t}}\rightarrow BU\mathfrak {{t}}$ is a quasi-isomorphism, there is a closed $c\in \mathsf {Def}( hoLie _2 \rightarrow BU\mathfrak {{t}})$, such that $U:=h'+c\in \mathsf {Def}( hoLie _2 \rightarrow C\mathfrak {{t}})$. Since $H^0(\mathsf {Def}( hoLie _2 \rightarrow C\mathfrak {{t}}))=0$, $c$ is in fact exact. Taking $U$ for the $U$ in Algorithm 2 of section 9, we see that the output $\gamma '$ of that algorithm is the image of $U$ after composition with $BU\mathfrak {{t}}\rightarrow {\mathsf {Gra}}$. The image of $h'$ after composition with $BU\mathfrak {{t}}\rightarrow {\mathsf {Gra}}$ is $\xi$, hence

$$\begin{aligned} \gamma ' = \xi +(\text {exact terms}). \end{aligned}$$

Appendix K: A short note on the directed version of the graph complex.

One may define a directed version of the graph complex $\mathsf {GC}_n$ by (i) taking directed instead of undirected edges, (ii) allowing bivalent vertices and (iii) requiring that a graph has at least one trivalent vertex.¹⁴ Call the resulting graph complexes $\mathsf {dGC}_n$.

Proposition K.1

$$\begin{aligned} H(\mathsf {dGC}_n) \cong H(\mathsf {GC}_n). \end{aligned}$$

There is even an explicit quasi-isomorphism of dg Lie algebras

$$\begin{aligned} \mathsf {GC}_n \rightarrow \mathsf {dGC}_n \end{aligned}$$

sending an undirected graph to a sum of directed graphs, obtained by interpreting each edge as the sum of edges in both directions.

Proof (Proof of the Proposition)

[Proof of the Proposition] Set up a spectral sequence on the number of non-bivalent vertices. The first differential produces one bivalent vertex. As in the undirected case, consider a graph with bivalent vertices as one with at least trivalent vertices and labelled edges, see Fig. 16. The first differential just changes the labels. The resulting subcomplex is (essentially) a tensor product the complex for one edge. The cohomology of the latter can be seen to have a single nontrivial cohomology class, represented by the sum of edges in both direcions (i.e., $\leftarrow + \rightarrow$). Hence the first convergent in the spectral sequence is $\mathsf {GC}_n$ and hence $\mathsf {GC}_n\rightarrow \mathsf {dGC}_n$ is a quasi-isomorphism. $\square$

Fig. 16

Passing from a graph with bivalent vertices to one with trivalent vertices only, but labelled edges. Note that we cheat a little, since one has to assign some directions, by some covention, to the edges on the right so as to interpret the labels correctly. However, for the argument in the proof this does not matter

Appendix L: The automorphism group of $hoe _n$

Let $\mathcal C$ be a coaugmented cooperad with zero differential, finite dimensional in each arity. We will consider the cobar construction $\Omega (\mathcal C)$ and its automorphism group ${{\mathrm{End}}}(\Omega (\mathcal C))$. First note that by functoriality of the cobar construction we have a map from the automorphism group of the coaugmented cooperad $\mathcal C$

$$\begin{aligned} \phi : {{\mathrm{End}}}(\mathcal C) \rightarrow {{\mathrm{End}}}(\Omega (\mathcal C)). \end{aligned}$$

Recall from [32, section 6.5] that elements of $\Omega (\mathcal C)$ may be understood as linear combinations of certain trees, whose vertices are decorated by elements of $\overline{\mathcal C}$. The operadic composition is grafting of trees. Hence there is a filtration by the number of vertices in a tree on the operad $\Omega (\mathcal C)$. Concretely, ${\mathcal {F}}^p\Omega (\mathcal C)$ is spanned by trees with $p$ or more vertices. By quasi-freeness any automorphism of $\Omega (\mathcal C)$ preserves this filtration. Clearly

$$\begin{aligned} \overline{\mathcal C} \cong (\Omega (\mathcal C)/{\mathcal {F}}^2 \Omega (\mathcal C))[1] \end{aligned}$$

and hence we have a map

$$\begin{aligned} \psi : {{\mathrm{End}}}(\Omega (\mathcal C))\rightarrow {{\mathrm{End}}}_{\mathbb {S}-\mathrm {mod}}(\mathcal C) \end{aligned}$$

where ${{\mathrm{End}}}_{\mathbb {S}-\mathrm {mod}}(\mathcal C)$ is the group of automorphisms of the $\mathbb {S}$-module $\mathcal C$. It is clear that $\psi \circ \phi$ agrees with the inclusion ${{\mathrm{End}}}(\mathcal C)\subset {{\mathrm{End}}}_{\mathbb {S}-\mathrm {mod}}(\mathcal C)$. We claim that the image of $\psi$ actually lands in ${{\mathrm{End}}}(\mathcal C)$. Indeed any automorphism $f\in {{\mathrm{End}}}(\Omega (\mathcal C))$ has to respect the differential. On generators, The differential is (see [32, 6.5.5]) just the infinitesimal decomposition (see [32, 6.1.7]) on $\mathcal C$, up to degree shifts.¹⁵ It follows that $\psi (f)$ has to respect this infinitesimal decomposition. But the latter generates all cocompositions in $\mathcal C$ and hence $\psi (f)$ has to respect the cooperad structure. (The coaugmentation is also respected by construction.) Thus we have have a decomposition

$$\begin{aligned} {{\mathrm{End}}}(\Omega (\mathcal C)) = {{\mathrm{End}}}(\mathcal C) \ltimes {{\mathrm{End}}}_1(\Omega (\mathcal C)) \end{aligned}$$

where we abbreviate

$$\begin{aligned} {{\mathrm{End}}}_1(\Omega (\mathcal C)) = \ker \phi . \end{aligned}$$

Lemma L.1

${{\mathrm{End}}}_1(\Omega (\mathcal C))$ is a pro-unipotent group.

Proof

Let $\mathcal P_n\subset \Omega (\mathcal C)$ be the suboperad spanned by trees all of whose vertices have $\le n$ children. Equivalently this sub-operad is generated by an $\mathcal S$ module obtained from $\mathcal C$ by setting to zero all $\mathcal C(N)$ for $N>n$. By quasi-freeness all automorphisms of $\Omega (\mathcal C)$ have to respect $\mathcal P_n$. Similarly all automorphisms of $\mathcal P_n$ have to respect $\mathcal P_{n-1}\subset \mathcal P_n$ etc. We hence have maps

$$\begin{aligned} {{\mathrm{End}}}(\mathcal P_{n-1}) \leftarrow {{\mathrm{End}}}(\mathcal P_n). \end{aligned}$$

Clearly $\lim _\leftarrow {{\mathrm{End}}}(\mathcal P_n) \cong {{\mathrm{End}}}(\Omega (\mathcal C))$. The filtration ${\mathcal {F}}$ from above induces filtrations on each $\mathcal P_n$. Denote by ${{\mathrm{End}}}_1(\mathcal P_n)\subset {{\mathrm{End}}}(\mathcal P_n)$ the subgroup of automorphisms fixing all generators modulo elements of ${\mathcal {F}}^2$. Then, in the same manner as above we have arrows

$$\begin{aligned} {{\mathrm{End}}}_1(\mathcal P_{n-1}) \leftarrow {{\mathrm{End}}}_1(\mathcal P_n) \end{aligned}$$

and $\lim _\leftarrow {{\mathrm{End}}}_1(\mathcal P_n) \cong {{\mathrm{End}}}_1(\Omega (\mathcal C))$.

We claim that each ${{\mathrm{End}}}_1(\mathcal P_n)$ is a unipotent algebraic group. Indeed by the finiteness assumption on $\mathcal C$ it is an algebraic subgroup of some $GL(\mathcal P_n(n))$. Furthermore, if $g\in {{\mathrm{End}}}_1(\mathcal P_n)$ then $(g-{ id })$ maps ${\mathcal {F}}^p\mathcal P_n$ into ${\mathcal {F}}^{p-1}\mathcal P_n$ and hence is a nilpotent element. $\square$

One may also verify that the Lie algebra of ${{\mathrm{End}}}_1(\Omega (\mathcal C))$ is given by the closed degree zero elements in ${{\mathrm{Der}}}'(\Omega (\mathcal C))$ (as defined in Sect. 2).

Remark L.2

(The special case $hoe _n$). Let us specialize to the case $\mathcal C=e_n^\vee$ relevant to this paper. In this case ${{\mathrm{End}}}(\mathcal C)={{\mathrm{End}}}(e_n^\vee )\cong \mathbb {K}^\times \times \mathbb {K}^\times$, the two factors acting by rescaling the two cogenerators. No (non-trivial) such rescaling is homotopy trivial.