2.1 The category $\mathbf {A}$ of Jacobi diagrams in handlebodies

Here, we briefly review the category $\mathbf {A}$ of Jacobi diagrams in handlebodies defined in [Reference Habiro and Massuyeau11]. We use the same notations as in [Reference Katada16].

For $n\geq 0$ , let be the oriented $1$ -manifold consisting of n arc components.

Let $I=[-1,1]$ . For $n\geq 0$ , let $U_n \subset \mathbb {R}^3$ denote the handlebody of genus n that is obtained from the cube $I^3$ by attaching n handles on the top square $I^2 \times \{1\}$ as depicted in Figure 1. We call $l := I \times \{0\} \times \{-1\}$ the bottom line of $U_n$ and $l':= I\times \{0\} \times \{1\}$ the upper line of $U_n$ . We call $S := I^2 \times \{-1\}$ the bottom square of $U_n$ .

Figure 1

The handlebody $U_n$ .

For $i\in [n]$ , let $x_i$ be a loop which goes through only the i-th handle of the handlebody $U_n$ just once, and let $x_i$ denote its homotopy class as well. In what follows, for loops $\gamma _1$ and $\gamma _2$ with base points on l, let $\gamma _2\gamma _1$ denote the loop that goes through $\gamma _1$ first and then goes through $\gamma _2$ . That is, we write a product of elements of the fundamental group of $U_n$ in the opposite order to the usual one. Let $H=H_1(U_n;\mathbb {Z})$ , and let $\bar {x}_i\in H$ be the homology class of $x_i$ . We have $H=\bigoplus _{i=1}^n \mathbb {Z}\bar {x}_i$ and $\pi _1(U_n)=\langle x_1,\cdots ,x_n\rangle $ . Let

$$ \begin{align*}V_n=H^1(U_n;\Bbbk)=\operatorname{Hom}(H,\Bbbk),\end{align*} $$

and let $\{v_1,\cdots ,v_n\}$ be the dual basis of $\{\bar {x}_1,\cdots ,\bar {x}_n\}$ .

The objects in $\mathbf {A}$ are nonnegative integers.

For $m,n\geq 0$ , the hom-set $\mathbf {A}(m,n)$ is the $\Bbbk $ -vector space spanned by $(m,n)$ -Jacobi diagrams modulo the STU relation. An $(m,n)$ -Jacobi diagram is a Jacobi diagram on $X_n$ mapped into $U_m$ in such a way that the endpoints of $X_n$ are uniformly distributed on the bottom line l of $U_m$ (see [Reference Habiro and Massuyeau11, Reference Katada16] for further details). We usually depict $(m,n)$ -Jacobi diagrams by drawing their images under the orthogonal projection of $\mathbb {R}^3$ onto $\mathbb {R}\times \{0\}\times \mathbb {R}$ .

The degree of an $(m,n)$ -Jacobi diagram is the degree of its Jacobi diagram. Let $\mathbf {A}_d(m,n)\subset \mathbf {A}(m,n)$ be the subspace spanned by $(m,n)$ -Jacobi diagrams of degree d. We have $\mathbf {A}(m,n)=\bigoplus _{d\geq 0} \mathbf {A}_d(m,n)$ .

The category $\mathbf {A}$ has a structure of a linear symmetric strict monoidal category. The tensor product on objects is addition. The monoidal unit is $0$ . The tensor product on morphisms is juxtaposition followed by horizontal rescaling and relabelling of indices. The symmetry is determined by

2.2 N-series and graded Lie algebras

Here, we briefly review the definition of an action of an N-series on a filtered vector space and the induced action of the graded Lie algebra on the graded vector space (see [Reference Katada16] for details).

An N-series $K_{\ast }=(K_n)_{n\geq 1}$ of a group K is a descending series

$$ \begin{align*}K=K_1\supset K_2\supset \cdots \end{align*} $$

such that $[K_n,K_m]\subset K_{n+m}$ for all $n,m\geq 1$ .

A morphism $f: G_{\ast }\rightarrow K_{\ast }$ between N-series is a group homomorphism $f: G_1 \rightarrow K_1$ such that we have $f(G_n)\subset K_n$ for all $n\geq 1.$

For a filtered vector space $W_{\ast }$ , set

$$ \begin{align*}\operatorname{Aut}_n(W_{\ast}):=\{\phi\in\operatorname{Aut}_{\mathbf{fVect}}(W_{\ast})\mid [\phi,w]\in W_{k+n}\text{ for all }w\in W_k, k\geq 0\}\quad (n\geq 1), \end{align*} $$

where $[\phi ,w]:=\phi (w)-w$ for $w\in W_k$ . We can easily check that $\operatorname {Aut}_{\ast }(W_{\ast }):=(\operatorname {Aut}_n(W_{\ast }))_{n\geq 1}$ is an N-series.

For an N-series $K_{\ast }$ , we have a graded Lie algebra $\mathrm {gr}(K_{\ast })=\bigoplus _{n\geq 1} K_n/K_{n+1}$ , where the Lie bracket is defined by the commutator.

For a graded vector space $W=\bigoplus _{k\geq 0}W_k$ , set

$$ \begin{align*}\operatorname{End}_n(W):=\{\phi\in\operatorname{End}(W)\mid \phi(W_k)\subset W_{k+n}\text{ for }k\geq 0\}\quad(n\geq 1). \end{align*} $$

We can check that $\operatorname {End}_{+}(W)=\bigoplus _{n\geq 1}\operatorname {End}_n(W)$ is a graded Lie algebra, where the Lie bracket is defined by

$$ \begin{align*}[f,g]:=f\circ g-g\circ f \quad \text{for} \quad f\in\operatorname{End}_k(W),g\in\operatorname{End}_l(W)\; (k,l\geq 1).\end{align*} $$

Proposition 2.3. An action of an N-series $K_{\ast }$ on a filtered vector space $W_{\ast }$ induces an action of the graded Lie algebra $\mathrm {gr}(K_{\ast })$ on the graded vector space $\mathrm {gr}(W_{\ast })$ , which is a morphism

$$ \begin{align*}\rho_{+}:\bigoplus_{n\geq 1}\operatorname{gr}^n(K_{\ast}) \rightarrow\bigoplus_{n\geq 1}\operatorname{End}_n(\mathrm{gr}(W_{\ast}))\end{align*} $$

defined by $\rho _{+}(gK_{n+1})([v]_{W_{k+1}})=[[g,v]]_{W_{k+n+1}}$ for $gK_{n+1}\in \operatorname {gr}^n(K_{\ast })$ , $[v]_{W_{k+1}}\in \operatorname {gr}^k(W_{\ast })$ .

The proof can be seen in Proposition 5.14 of [Reference Katada16].

2.3 Contents of the previous paper

Here, we briefly review the notations and contents of the previous paper [Reference Katada16]. Let $\operatorname {Aut}(F_n)$ denote the automorphism group of the free group $F_n$ of rank n and $\operatorname {GL}(n;\mathbb {Z})$ the general linear group of degree n. Let $\operatorname {IA}(n)$ denote the IA-automorphism group of $F_n$ , that is the kernel of the canonical surjection

$$ \begin{align*}\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(H_1(F_n;\mathbb{Z}))\cong \operatorname{GL}(n;\mathbb{Z}).\end{align*} $$

Let $\Gamma _{\ast }(\operatorname {IA}(n))=(\Gamma _{r}(\operatorname {IA}(n)))_{r\geq 1}$ denote the lower central series of $\operatorname {IA}(n)$ , and $\mathrm {gr}(\operatorname {IA}(n))=\bigoplus _{r\geq 1}\operatorname {gr}^r(\operatorname {IA}(n))$ the associated graded Lie algebra, where $\operatorname {gr}^r(\operatorname {IA}(n))=\Gamma _{r}(\operatorname {IA}(n))/\Gamma _{r+1}(\operatorname {IA}(n))$ .

Let $A_d(n)=\mathbf {A}_d(0,n)$ denote the $\Bbbk $ -vector space of Jacobi diagrams of degree d on $X_n$ . We consider a filtration for $A_d(n)$

$$ \begin{align*}A_d(n)=A_{d,0}(n)\supset A_{d,1}(n)\supset\cdots\supset A_{d,2d-2}(n)\supset A_{d,2d-1}(n)=0 \end{align*} $$

such that $A_{d,k}(n)\subset A_d(n)$ is the subspace spanned by Jacobi diagrams with at least k trivalent vertices. Hence, $A_d(n)$ is a filtered vector space.

Let $\mathbf {F}$ denote the category of finitely generated free groups and $\mathbf {fVect}$ the category of filtered vector spaces over $\Bbbk $ .

We have a $\Bbbk $ -vector space isomorphism

$$ \begin{align*}Z:\Bbbk\mathbf{F}^{\mathrm{op}}(m,n)\xrightarrow{\cong} \mathbf{A}_0(m,n)\end{align*} $$

from the hom-set $\Bbbk \mathbf {F}^{\mathrm {op}}(m,n)$ of the $\Bbbk $ -linearization of the opposite category of $\mathbf {F}$ to the degree $0$ part of the hom-set $\mathbf {A}(m,n)$ [Reference Habiro and Massuyeau11]. We define a functor

$$ \begin{align*}A_d:\mathbf{F}^{\mathrm{op}}\rightarrow \mathbf{fVect}\end{align*} $$

by $A_d(n)=\mathbf {A}_d(0,n)$ for an object $n\in \mathbb {N}$ and $A_d(f)=Z(f)_{\ast }$ for a morphism $f\in \mathbf {F}^{\mathrm {op}}(m,n)$ , where $Z(f)_{\ast }$ denotes the post-composition with $Z(f)$ . The functor $A_d$ is a polynomial functor of degree $2d$ in the sense of [Reference Hartl, Pirashvili and Vespa12, Reference Powell and Vespa20] (see Remark 3.1 of [Reference Katada16]). By restricting this functor to the automorphism group, we obtain an action of the opposite group $\operatorname {Aut}(F_n)^{\mathrm {op}}$ of $\operatorname {Aut}(F_n)$ on $A_d(n)$ for each $n\geq 0$ . We consider this action as a right action of $\operatorname {Aut}(F_n)$ on $A_d(n)$ . The $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ induces an action on $A_d(n)$ of the outer automorphism group $\operatorname {Out}(F_n)$ of $F_n$ (see Theorem 5.1 in [Reference Katada16]).

On the other hand, the associated graded vector space $\mathrm {gr}(A_d(n))$ of $A_d(n)$ is identified via the PBW map [Reference Bar-Natan2, Reference Bar-Natan3]

(2.1) $$ \begin{align} \theta_{d,n}:\mathrm{gr}(A_d(n))\xrightarrow{\cong} B_d(n) \end{align} $$

with the graded $\Bbbk $ -vector space $B_d(n)=\bigoplus _{k\geq 0}B_{d,k}(n)= \bigoplus _{k=0}^{2d-2}B_{d,k}(n)$ of $V_n$ -colored open Jacobi diagrams of degree d, where the grading is determined by the number of trivalent vertices. Note that we have $\theta _{d,n}=\bigoplus _{k}\theta _{d,n,k}$ , where

$$ \begin{align*}\theta_{d,n,k}:\operatorname{gr}^k(A_d(n))\xrightarrow{\cong} B_{d,k}(n).\end{align*} $$

Let $\mathbf {FAb}$ denote the category of finitely generated free abelian groups and $\mathbf {gVect}$ the category of graded vector spaces over $\Bbbk $ .

We define a functor

$$ \begin{align*}B_d: \mathbf{FAb}^{\mathrm{op}}\rightarrow \mathbf{gVect}\end{align*} $$

by sending an object $n\in \mathbb {N}$ to the graded vector space $B_d(n)$ and a morphism $f\in \mathbf {FAb}^{\mathrm {op}}(m,n)=\operatorname {Mat}(m,n;\mathbb {Z})$ to $B_d(f)$ , which is a right action on each coloring, where we consider an element of $V_n$ as a $(1\times n)$ -matrix. By restricting this functor to the automorphism group, we obtain an action of the opposite group $\operatorname {GL}(n;\mathbb {Z})^{\mathrm {op}}$ of $\operatorname {GL}(n;\mathbb {Z})$ on $B_d(n)$ for each $n\geq 0$ . We consider this action as a right action of $\operatorname {GL}(n;\mathbb {Z})$ on $B_d(n)$ . Note that the $\operatorname {GL}(n;\mathbb {Z})$ -action on $B_d(n)$ naturally extends to a $\operatorname {GL}(V_n)$ -action on $B_d(n)$ .

Proposition 2.4 see Proposition 3.2 of [Reference Katada16]

For $d\geq 0$ , the PBW maps equation (2.1) give a natural isomorphism

$$ \begin{align*}\theta_d:\mathrm{gr}\circ A_d\overset{\cong}\Rightarrow B_d\circ\mathrm{ab}^{\mathrm{op}},\end{align*} $$

where $\mathrm {ab}^{\mathrm {op}}$ denotes the opposite functor of the abelianization functor and $\mathrm {gr}$ denote the functor that sends a filtered vector space to its associated graded vector space.

By this proposition, it turns out that the $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ , which is an action of an extended N-series on a filtered vector space, induces two actions on $B_d(n)$ , which form an action of an extended graded Lie algebra on a graded vector space (see Theorem 5.15 of [Reference Katada16] and [Reference Habiro and Massuyeau10] for extended N-series and extended graded Lie algebras). One of them is the $\operatorname {GL}(n;\mathbb {Z})$ -action, and the other of them is an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ on the graded vector space $B_d(n)$ , which consists of $\operatorname {GL}(n;\mathbb {Z})$ -module homomorphisms

(2.2) $$ \begin{align} [\cdot,\cdot]: B_{d,k}(n)\otimes \operatorname{gr}^r(\operatorname{IA}(n)) \rightarrow B_{d,k+r}(n) \end{align} $$

for $k\geq 0, r\geq 1$ (see Proposition 5.10 and Theorem 5.15 of [Reference Katada16]). By using these two actions on $B_d(n)$ , we obtained an indecomposable decomposition of $A_2(n)$ as $\operatorname {Aut}(F_n)$ -modules (see Theorem 6.9 of [Reference Katada16]).

2.4 Hopf algebra in a symmetric strict monoidal category

We review the definition of a Hopf algebra in a symmetric strict monoidal category. Let $\mathcal {C}=(\mathcal {C},\otimes ,I,P)$ be a symmetric strict monoidal category. A Hopf algebra in $\mathcal {C}$ is an object H in $\mathcal {C}$ equipped with morphisms

$$ \begin{align*}\mu:H\otimes H\rightarrow H, \quad \eta:I\rightarrow H, \quad \Delta:H\rightarrow H\otimes H, \quad \epsilon:H\rightarrow I,\quad S:H\rightarrow H, \end{align*} $$

called the multiplication, unit, comultiplication, counit and antipode, respectively, satisfying

1. (1) $\mu (\mu \otimes \operatorname {id}_H)=\mu (\operatorname {id}_H \otimes \mu ),\quad \mu (\eta \otimes \operatorname {id}_H)=\operatorname {id}_H=\mu (\operatorname {id}_H\otimes \eta ),$

2. (2) $(\Delta \otimes \operatorname {id}_H)\Delta =(\operatorname {id}_H\otimes \Delta )\Delta ,\quad (\epsilon \otimes \operatorname {id}_H)\Delta =\operatorname {id}_H=(\operatorname {id}_H\otimes \epsilon )\Delta ,$

3. (3) $\epsilon \eta =\operatorname {id}_I, \quad \epsilon \mu =\epsilon \otimes \epsilon ,\quad \Delta \eta =\eta \otimes \eta ,$

4. (4) $\Delta \mu =(\mu \otimes \mu )(\operatorname {id}_H\otimes P_{H,H}\otimes \operatorname {id}_H)(\Delta \otimes \Delta ),$

5. (5) $\mu (\operatorname {id}_H\otimes S)\Delta =\mu (S\otimes \operatorname {id}_H)\Delta =\eta \epsilon .$

A Hopf algebra H is said to be cocommutative if $P_{H,H}\Delta =\Delta .$

Define $\mu _n : H^{\otimes {n}} \otimes H^{\otimes {n}} \rightarrow H^{\otimes {n}}$ and $\Delta _m : H^{\otimes {m}} \rightarrow H^{\otimes {m}} \otimes H^{\otimes {m}}$ inductively by

$$ \begin{align*}\mu_0=\operatorname{id}_I, \quad \mu_{n+1}=(\mu_n \otimes \mu)(\operatorname{id}_{H^{\otimes{n}}} \otimes P_{H,H^{\otimes{n}}} \otimes \operatorname{id}_H)\end{align*} $$

for $n\geq 0$ and by

$$ \begin{align*}\Delta_0=\operatorname{id}_I,\quad \Delta_{m+1}=(\operatorname{id}_{H^{\otimes{m}}} \otimes P_{H^{\otimes{m}},H} \otimes \operatorname{id}_H)(\Delta_m \otimes \Delta) \end{align*} $$

for $m\geq 0$ .

For morphisms $f,\ f': H^{\otimes {m}} \rightarrow H^{\otimes {n}}$ , $m,n\geq 0$ , the convolution $f\ast f'$ of f and $f'$ is defined by

$$ \begin{align*}f\ast f':= \mu_n (f\otimes f')\Delta_m. \end{align*} $$

The category $\mathbf {A}$ has a cocommutative Hopf algebra with the object $1$ , where

2.5 Lie algebra in a linear symmetric strict monoidal category

We review the definition of a Lie algebra in a linear symmetric strict monoidal category. Let $\mathcal {C}=(\mathcal {C},\otimes ,I,P)$ be a linear symmetric strict monoidal category. A Lie algebra in $\mathcal {C}$ is an object L in $\mathcal {C}$ equipped with a morphism

$$ \begin{align*}[\cdot,\cdot]:L\otimes L\rightarrow L\end{align*} $$

satisfying

1. (1) $[\cdot ,\cdot ](\operatorname {id}_{L\otimes L}+P_{L,L})=0$ ,

2. (2) $[\cdot ,\cdot ](\operatorname {id}_L\otimes [\cdot ,\cdot ])(\operatorname {id}_{L^{\otimes 3}}+\sigma +\sigma ^2)=0,$ where $\sigma =(1,2,3):L^{\otimes 3}\rightarrow L^{\otimes 3}$ .

3.1 Andreadakis filtration $\mathcal {A}_{\ast }(n)$ of $\operatorname {Aut}(F_n)$

In what follows, we consider the left action of $\operatorname {Aut}(F_n)$ on $F_n$ . Let $\Gamma _r:=\Gamma _r(F_n)$ denote the r-th term of the lower central series of the free group $F_n$ of rank n. Let $\mathcal {L}_r(n) :=\Gamma _r/\Gamma _{r+1}$ for $r\geq 1$ . Note that $H=\mathcal {L}_1(n)$ and that $\mathcal {L}_r(n)$ is the degree r part of the free Lie algebra $\mathcal {L}_{\ast }(n)$ on H.

For $r\geq 0$ , the left action of $\operatorname {Aut}(F_n)$ on each nilpotent quotient $F_n/\Gamma _{r+1}$ induces a group homomorphism

$$ \begin{align*}\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(F_n/\Gamma_{r+1}).\end{align*} $$

Set

$$ \begin{align*}\mathcal{A}_{r}(n) := \ker(\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(F_n/\Gamma_{r+1})) \lhd \operatorname{Aut}(F_n).\end{align*} $$

Then we have a filtration, which is called the Andreadakis filtration of $\operatorname {Aut}(F_n)$ :

$$ \begin{align*}\operatorname{Aut}(F_n)=\mathcal{A}_0(n)\supset\mathcal{A}_1(n)=\operatorname{IA}(n)\supset\mathcal{A}_2(n)\supset\cdots.\end{align*} $$

For $r\geq 1$ , the Johnson homomorphism

$$ \begin{align*}\tau_r:\mathrm{gr}^r(\mathcal{A}_{\ast}(n))\hookrightarrow \operatorname{Hom}(H,\mathcal{L}_{r+1}(n))\end{align*} $$

is the injective homomorphism induced by the group homomorphism

$$ \begin{align*}\tau^{\prime}_{r}:\mathcal{A}_r(n)\rightarrow\operatorname{Hom}(H,\mathcal{L}_{r+1}(n))\end{align*} $$

defined by

$$ \begin{align*}\tau^{\prime}_{r}(f)(x\Gamma_2):=f(x)x^{-1}\Gamma_{r+2}\quad\text{for}\ f\in\mathcal{A}_r(n), x\in F_n.\end{align*} $$

3.2 The target group of the Johnson homomorphism

The target group $\operatorname {Hom}(H,\mathcal {L}_{r+1}(n))\cong H^{\ast }\otimes \mathcal {L}_{r+1}(n)$ of the Johnson homomorphism is identified with the degree r part $\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ of the free Lie algebra $\mathcal {L}_{\ast }(n)$ and with the tree module $T_r(n)$ via abelian group isomorphisms

(3.1) $$ \begin{align} H^{\ast}\otimes \mathcal{L}_{r+1}(n)\cong \operatorname{Der}_r(\mathcal{L}_{\ast}(n)) \cong T_r(n). \end{align} $$

Here, we briefly review the derivation Lie algebra and the tree module. (See [Reference Satoh22] for details.)

A derivation f of $\mathcal {L}_{\ast }(n)$ is a $\mathbb {Z}$ -linear map $f:\mathcal {L}_{\ast }(n)\rightarrow \mathcal {L}_{\ast }(n)$ such that $f([a,b])=[f(a),b]+[a,f(b)]$ for any $a,b\in \mathcal {L}_{\ast }(n)$ . The derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ of the Lie algebra $\mathcal {L}_{\ast }(n)$ is the set of all derivations of $\mathcal {L}_{\ast }(n)$ . The degree r part $\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ of the derivation Lie algebra is defined to be

$$ \begin{align*}\operatorname{Der}_r(\mathcal{L}_{\ast}(n))=\{f\in \operatorname{Der}(\mathcal{L}_{\ast}(n))\mid f(a)\in \mathcal{L}_{r+1}(n) \text{ for any }a\in H\}.\end{align*} $$

Then we have $\operatorname {Der}(\mathcal {L}_{\ast }(n))=\bigoplus _{r\geq 0}\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ and abelian group isomorphisms

$$ \begin{align*}\operatorname{Der}_r(\mathcal{L}_{\ast}(n))\cong \operatorname{Hom}(H,\mathcal{L}_{r+1}(n))\cong H^{\ast}\otimes \mathcal{L}_{r+1}(n).\end{align*} $$

We call a connected Jacobi diagram with no cycle a trivalent tree. For $r\geq 0$ , a trivalent tree is called a rooted trivalent tree of degree r if it has one univalent vertex (called the root) that is colored by an element of $H^{\ast }$ and $r+1$ univalent vertices (called leaves) that are colored by elements of H. Let $T_r(n)$ denote the $\mathbb {Z}$ -module spanned by rooted trivalent trees of degree r modulo the AS, IHX and multilinearity relations. We have an abelian group isomorphism

$$ \begin{align*}\Phi:H^{\ast}\otimes\mathcal{L}_{r+1}(n)\xrightarrow{\cong} T_{r}(n)\end{align*} $$

defined by

for $v_i\in H^{\ast }$ , $[\bar {x}_{i_1},\cdots ,[\bar {x}_{i_r},\bar {x}_{i_{r+1}}]\cdots ]\in \mathcal {L}_{r+1}(n)$ .

3.3 Andreadakis filtration $\mathcal {E}_{\ast }(n)$ of $\operatorname {End}(F_n)$

We extend the above construction to the endomorphism monoid $\operatorname {End}(F_n)$ of $F_n$ . For $r\geq 0$ , consider the canonical map

$$ \begin{align*}\rho_r:\operatorname{End}(F_n)\rightarrow \operatorname{End}(F_n/\Gamma_{r+1})\end{align*} $$

and set $\mathcal {E}_r(n):=\ker (\rho _r)$ . Then we have a filtration of monoids

$$ \begin{align*}\operatorname{End}(F_n)=\mathcal{E}_0(n)\supset\mathcal{E}_1(n)\supset\cdots,\end{align*} $$

and we call $\mathcal {E}_{\ast }(n)=(\mathcal {E}_{r}(n))_{r\geq 0}$ the Andreadakis filtration of $\operatorname {End}(F_n)$ .

For $f\in \operatorname {End}(F_n)$ and $x,y\in F_n$ , set

$$ \begin{align*}[f,x]:=f(x)x^{-1},\quad {}^y x=yxy^{-1},\end{align*} $$

and for a subset $T\subset F_n$ , set

$$ \begin{align*}[f,T]=\{[f,x]\in F_n \mid x\in T\}.\end{align*} $$

We can easily check the following lemma.

Lemma 3.1.

$$ \begin{align*}f\in \mathcal{E}_{r}(n)\quad \Leftrightarrow\quad [f, F_n]\subset \Gamma_{r+1}\quad \Leftrightarrow\quad [f,x_i]\in \Gamma_{r+1}\; (\text{for any }i\in [n]).\end{align*} $$

For subsets $S\subset \operatorname {End}(F_n)$ and $T\subset F_n$ , let $[S,T]$ denote the subgroup of $F_n$ generated by the elements $[f,x]$ for $f\in S, x\in T$ .

Lemma 3.2. We have

$$ \begin{align*}[\mathcal{E}_r(n),\Gamma_k]\subset \Gamma_{k+r} \end{align*} $$

for $r\geq 0, k\geq 1$ .

Proof. It is well known that $[\mathcal {A}_r(n),\Gamma _k]\subset \Gamma _{k+r}$ by Andreadakis [Reference Andreadakis1]. The same proof can be applied to $\mathcal {E}_r(n)$ . We use induction on k. When $k=1$ , we have $[\mathcal {E}_r(n),F_n]\subset \Gamma _{r+1}$ by the definition of $\mathcal {E}_r(n)$ . Suppose that $[\mathcal {E}_r(n),\Gamma _{k-1}]\subset \Gamma _{k-1+r}$ . We will show that $[\mathcal {E}_r(n),\Gamma _{k}]\subset \Gamma _{k+r}$ . Let $f\in \mathcal {E}_r(n)$ . Recall that $\Gamma _k$ is generated by the commutator $[x,y]$ with $x\in \Gamma _{k-1},y\in F_n$ . We can check that for $x\in \Gamma _{k-1},y\in F_n$ , we have

$$ \begin{align*}[f,[x,y]]={}^{[f,y]}([[f,y]^{-1},f(x)]\cdot [[f,x],{}^{x}y]\cdot[[x,y],[f,y]^{-1}])\in \Gamma_{k+r}. \end{align*} $$

For $z,w\in \Gamma _k$ , we have

$$ \begin{gather*} [f,zw]=[f,z]\cdot{}^{z}[f,w]\equiv [f,z][f,w] \quad(\bmod\:\Gamma_{k+r+1}), \end{gather*} $$

and by letting $w=z^{-1}$ , we have

$$ \begin{gather*} [f,z^{-1}]\equiv [f,z]^{-1}\quad(\bmod\:\Gamma_{k+r+1}). \end{gather*} $$

Therefore, we have $[f,z]\in \Gamma _{k+r}$ for any $z\in \Gamma _k$ .

Define a map

$$ \begin{align*}\sigma:\operatorname{End}(F_n)\rightarrow \operatorname{End}(F_n)\end{align*} $$

by $\sigma (f)=\tilde {f}$ for $f\in \operatorname {End}(F_n)$ , where

$$ \begin{align*}\tilde{f}(x_i)=[f,x_i]^{-1}x_i=x_i f(x_i)^{-1} x_i\end{align*} $$

for $i\in [n]$ .

Lemma 3.3. We have

(3.2) $$ \begin{align} \sigma^2=\operatorname{id}_{\operatorname{End}(F_n)} \end{align} $$

(3.3) $$ \begin{align} f\in \mathcal{E}_r(n)\quad\Rightarrow\quad \sigma(f)\in \mathcal{E}_r(n) \end{align} $$

(3.4) $$ \begin{align} f\in \mathcal{E}_r(n)\quad\Rightarrow\quad f\sigma(f),\: \sigma(f) f\in \mathcal{E}_{2r}(n). \end{align} $$

Proof. We have equation (3.2) since for any $f\in \operatorname {End}(F_n)$ and $i\in [n]$ , we have

$$ \begin{align*}\sigma^2(f)(x_i)=x_i\tilde{f}(x_i)^{-1}x_i=x_ix_i^{-1}f(x_i)x_i^{-1}x_i=f(x_i).\end{align*} $$

We have equation (3.3) since, for any $f\in \mathcal {E}_r(n)$ and $i\in [n]$ , we have

$$ \begin{align*}[\tilde{f}, x_i]=[f,x_i]^{-1}\in \Gamma_{r+1}. \end{align*} $$

We prove equation (3.4). Let $f\in \mathcal {E}_r(n)$ . We have

$$ \begin{align*}[f\tilde{f},x_i]=f([\tilde{f},x_i])[f,x_i]=f([f,x_i]^{-1})[f,x_i]=[f,[f,x_i]^{-1}]\in \Gamma_{2r+1}\end{align*} $$

for any $i\in [n]$ . Thus, we have

(3.5) $$ \begin{align} f\tilde{f}\in\mathcal{E}_{2r}(n). \end{align} $$

By equation (3.3), we have $\tilde {f}\in \mathcal {E}_r(n)$ , and by equations (3.2) and (3.5),

$$ \begin{align*}\tilde{f}f=\tilde{f}\tilde{\tilde{f}}\in\mathcal{E}_{2r}(n).\end{align*} $$

For $N\geq r\geq 0$ , we define an equivalence relation $\sim _{N}$ on the monoid $\mathcal {E}_r(n)$ by

$$ \begin{align*}f\sim_{N}g \quad \overset{\text{def}}\Leftrightarrow \quad [f,x]\equiv[g,x]\; (\bmod\: \Gamma_{N+1}) \quad \text{ for any }x\in F_n \end{align*} $$

for $f,g\in \mathcal {E}_r(n)$ . Thus, we have

$$ \begin{align*}f\sim_{N} \operatorname{id}_{F_n} \quad \Leftrightarrow \quad [f,x]\in \Gamma_{N+1} \quad \text{ for any }x\in F_n \quad \Leftrightarrow \quad f\in \mathcal{E}_N(n).\end{align*} $$

Lemma 3.4. Let $r\geq 1$ . For $f\in \mathcal {E}_r(n)$ , define $f^{R}_{N}$ and $f^{L}_{N}$ for $N\geq r+1$ inductively by

$$ \begin{align*}f^{R}_{N}= \begin{cases} \tilde{f} & (N=r+1)\\ f^{R}_{N-1}\widetilde{f f^{R}_{N-1}} & (N\geq r+2), \end{cases} \end{align*} $$

$$ \begin{align*}f^{L}_{N}= \begin{cases} \tilde{f} & (N=r+1)\\ \widetilde{f^{L}_{N-1} f}f^{L}_{N-1} & (N\geq r+2). \end{cases} \end{align*} $$

Then we have

$$ \begin{gather*} f^{R}_{N}\in \mathcal{E}_r(n),\quad f f^{R}_{N}\in \mathcal{E}_N(n),\quad f^{R}_{N}\sim_{N-1} f^{R}_{N-1},\\ f^{L}_{N}\in \mathcal{E}_r(n),\quad f^{L}_{N} f\in \mathcal{E}_N(n),\quad f^{L}_{N}\sim_{N-1} f^{L}_{N-1}. \end{gather*} $$

Proposition 3.5. For $N\geq 1$ , we have a filtration of groups

$$ \begin{align*}\mathcal{E}_1(n)/\sim_{N} \;\supset \mathcal{E}_2(n)/\sim_{N} \;\supset \cdots \supset \mathcal{E}_{N-1}(n)/\sim_{N} \;\supset \mathcal{E}_N(n)/\sim_{N}\;=1.\end{align*} $$

Moreover, this is an N-series.

Proof. Firstly, we show that $\mathcal {E}_r(n)/\sim _{N}$ is a group for each $r\geq 1$ . For $f,f',g\in \mathcal {E}_r(n)$ such that $f\sim _N f'$ , we can easily check that $fg\sim _N f'g$ and $gf\sim _N gf'$ . Thus, the composition makes the set $\mathcal {E}_r(n)/\sim _{N}$ a monoid. For $[f]\in \mathcal {E}_r(n)/\sim _{N}$ , by Lemma 3.4, it follows that $[f][f^{R}_N]=[f^{L}_N][f]=1 \in \mathcal {E}_r(n)/\sim _{N}$ . Since $\mathcal {E}_r(n)/\sim _{N}$ is a monoid, we have $[f^{R}_N]=[f^{L}_N]$ , and this is the inverse of $[f]$ . Therefore, $\mathcal {E}_r(n)/\sim _{N}$ is a group for each $r\geq 1$ .

Since $\mathcal {E}_r(n)\supset \mathcal {E}_{r+1}(n)$ , we have $\mathcal {E}_r(n)/\sim _{N} \; \supset \mathcal {E}_{r+1}(n)/\sim _{N}$ . Secondly, we show that the descending series is an N-series. It suffices to show that, for $f\in \mathcal {E}_r(n), g\in \mathcal {E}_s(n)$ , we have

$$ \begin{align*}[[f],[g]]=[f][g][f]^{-1}[g]^{-1}=[fgf^{R}_{N}g^{R}_{N}]\in\mathcal{E}_{r+s}(n)/\sim_{N}. \end{align*} $$

Note that, by Lemma 3.4, we can take $f^{R}_{N}, g^{R}_{N}\in \mathcal {E}_r(n)$ such that $ff^{R}_{N}, gg^{R}_{N}\in \mathcal {E}_{N}(n)\cap \mathcal {E}_{r+s}(n)$ . By commutator calculus, for $x\in F_n$ , we have

$$ \begin{align*}[fg,x]=[f,[g,x]]\;[g,x]\;[f,x]\equiv [g,x]\;[f,x] \quad (\bmod\: \Gamma_{r+s+1}), \end{align*} $$

$$ \begin{align*}[g,[g^{R}_{N},x]\;[f^{R}_{N},x]]=[g,[g^{R}_{N},x]]\; {}^{[g^{R}_{N},x]}[g,[f^{R}_{N},x]]\equiv [g,[g^{R}_{N},x]] \quad (\bmod\: \Gamma_{r+s+1}). \end{align*} $$

Similarly, we have

$$ \begin{align*}[f^{R}_{N} g^{R}_{N},x]\equiv [g^{R}_{N},x]\;[f^{R}_{N},x]\quad (\bmod\: \Gamma_{r+s+1}),\end{align*} $$

$$ \begin{align*}[f,[g^{R}_{N},x]\;[f^{R}_{N},x]]\equiv [f,[f^{R}_{N},x]]\quad (\bmod\: \Gamma_{r+s+1}). \end{align*} $$

Thus, we have

$$ \begin{gather*} \begin{aligned} [fg,[f^{R}_{N} g^{R}_{N},x]]&\equiv [g,[f^{R}_{N} g^{R}_{N},x]]\;[f,[f^{R}_{N} g^{R}_{N},x]]\\ &\equiv [g,[g^{R}_{N},x]\;[f^{R}_{N},x]]\;[f,[g^{R}_{N},x]\;[f^{R}_{N},x]]\\ &\equiv [g,[g^{R}_{N},x]]\;[f,[f^{R}_{N},x]] \quad (\bmod\: \Gamma_{r+s+1}). \end{aligned} \end{gather*} $$

Therefore, we have

$$ \begin{gather*} \begin{aligned} [f g f^{R}_{N} g^{R}_{N},x]&= [fg,[f^{R}_{N} g^{R}_{N},x]]\;[f^{R}_{N} g^{R}_{N},x]\;[fg,x]\\ &\equiv [g,[g^{R}_{N},x]]\;[f,[f^{R}_{N},x]]\;[g^{R}_{N},x]\;[f^{R}_{N},x]\;[g,x]\;[f,x]\\ &\equiv [g,[g^{R}_{N},x]]\;[g^{R}_{N},x]\;[g,x]\;[f,[f^{R}_{N},x]]\;[f^{R}_{N},x]\;[f,x]\\ &=[gg^{R}_{N},x]\;[ff^{R}_{N},x]\\ &\equiv 1 \quad (\bmod\: \Gamma_{r+s+1}), \end{aligned} \end{gather*} $$

and the proof is complete.

For $N\geq r\geq 1$ , we have a canonical projection

$$ \begin{align*}p_{N+1}:\mathcal{E}_r(n)/\sim_{N+1}\twoheadrightarrow\mathcal{E}_r(n)/\sim_{N}.\end{align*} $$

Let $\hat {\mathcal {E}}_r(n)$ denote the projective limit $\underset {N}{\varprojlim } (\mathcal {E}_r(n)/\sim _N)$ and

$$ \begin{align*}\pi_{N}:\hat{\mathcal{E}}_r(n)\twoheadrightarrow\mathcal{E}_r(n)/\sim_N \end{align*} $$

denote the projection. By Proposition 3.5, we have a descending series of groups

$$ \begin{align*}\hat{\mathcal{E}}_1(n)\supset \hat{\mathcal{E}}_2(n)\supset \cdots\end{align*} $$

satisfying

$$ \begin{align*}\bigcap_{r\geq 1} \hat{\mathcal{E}}_r(n)=\{\operatorname{id}\}.\end{align*} $$

We have a graded Lie algebra $\mathrm {gr}(\hat {\mathcal {E}}_{\ast }(n))$ associated to the N-series $\hat {\mathcal {E}}_{\ast }(n)$ . Let $\operatorname {gr}^r(\mathcal {E}_{\ast }(n)):=\mathcal {E}_r(n)/\sim _{r+1}$ for $r\geq 1$ and $\mathrm {gr}(\mathcal {E}_{\ast }(n)):=\bigoplus _{r\geq 1}\operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ .

Proposition 3.7. We have a group isomorphism

$$ \begin{align*}\bar{\pi}_{r+1}:\operatorname{gr}^r(\hat{\mathcal{E}}_{\ast}(n))\xrightarrow{\cong}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\end{align*} $$

induced by the projection $\pi _{r+1}:\hat {\mathcal {E}}_{r}(n)\rightarrow \operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ . Therefore, $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ is a graded Lie algebra.

3.4 Johnson homomorphism of $\operatorname {End}(F_n)$

For $r\geq 1$ , by using Lemma 3.2, we can define a monoid homomorphism

$$ \begin{align*}\tilde{\tau}^{\prime}_{r}:\mathcal{E}_r(n)\rightarrow\operatorname{Hom}(H, \mathcal{L}_{r+1}(n))\end{align*} $$

by $\tilde {\tau }^{\prime }_{r}(f)(x\Gamma _2):=[f,x]\Gamma _{r+2}$ for $f\in \mathcal {E}_r(n), x\in F_n$ . It is easily checked that the monoid homomorphism $\tilde {\tau }^{\prime }_{r}$ induces an injective group homomorphism

$$ \begin{align*}\tilde{\tau}_r:\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\hookrightarrow\operatorname{Hom}(H,\mathcal{L}_{r+1}(n)). \end{align*} $$

We call it the r-th Johnson homomorphism of $\operatorname {End}(F_n)$ .

Proof. It suffices to show that $\tilde {\tau }_r$ is surjective. For any $\varphi \in \operatorname {Hom}(H,\mathcal {L}_{r+1}(n))$ , we fix a representative of $\varphi (x_i\Gamma _2)\in \mathcal {L}_{r+1}(n)$ and write it $\varphi (x_i)\in \Gamma _{r+1},$ for $i\in [n]$ . Define $\psi \in \operatorname {End}(F_n)$ by

$$ \begin{align*}\psi(x_i)= \varphi(x_i)x_i \text{ for }i\in[n].\end{align*} $$

It turns out that $[\psi ,x]\Gamma _{r+2}=\varphi (x\Gamma _{2})\in \mathcal {L}_{r+1}(n)$ for any $x\in F_n$ by induction on the word length of $x\in F_n$ . Therefore, we have $\tilde {\tau }_r(\psi )=\varphi $ , and thus the map $\tilde {\tau }_r$ is surjective.

Then we obtain the following commutative diagram:

Remark 3.9. It is well known that the Andreadakis filtration $\mathcal {A}_{\ast }(n)$ of $\operatorname {Aut}(F_n)$ includes the lower central series of $\operatorname {IA}(n)$ :

$$ \begin{align*}\Gamma_r(\operatorname{IA}(n))\subset\mathcal{A}_{r}(n). \end{align*} $$

We have $\mathcal {A}_1(n)=\operatorname {IA}(n)$ by definition. Andreadakis [Reference Andreadakis1] conjectured that

(3.6) $$ \begin{align} \mathcal{A}_r(n)=\Gamma_r(\operatorname{IA}(n)) \end{align} $$

for all $r\geq 2,n\geq 2$ . Andreadakis [Reference Andreadakis1] ( $n=3$ ) and Kawazumi [Reference Kawazumi17] (for any n) showed that equation (3.6) holds for $r=2$ . Moreover, Andreadakis [Reference Andreadakis1] showed that the first Johnson homomorphism $\tau _1$ of $\operatorname {Aut}(F_n)$ is an isomorphism. Therefore, we have abelian group isomorphisms

(3.7) $$ \begin{align} \operatorname{gr}^1(\operatorname{IA}(n))\cong \operatorname{Hom}(H,\mathcal{L}_2(n))\cong \operatorname{gr}^1(\mathcal{E}_{\ast}(n)). \end{align} $$

Recently, Satoh [Reference Satoh23] showed that equation (3.6) holds for $r=3$ . On the other hand, Bartholdi [Reference Bartholdi5] showed that

$$ \begin{align*}(\mathcal{A}_{5}(3)/ \Gamma_5(\operatorname{IA}(3)))\otimes \mathbb{Q}\cong \mathbb{Q}^{\oplus 3},\end{align*} $$

which is a counterexample of the Andreadakis conjecture. Now, the Andreadakis conjecture remains open for $n\gg r$ .

3.5 The derivation Lie algebra

By equation (3.1) and Proposition 3.8, we have abelian group isomorphisms

$$ \begin{align*}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\cong H^{\ast}\otimes \mathcal{L}_{r+1}(n)\cong \operatorname{Der}_r(\mathcal{L}_{\ast}(n)) \cong T_r(n).\end{align*} $$

We write $\tilde {\tau }_r:\operatorname {gr}^r(\mathcal {E}_{\ast }(n))\xrightarrow {\cong } \operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ as well.

Proposition 3.10. The abelian group isomorphism

$$ \begin{align*}\tilde{\tau}=\bigoplus_{r\geq 1} \tilde{\tau}_r:\mathrm{gr}(\mathcal{E}_{\ast}(n))\xrightarrow{\cong} \operatorname{Der}(\mathcal{L}_{\ast}(n))\end{align*} $$

is an isomorphism of graded Lie algebras.

Proof. We only need to check that the Lie bracket of $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ is sent to the Lie bracket of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ . For $f\in \hat {\mathcal {E}}_r(n),g\in \hat {\mathcal {E}}_s(n)$ and $x\in F_n$ , we have

$$ \begin{gather*} \begin{aligned} [\tilde{\tau}_r([f]),\tilde{\tau}_s([g])](x\Gamma_2)&=\tilde{\tau}_r([f])\tilde{\tau}_s([g])(x\Gamma_2)-\tilde{\tau}_s([g])\tilde{\tau}_r([f])(x\Gamma_2)\\ &=[f,[g,x]]\;[g,[f,x]]^{-1}=[[f,g],x]\in \mathcal{L}_{r+s+1}(n). \end{aligned} \end{gather*} $$

On the other hand, we have

$$ \begin{align*}\tilde{\tau}_{r+s}([[f,g]])(x\Gamma_2)=[[f,g],x]\in \mathcal{L}_{r+s+1}(n).\end{align*} $$

Therefore, $\tilde {\tau }$ is an isomorphism of graded Lie algebras.

Remark 3.11. The tree module $\bigoplus _{r\geq 1}T_r(n)$ also has a graded Lie algebra structure which is induced by the Lie algebra structure of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ . The Lie bracket

$$ \begin{align*}[\cdot,\cdot]:T_r(n)\times T_s(n)\rightarrow T_{r+s}(n)\end{align*} $$

is defined by the difference between two linear sums obtained by contracting the root of one of the trees and the leaves of the other tree

4.1 Bracket map $[\cdot ,\cdot ]: B_{d,k}(n)\otimes \operatorname {gr}^r(\mathcal {E}_{\ast }(n))\rightarrow B_{d,k+r}(n)$

We have a right $\operatorname {End}(F_n)$ -action on $A_d(n)$ by letting

$$ \begin{align*}u\cdot g:=A_d(g)(u)\end{align*} $$

for $u\in A_d(n), g\in \operatorname {End}(F_n)$ . We define

(4.1) $$ \begin{align} [\cdot,\cdot]:A_d(n)\times\operatorname{End}(F_n)\rightarrow A_d(n) \end{align} $$

by $[u,g]:=u\cdot g -u$ for $u\in A_d(n),g\in \operatorname {End}(F_n)$ , which we call the bracket map.

Theorem 4.1. The N-series $\hat {\mathcal {E}}_{\ast }(n)$ acts on the right on the filtered vector space $A_d(n)$ . That is, we have

$$ \begin{align*}[A_{d,k}(n),\mathcal{E}_r(n)]\subset A_{d,k+r}(n)\end{align*} $$

for any $r\geq 1$ .

Note that we have $[A_{d,k}(n),\Gamma _{r}(\operatorname {IA}(n))]\subset A_{d,k+r}(n)$ (see Lemma 5.7 in [Reference Katada16]). We will prove Theorem 4.1 in Section 4.4.

By using Theorem 4.1, we can extend the bracket map

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n)\otimes\operatorname{gr}^r(\operatorname{IA}(n))\rightarrow B_{d,k+r}(n)\end{align*} $$

to $\operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ .

Corollary 4.2. Let $r\geq 1$ . The bracket map (4.1) induces a $\Bbbk $ -linear map

$$ \begin{align*} [\cdot,\cdot]:B_{d,k}(n)\otimes\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow B_{d,k+r}(n). \end{align*} $$

We can also extend the $\operatorname {GL}(n;\mathbb {Z})$ -module map

$$ \begin{align*}\beta_{d,k}^r:\operatorname{gr}^r(\operatorname{IA}(n))\rightarrow\operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n))\end{align*} $$

defined by $\beta _{d,k}^r(g)(u)=[u,g]$ for $g\in \operatorname {gr}^r(\operatorname {IA}(n)),u\in B_{d,k}(n)$ to a group homomorphism

$$ \begin{align*}\tilde{\beta}_{d,k}^r:\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow\operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n)), \end{align*} $$

which $\beta _{d,k}^r$ factors through. That is, we have $\beta _{d,k}^r=\tilde {\beta }_{d,k}^r i$ , where the map $i:\operatorname {gr}^r(\operatorname {IA}(n))\rightarrow \operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ is induced by the inclusion map $\Gamma _{r}(\operatorname {IA}(n))\hookrightarrow \mathcal {E}_{r}(n)$ .

Remark 4.3. The right action of the N-series $\hat {\mathcal {E}}_{\ast }(n)$ on $A_d(n)$ induces an action of the graded Lie algebra $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ on the graded vector space $B_{d}(n)$ :

$$ \begin{align*}\mathrm{gr}(\mathcal{E}_{\ast}(n))\xrightarrow{\cong}\mathrm{gr}(\hat{\mathcal{E}}_{\ast}(n))\rightarrow \bigoplus_{r\geq 1}\operatorname{End}_r(B_{d}(n)),\end{align*} $$

which is given by the group homomorphisms $\tilde {\beta }_{d,k}^r$ . This induced action can be regarded as an action of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on the graded vector space $B_d(n)$ by the identification in Section 3.5.

4.2 The category $\mathbf {A}^{L}$ of extended Jacobi diagrams in handlebodies

The category $\mathbf {A}$ has a cocommutative Hopf algebra with the underlying object $1$ , which we recalled in Section 2.4. Moreover, the morphisms of the category $\mathbf {A}$ have Jacobi diagrams, and the STU relations correspond to relations of Lie algebras. In a proof of Theorem 4.1, we use graphical computations which deal with the Hopf algebra structure and the Lie algebra structure. For this purpose, we extend the category $\mathbf {A}$ to another category $\mathbf {A}^{L}$ which includes the Hopf algebra structure and the Lie algebra structure. In Appendix A, we give an expected presentation of the category $\mathbf {A}^{L}$ .

Construct the category $\mathbf {A}^{L}$ as follows. The set of objects of $\mathbf {A}^{L}$ is the free monoid generated by two objects H and L, where multiplication is denoted by $\otimes $ . The category $\mathbf {A}^{L}$ includes the category $\mathbf {A}$ as a full subcategory with the free monoid generated by H as the set of objects. (On the other hand, the full subcategory with the free monoid generated by L is isomorphic to a category in [Reference Hinich and Vaintrob13]. See Remark A.4.) In the category $\mathbf {A}^{L}$ , we consider diagrams that are obtained from Jacobi diagrams in handlebodies by attaching univalent vertices of the Jacobi diagrams to the bottom line l and the upper line $l'$ .

Example 4.4. Here is a morphism in $\mathbf {A}^{L}(H\otimes L\otimes H\otimes L\otimes H,H\otimes L^{\otimes 2}\otimes H)$ :

As depicted in Figure 2, the objects H and L in the source of a morphism of $\mathbf {A}^{L}$ correspond to a handle of the handlebody and a univalent vertex attached to the upper line $l'$ , respectively.

Figure 2

Source of a morphism

As depicted in Figure 3, the objects H and L in the target of a morphism of $\mathbf {A}^{L}$ correspond to an arc component mapped into the handlebody and a univalent vertex attached to the bottom line l, respectively.

Figure 3

Target of a morphism

In the category $\mathbf {A}^{L}$ , the object H is considered as a Hopf algebra and L is considered as a Lie algebra. See Section 4.3 and Appendix A.

To define morphisms of the category $\mathbf {A}^{L}$ precisely, we give the following definition.

Note that an $(X_m,\emptyset )$ -diagram is a Jacobi diagram on $X_m$ .

For an object $w=H^{\otimes m_1}\otimes L^{\otimes n_1}\otimes \cdots \otimes H^{\otimes m_r}\otimes L^{\otimes n_r}\in \mathbf {A}^{L}$ , let $m:=\sum _{i=1}^r m_i$ and $n:=\sum _{i=1}^r n_i$ . For $p\geq 0$ , let $[p]^{+}:=\{1^{+},\cdots ,p^{+}\}$ and $[p]^{-}:=\{1^{-},\cdots ,p^{-}\}$ be two copies of $[p]$ .

We identify two $(w,w')$ -diagrams if they are homotopic in $U_m$ relative to the endpoints of $X_m'\cup D$ . In what follows, we simply write D for a $(w,w')$ -diagram. For objects w and $w'$ , the hom-set $\mathbf {A}^{L}(w,w')$ is the $\Bbbk $ -vector space spanned by $(w,w')$ -diagrams modulo the STU, AS and IHX relations.

The composition of $\mathbf {A}^{L}$ is defined in a similar way to that of the category $\mathbf {A}$ . We can define a square diagram for an $(w,w')$ -diagram similarly. Let D be a diagram in $\mathbf {A}^{L}(w, w')$ and $D'$ a diagram in $\mathbf {A}^{L}(w', w'')$ . Deform $D'$ to have only the parallel copies of the handle cores in each handle. Then the composition $D'\circ D$ is a diagram obtained by stacking the cabling of D on top of the square presentation of $D'$ .

The identity morphism $\operatorname {id}_{H^{\otimes m_1}\otimes L^{\otimes n_1}\otimes \cdots \otimes H^{\otimes m_r}\otimes L^{\otimes n_r}}$ is the following diagram:

We can naturally extend the linear symmetric strict monoidal structure of $\mathbf {A}$ to the category $\mathbf {A}^{L}$ , where the tensor product is defined to be the juxtaposition of the handlebodies.

Note that the symmetries in $\mathbf {A}^{L}$ are determined by

The degree of a $(w,w')$ -diagram is defined by

$$ \begin{align*}\frac{1}{2}\#\{\text{vertices}\}-\#\{\text{univalent vertices attached to the upper line }l'\}.\end{align*} $$

Let $\mathbf {A}^{L}_{d}(w,w')\subset \mathbf {A}^{L}(w,w')$ be the subspace spanned by $(w,w')$ -diagrams of degree d. We have $\mathbf {A}^{L}(w,w')=\bigoplus _{d\geq 0}\mathbf {A}^{L}_{d}(w,w')$ . Since we have

$$ \begin{align*}\mathbf{A}^{L}_{d'}(w',w'')\circ \mathbf{A}^{L}_{d}(w,w')\subset \mathbf{A}^{L}_{d+d'}(w,w'')\end{align*} $$

and

$$ \begin{align*}\mathbf{A}^{L}_{d'}(w,w')\otimes \mathbf{A}^{L}_{d}(z,z')\subset \mathbf{A}^{L}_{d+d'}(w\otimes z,w'\otimes z')\end{align*} $$

for any $w,w',w'',z,z'\in \mathbf {A}^{L}$ , this grading is an $\mathbb {N}$ -grading on $\mathbf {A}^{L}$ . Note that we have $\mathbf {A}_d(m,n)= \mathbf {A}^{L}_d(H^{\otimes m},H^{\otimes n})$ for $m,n\geq 0$ .

4.3 Relations for morphisms in $\mathbf {A}^{L}$

Here, we observe some relations for morphisms of $\mathbf {A}^{L}$ , which we use in the proof of Theorem 4.1.

The cocommutative Hopf algebra $(H,\mu ,\eta ,\Delta ,\epsilon ,S)$ in $\mathbf {A}$ naturally induces a cocommutative Hopf algebra in $\mathbf {A}^{L}$ such that

Additionally, the triple $(L,[\cdot ,\cdot ],c_{L})$ is a Lie algebra with a symmetric invariant $2$ -tensor in $\mathbf {A}^{L}$ (see Appendix A.2), where

Moreover, $\mathbf {A}^{L}$ has two morphisms

The degree of the morphism $c_{L}$ is $1$ and that of the others of the above morphisms is $0$ .

The iterated multiplications

and the iterated comultiplications

for $q\geq 0$ are inductively defined by

$$ \begin{align*}\mu^{[0]}=\eta,\quad\mu^{[1]}=\operatorname{id}_H, \quad \mu^{[q+1]}=\mu(\mu^{[q]}\otimes\operatorname{id}_H) \quad(q\geq 1),\end{align*} $$

$$ \begin{align*}\Delta^{[0]}=\epsilon,\quad\Delta^{[1]}=\operatorname{id}_H, \quad \Delta^{[q+1]}=(\Delta^{[q]}\otimes\operatorname{id}_H)\Delta \quad(q\geq 1).\end{align*} $$

Let

which denotes the adjoint action, and

(4.2)

which denotes the commutator.

Let $\mathfrak {g}$ be a Lie algebra and $U=U(\mathfrak {g})$ be the universal enveloping algebra. We have a filtration $F_{\ast }(U)$ of U induced by the usual filtration of the tensor algebra $T(\mathfrak {g})$ of $\mathfrak {g}$ . Since U has a cocommutative Hopf algebra structure, we can define the commutator operator

$$ \begin{align*}comm:U^{\otimes 2}\rightarrow U \end{align*} $$

in a similar way as equation (4.2). For $x_1,\cdots ,x_m,y_1,\cdots ,y_n\in \mathfrak {g}$ , we have

$$ \begin{align*}comm(x_1\cdots x_m,y_1\cdots y_n)\in F_{\min(m,n)}(U).\end{align*} $$

The following lemma is a diagrammatic version of this fact.

For example, we have

Proof of Lemma 4.9

By using Lemma 4.8 (2) and

, we have

By Lemma 4.8 (1), it suffices to consider

. By Lemma 4.8 (3), we have

Thus, when $p=0$ , we have $D=0$ . When $p\geq 1$ , by Lemma 4.8 (4), we have

Note that the last term is a $\mathbb {Z}$ -linear sum of unions of tree diagrams with m trivalent vertices. Therefore, the first equality of (1) follows. If $m=n=1$ , then the equality follows from the case where $m=p=1, q=0$ . The second equality of (1) follows similarly.

The first equality of (2) follows from The second equality follows similarly.

We have (3) because

4.4 Proof of Theorem 4.1

In this subsection, we prove Theorem 4.1.

For any $y_1,\cdots ,y_r\in F_n$ , we call $[y_1,\cdots ,[y_{r-1},y_r]]\in \Gamma _r$ an r-fold commutator.

For $i\in [n]$ , define $d_i\in \operatorname {End}(F_n)=\mathbf {F}^{\mathrm {op}}(n,n)$ by

$$ \begin{align*}d_i(x_i)=[y_1,\cdots,[y_{r},y_{r+1}]]^{\epsilon},\quad d_i(x_j)=1 \quad (j\neq i)\end{align*} $$

for $y_1,\cdots ,y_{r+1}\in F_n, \epsilon \in \{\pm 1\}$ , which we call an $(r+1)$ -fold commutator at i. Via the isomorphism $\Bbbk \mathbf {F}^{\mathrm {op}}(n,n)\cong \mathbf {A}_0(n,n)$ , we identify $d_i\in \mathbf {F}^{\mathrm {op}}(n,n)$ with a morphism of the following form

which we also call an $(r+1)$ -fold commutator at i, where each

depicts S or $\operatorname {id}_H$ , and $q_k,p_l\geq 0$ satisfy $\sum _{k=1}^{n}q_k=\sum _{l=1}^{r+1}p_l$ .

Claim 1. An element $g\in \mathcal {E}_r(n)$ can be written as a convolution product

$$ \begin{align*}g= d_{1,1}\ast\cdots\ast d_{1,l_1}\ast\cdots\ast d_{n,1}\ast\cdots\ast d_{n,l_n}\ast\operatorname{id}_{H^{\otimes n}},\end{align*} $$

where $d_{i,j}$ is an $(r+1)$ -fold commutator at i for $i\in [n]$ ( $l_i\geq 0, 1\leq j\leq l_i$ ).

Proof of Theorem 4.1

We show that $[A_{d,k}(n),\mathcal {E}_{r}(n)]\subset A_{d,k+r}(n)$ . We can write an element of $A_{d,k}(n)\subset \mathbf {A}^{L}(I,H^{\otimes n})$ as a linear sum of the following diagrams:

where D is a Jacobi diagram with at least k trivalent vertices. Let $g\in \mathcal {E}_{r}(n)$ . By Claim 1, we can write g as a convolution product

$$ \begin{align*}g=d_{1,1}\ast\cdots\ast d_{1,l_1}\ast\cdots\ast d_{n,1}\ast\cdots\ast d_{n,l_n}\ast \operatorname{id}_{H^{\otimes n}},\end{align*} $$

where $d_{i,j}\in \mathbf {A}_0(n,n)$ is an $(r+1)$ -fold commutator at i. Let $l=1+\sum _{i=1}^{n}{l_i}$ .

By using

we have

Here, each

is once connected to all of the diagrams $d_{1,1}, \cdots , d_{n,l_n}$ and $\operatorname {id}_{H^{\otimes n}}$ . Since we have

by Lemma 4.8 (2), the element $u\cdot g$ is a linear sum of diagrams of shape

, where

denotes

or

. If all

that are connected to $\operatorname {id}_{H^{\otimes n}}$ are

, then it is easily checked that the corresponding summand is just u by using Lemma 4.9 (3). Otherwise, at least one of

that are connected to diagrams $d_{1,1},\cdots ,d_{n,l_n}$ are

. By using Lemma 4.9, it follows that each summand is a linear sum of diagrams with at least $k+r$ trivalent vertices. Therefore, we have $[u,g]=u\cdot g-u \in A_{d,k+r}(n)$ .

5.1 Preliminaries to computation

Let $N\geq 1$ . We briefly review the construction of the irreducible representations of the symmetric group $\mathfrak {S}_N$ . See Fulton–Harris [Reference Fulton6] and Sagan [Reference Sagan21] for basic facts of representation theory of $\mathfrak {S}_N$ . Let $\lambda =(\lambda _1,\cdots ,\lambda _l)$ be a partition of N, and write $\lambda \vdash N$ . A Young diagram of $\lambda $ consists of $\lambda _i$ boxes in the i-th row for $i\in [l]$ such that the rows of boxes are lined up on the left. A $\lambda $ -tableau is a numbering of the boxes by the integers in $[N]$ . We call a $\lambda $ -tableau standard if the numbering increases in each row and in each column. The canonical $\lambda $ -tableau is a standard tableau whose numbering starts from the first row from left to right and then the second row from left to right and so on.

Let $t_0$ be the canonical $\lambda $ -tableau. Define $R_{t_0}$ (resp. $C_{t_0}$ ) to be the subgroup of $\mathfrak {S}_N$ that preserves each row (resp. column) of $t_0$ . We define

$$ \begin{align*}a_{\lambda}:=\sum_{\sigma\in R_{t_0}}\sigma,\quad b_{\lambda}:=\sum_{\sigma\in C_{t_0}}\mathrm{sgn}(\sigma)\sigma\in\Bbbk\mathfrak{S}_N.\end{align*} $$

For each $\lambda \vdash N$ , the Young symmetrizer $c_{\lambda }$ is defined by

(5.1) $$ \begin{align} c_{\lambda}=b_{\lambda}a_{\lambda}\in\Bbbk\mathfrak{S}_{N}. \end{align} $$

The Specht module $S^{\lambda }$ , which is an irreducible representation of $\mathfrak {S}_N$ corresponding to $\lambda $ , can be constructed as

$$ \begin{align*}S^{\lambda}=\Bbbk\mathfrak{S}_N\cdot c_{\lambda}. \end{align*} $$

Lemma 5.1. We have the following decomposition of $\Bbbk \mathfrak {S}_N$ -bimodules

$$ \begin{gather*} \begin{aligned} \Bbbk\mathfrak{S}_{N}=\bigoplus_{\lambda\vdash N}\Bbbk\mathfrak{S}_{N}\cdot c_{\lambda}\cdot\Bbbk\mathfrak{S}_{N}. \end{aligned} \end{gather*} $$

For $N',N''\geq 0$ , let $N=N'+N''$ . For $\mu \vdash N',\nu \vdash N''$ , let $S^{\mu }\diamond S^{\nu }$ denote the representation of $\mathfrak {S}_N$ induced from the tensor product representation $S^{\mu }\boxtimes S^{\nu }$ of $\mathfrak {S}_{N'}\times \mathfrak {S}_{N''}$ by the inclusion of $\mathfrak {S}_{N'}\times \mathfrak {S}_{N''}$ in $\mathfrak {S}_N$ . By the Littlewood–Richardson rule, we have

$$ \begin{align*}S^{\mu}\diamond S^{\nu}=\bigoplus_{\lambda\vdash N} (S^{\lambda})^{LR_{\mu,\nu}^{\lambda}}, \end{align*} $$

where $LR_{\mu ,\nu }^{\lambda }$ denotes the Littlewood–Richardson coefficient. We have the following lemma by using basic facts of representation theory of $\mathfrak {S}_N$ .

Lemma 5.2. Let $N=N'+N''$ for $N',N''\geq 0$ . Let $\lambda \vdash N,\mu \vdash N',\nu \vdash N''$ , respectively. We have

$$ \begin{align*}\dim_{\Bbbk}((c_{\mu}\diamond c_{\nu})\cdot \Bbbk\mathfrak{S}_N\cdot c_{\lambda})= LR_{\mu,\nu}^{\lambda}.\end{align*} $$

In particular, if the Littlewood–Richardson coefficient $LR_{\mu ,\nu }^{\lambda }=0$ , then we have

$$ \begin{align*}(c_{\mu}\diamond c_{\nu})\cdot \Bbbk\mathfrak{S}_N\cdot c_{\lambda}=0. \end{align*} $$

5.2 Contraction map

We have an isomorphism of $\operatorname {GL}(V_n)$ -modules

(5.2) $$ \begin{align} B_{d,k}(n)\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} D_{d,k}, \end{align} $$

where $D_{d,k}$ is the $\Bbbk $ -vector space spanned by $[2d-k]$ -colored open Jacobi diagrams of degree d such that the map $\{\text {univalent vertices of }D\}\rightarrow [2d-k]$ that gives the coloring of D is a bijection. Thus, any element of $B_{d,k}(n)$ can be written in the form

$$ \begin{align*}u(w_1,\cdots,w_{2d-k}):=(w_1\otimes\cdots\otimes w_{2d-k})\otimes u \end{align*} $$

for $u\in D_{d,k}$ and $w_1,\cdots , w_{2d-k}\in V_n$ .

For $\lambda \vdash 2d-k$ , let $B_{d,k}(n)_\lambda $ be the isotypic component of $B_{d,k}(n)$ corresponding to $\lambda $ ; that is,

$$ \begin{align*}B_{d,k}(n)_\lambda\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}}\Bbbk\mathfrak{S}_{2d-k}c_{\lambda}D_{d,k}.\end{align*} $$

We have $B_{d,k}(n)=\bigoplus _{\lambda \vdash 2d-k}B_{d,k}(n)_{\lambda }$ .

We define a contraction map

$$ \begin{align*}c:B_{d,k}(n)\otimes T_r(n) \rightarrow B_{d,k+r}(n),\end{align*} $$

which is an analogue of the contraction map defined in Appendix B of [Reference Fulton6].

Let $p\geq q$ . For $I=(i_1,\cdots ,i_{q})$ such that $i_1,\cdots ,i_{q}$ are distinct elements of $[p]$ , define a contraction map

$$ \begin{align*}c^I:V_n^{\otimes p}\otimes (V_n^{\ast})^{\otimes q}\rightarrow V_n^{\otimes (p-q)} \end{align*} $$

by

$$ \begin{align*}c^I((w_1\otimes\cdots\otimes w_p)\otimes(y_1\otimes\cdots\otimes y_q))= \left(\prod_{j=1}^{q}\langle w_{i_{j}},y_j\rangle\right) w_1\otimes\cdots\hat{w}_{i_1}\cdots\hat{w}_{i_q}\cdots\otimes w_{p}, \end{align*} $$

where $\hat {w}_{i_1}\cdots \hat {w}_{i_q}$ denotes the omission of $w_{i_1},\cdots ,w_{i_q}$ and where $\langle -,-\rangle :V_n\otimes V_n^{\ast }\rightarrow \Bbbk $ denotes the dual pairing. (See [Reference Fulton6] for details.)

We next consider a diagrammatic version of the above contraction map $c^I$ . Let $2d-k\geq r+1$ . For $I=(i_1,\cdots ,i_{r+1})\in [2d-k]^{r+1}$ such that $i_1,\cdots ,i_{r+1}$ are distinct, we define a linear map

$$ \begin{align*}c^I:B_{d,k}(n)\otimes T_r(n) \rightarrow B_{d,k+r}(n)\end{align*} $$

by contracting colorings of a Jacobi diagram and leaves of a rooted trivalent tree; that is,

where

We define a contraction map

$$ \begin{align*}c:B_{d,k}(n)\otimes T_r(n) \rightarrow B_{d,k+r}(n) \end{align*} $$

by $c=\sum _{I=(i_1,\cdots ,i_{r+1})\in [2d-k]^{r+1}:\text {distinct}}c^I$ . By using the contraction map c, we define a map

$$ \begin{align*}\gamma_{d,k}^r:T_r(n)\rightarrow \operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n))\end{align*} $$

by $\gamma _{d,k}^r(g)(u'):=c(u'\otimes g)$ for $g\in T_r(n), u'=u(w_1,\cdots ,w_{2d-k})\in B_{d,k}(n)$ .

5.3 Vanishing conditions for the contraction map

Here, we observe that the contraction map vanishes under certain specific conditions.

For $r\geq 0$ , a trivalent tree is called a based trivalent tree of degree r if it has one distinguished univalent vertex with no coloring (called a base) and $r+1$ univalent vertices (called leaves) that are colored by distinct elements of $[r+1]$ . (Note that a based trivalent tree is different from a rooted trivalent tree.) Let $L_{r}$ denote the $\mathbb {Z}$ -module spanned by based trivalent trees of degree r modulo the AS and IHX relations. The symmetric group $\mathfrak {S}_{r+1}$ acts on the $\mathbb {Z}$ -module $L_{r}$ by the action on colorings of based trivalent trees. Then we have

$$ \begin{align*}\mathcal{L}_{r+1}(n)\cong H^{\otimes (r+1)}\otimes_{\mathbb{Z}\mathfrak{S}_{r+1}} L_{r}.\end{align*} $$

On the other hand, $\mathcal {L}_{r+1}(n)$ has a $\operatorname {GL}(n;\mathbb {Z})$ -module structure by the standard action on each factor. (See [Reference Fulton and Harris7] for representation theory of $\operatorname {GL}(n;\mathbb {Z})$ .) For $\mu \vdash r+1$ , let $\mathcal {L}_{r+1}(n)_{\mu }$ denote the isotypic component of $\mathcal {L}_{r+1}(n)$ corresponding to $\mu $ ; that is,

$$ \begin{align*}\mathcal{L}_{r+1}(n)_{\mu}\cong H^{\otimes (r+1)}\otimes_{\mathbb{Z}\mathfrak{S}_{r+1}} \mathbb{Z}\mathfrak{S}_{r+1}c_{\mu} L_{r}.\end{align*} $$

We have $\mathcal {L}_{r+1}(n)=\bigoplus _{\mu \vdash r+1}\mathcal {L}_{r+1}(n)_{\mu }$ .

For partitions $\lambda $ and $\mu $ , we write $\lambda \nsupseteq \mu $ if the Young diagram of $\lambda $ does not contain that of $\mu $ .

Proposition 5.3. For $2d-k\geq r+1$ , let $\lambda \vdash 2d-k$ and $\mu \vdash r+1$ . We have

$$ \begin{align*}c(B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_{r+1}(n)_{\mu})) \subset \bigoplus_{\rho:LR^{\lambda}_{\mu,\nu}LR^{\rho}_{\nu,(1)}\neq0 \text{ for some }\nu} B_{d,k+r}(n)_{\rho}. \end{align*} $$

In particular, if $\lambda \nsupseteq \mu $ , then we have

$$ \begin{align*}c(B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_{r+1}(n)_{\mu}))=0.\end{align*} $$

Proof. Any element of $B_{d,k}(n)_{\lambda }$ is a linear sum of $(c_{\lambda }\cdot u)(w_1,\cdots ,w_{2d-k})$ , where $u(w_1,\cdots ,w_{2d-k})\in B_{d,k}(n)$ . Any element of $L_{r}$ is a linear sum of

for $\pi \in \mathfrak {S}_{r+1}$ . Thus, any element of $H^{\ast }\otimes \mathcal {L}_{r+1}(n)_{\mu }$ is a linear sum of $w\otimes ((y_1\otimes \cdots \otimes y_{r+1})\otimes c_{\mu }\cdot L)$ for $w\in H^{\ast }, y_1,\cdots ,y_{r+1}\in H$ .

For any $I=(i_1,\cdots ,i_{r+1})\in [2d-k]^{r+1}$ such that $i_1,\cdots ,i_{r+1}$ are distinct, we have

$$ \begin{align*}c^I((c_{\lambda}\cdot u)(w_1,\cdots,w_{2d-k})\otimes (w\otimes ((y_1 \otimes \cdots\otimes y_{r+1})\otimes c_{\mu}\cdot L))) =\prod_{j=1}^{r+1}\langle w_{i_j}, y_j \rangle D, \end{align*} $$

where

Let $l=2d-k-r-1$ . By Lemma 5.1, we have

$$ \begin{align*}\operatorname{id}_l=\sum_{\nu\vdash l,1\leq i\leq \dim{S^{\nu}}}\tau_{i,1} c_{\nu}\tau_{i,2},\end{align*} $$

where $\tau _{i,1},\tau _{i,2}\in \Bbbk \mathfrak {S}_l$ . Thus, we have

If $LR^{\lambda }_{\mu ,\nu }=0$ for any $\nu \vdash l$ , then we have $D=0$ by Lemma 5.2. Otherwise, since we have

$$ \begin{align*}\operatorname{id}_1\otimes c_{\nu}\in \bigoplus_{\rho\vdash l+1} (S^{\rho})^{LR^{\rho}_{\nu,(1)}}, \end{align*} $$

by the Littlewood–Richardson rule, it follows that

$$ \begin{align*}D\in\bigoplus_{\rho:LR^{\lambda}_{\mu,\nu}LR^{\rho}_{\nu,(1)}\neq0 \text{ for some }\nu}\left( B_{d,k+r}(n)\right)_{\rho}. \end{align*} $$

If $\lambda \nsupseteq \mu $ , then $LR_{\mu ,\nu }^{\lambda }=0$ for any $\nu \vdash l$ . Thus, we have

$$ \begin{align*}c(B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_{r+1}(n)_{\mu}))=0. \end{align*} $$

Remark 5.4. Note that we have $\mathcal {L}_2(n)=\mathcal {L}_2(n)_{(1^2)}$ . Thus, the restriction

$$ \begin{align*}c:B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_2(n)_{(1^2)})\rightarrow B_{d,k+1}(n)_{\rho}\end{align*} $$

of the contraction map vanishes unless $\rho $ can be obtained from $\lambda $ by taking away one box from each of two different rows of $\lambda $ and then by adding one box.

7.1 Decomposition of $B_d(n)$ with respect to connected parts

Note that $D_{d,k}^c\neq 0$ if and only if $d-1\leq k\leq 2d-2$ because each element of $D_{d,k}^c$ has at least two univalent vertices and is connected. For $d\geq 1,k\geq 0$ , the pair $(d,k)$ is called a good pair if $d-1\leq k\leq 2d-2$ . We consider the following decomposition of a pair $(d,k)$ to consider the decomposition of an element of $D_{d,k}$ into the connected parts.

Definition 7.1. Let $d,\ k\geq 0$ . A decomposition of $(d,k)$ into good pairs is a sequence of triples of integers

$$ \begin{align*}\pi = ((a_1,d_1, k_1),\cdots, (a_l,d_l, k_l)) \end{align*} $$

such that $(d_i,k_i)$ are good pairs, $a_i\geq 1$ ,

$$ \begin{align*}\sum_{i=1}^l a_i d_i = d, \quad \sum_{i=1}^l a_i k_i = k, \end{align*} $$

and

$$ \begin{align*}\quad(d_1,k_1)>(d_2,k_2)>\cdots>(d_l,k_l) \end{align*} $$

in the lexicographical order.

Let $\Pi (d,k)$ be the set of all decompositions of $(d,k)$ into good pairs.

For example, we have

(7.1) $$ \begin{align} \Pi(4,2)=\{((1,3,2),(1,1,0)),((1,2,2),(2,1,0)),((2,2,1))\}. \end{align} $$

For any diagram $K\in D_{d,k}$ , we can assign a decomposition of $(d,k)$ into good pairs such that $d_i$ and $k_i$ correspond to the degree and the number of trivalent vertices of each connected component of K, respectively, and $a_i$ corresponds to the multiplicity of $(d_i,k_i)$ . We call a coloring of $K=\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}\in D_{d,k}$ standard if the set of colorings of $K_{i}^{(j)}\in D_{d_i,k_i}^c$ is

$$ \begin{align*}\left\{\sum_{p=1}^{i-1}(2d_p-k_p)a_p +(j-1)(2d_i-k_i)+1 ,\cdots, \sum_{p=1}^{i-1}(2d_p-k_p)a_p +j(2d_i-k_i)\right\}\end{align*} $$

for each $i\in [l],j\in [a_i]$ .

Theorem 7.2. For $d,k,n\geq 0$ , we have an isomorphism of $\operatorname {GL}(V_n)$ -modules

(7.2) $$ \begin{align} B_{d,k}(n) \cong \bigoplus_{\pi =((a_1,d_1, k_1),\cdots,(a_l,d_l, k_l))\in\Pi(d,k)} \left(\bigotimes_{i=1}^l \operatorname{Sym}^{a_i}(B_{d_i,k_i}^c(n))\right). \end{align} $$

To prove this, we need the following proposition.

Proposition 7.3. Let $d,k\geq 0$ . We have an isomorphism of $\mathfrak {S}_{2d-k}$ -modules

(7.3) $$ \begin{align} D_{d,k}\cong \bigoplus_{\pi =((a_1,d_1,k_1),\cdots,(a_l,d_l,k_l))\in\Pi(d,k)} \operatorname{Ind}_{\prod_{i=1}^l(\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\mathfrak{S}_{2d-k}} (\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i}), \end{align} $$

where $\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i}={\mathfrak {S}_{2d_i-k_i}}^{a_i}\rtimes \mathfrak {S}_{a_i}\subset \mathfrak {S}_{(2d_i-k_i)a_i}$ is the wreath product.

For example, we have an isomorphism of $\mathfrak {S}_6$ -modules for $(d,k)=(4,2)$ , which corresponds to equation (7.1),

$$ \begin{align*}D_{4,2}\cong \operatorname{Ind}_{\mathfrak{S}_{4}\times \mathfrak{S}_{2}}^{\mathfrak{S}_{6}} (D_{3,2}^c \otimes D_{1,0}^c) \oplus \operatorname{Ind}_{\mathfrak{S}_{2}\times (\mathfrak{S}_{2}\wr\mathfrak{S}_{2})}^{\mathfrak{S}_{6}} (D_{2,2}^c \otimes (D_{1,0}^c)^{\otimes 2}) \oplus \operatorname{Ind}_{\mathfrak{S}_{3}\wr\mathfrak{S}_{2}}^{\mathfrak{S}_{6}} (D_{2,1}^c)^{\otimes 2}. \end{align*} $$

For example,

and

Via the above isomorphism, the element

corresponds to the element

Proof of Proposition 7.3 Let $D^{\prime }_{d,k}$ denote the right-hand side of equation (7.3).

For any coset $\sigma \in \mathfrak {S}_{2d-k}/\prod _{i=1}^l (\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i})$ , we fix a representative $\tilde {\sigma }\in \mathfrak {S}_{2d-k}$ of $\sigma $ .

Any element of $D^{\prime }_{d,k}$ can be written uniquely as a linear sum of

$$ \begin{align*}K=\tilde{\sigma}\cdot\bigotimes_{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}, \end{align*} $$

where $K_{i}^{(j)}\in D_{d_i,k_i}^c$ . We assign $\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i}K_{i}^{(j)}$ a standard coloring in $[2d-k]$ according to the order of the colorings in $\bigsqcup _{i=1}^{l} [2d_i-k_i]^{a_i}$ of $\bigotimes _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ . For example, if

then the corresponding coloring of $\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i}K_{i}^{(j)}$ is

Define a map $\Psi :D^{\prime }_{d,k}\rightarrow D_{d,k}$ by

$$ \begin{align*}\Psi(K)=\tilde{\sigma}\cdot \bigsqcup_{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}, \end{align*} $$

where $\tilde {\sigma }\in \mathfrak {S}_{2d-k}$ acts on the colorings in $[2d-k]$ . We can check that the map $\Psi $ is an $\mathfrak {S}_{2d-k}$ -module map.

We need to check that $\Psi $ is bijective. If we have $\Psi (K)=\Psi (L)$ for $K=\tilde {\sigma }\cdot \bigotimes _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ , $L=\tilde {\tau }\cdot \bigotimes _{1\leq i\leq l, 1\leq j\leq a_i} L_{i}^{(j)}$ , then we have $\sigma =\tau $ by looking at the set of colorings of each connected component. Since we fix the representatives of cosets of $\mathfrak {S}_{2d-k}/\prod _{i=1}^l (\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i})$ , we have $\tilde {\sigma }=\tilde {\tau }$ . Thus, we have $K=L$ and $\Psi $ is injective. For any element $K\in D_{d,k}$ , we can take $\sigma \in \mathfrak {S}_{2d-k}/\prod _{i=1}^l (\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i})$ such that $K=\tilde {\sigma }\cdot \bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ , where $K_{i}^{(j)}\in D_{(d_i,k_i)}^c$ and $\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ has a standard coloring. Therefore, $\Psi $ is surjective.

Proof of Theorem 7.2 By Proposition 7.3, we have

$$ \begin{gather*} \begin{aligned} B_{d,k}(n) &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}}D_{d,k} \\ &\cong\bigoplus_{\pi\in \Pi(d,k)} \left(V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l (\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\mathfrak{S}_{2d-k}} \left(\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i} \right) \right). \end{aligned} \end{gather*} $$

Moreover, we can check equation (7.2) as follows.

$$ \begin{gather*} \begin{aligned} &V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l (\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\mathfrak{S}_{2d-k}} \left(\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i} \right)\\ &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)}}^{\mathfrak{S}_{2d-k}} \left( \operatorname{Ind}_{\prod_{i=1}^l (\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)}} \left(\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)}}^{\mathfrak{S}_{2d-k}} \left( \bigotimes_{i=1}^l \operatorname{Ind}_{\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i}}^{\mathfrak{S}_{a_i(2d_i-k_i)}} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk(\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)})} \left( \bigotimes_{i=1}^l \operatorname{Ind}_{\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i}}^{\mathfrak{S}_{a_i(2d_i-k_i)}} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong \bigotimes_{i=1}^l \left( V_n^{\otimes a_i(2d_i-k_i)}\otimes_{\Bbbk\mathfrak{S}_{a_i(2d_i-k_i)}} \left( \operatorname{Ind}_{\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i}}^{\mathfrak{S}_{a_i(2d_i-k_i)}} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right) \right)\\ &\cong \bigotimes_{i=1}^l \left( V_n^{\otimes a_i(2d_i-k_i)}\otimes_{\Bbbk(\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong \bigotimes_{i=1}^l \operatorname{Sym}^{a_i}\left( V_n^{\otimes(2d_i-k_i)}\otimes_{\Bbbk\mathfrak{S}_{2d_i-k_i}} D_{d_i,k_i}^c \right)\\ &\cong \bigotimes_{i=1}^l \operatorname{Sym}^{a_i}\left(B_{d_i,k_i}^c(n)\right). \end{aligned} \end{gather*} $$

7.2 Irreducible decomposition of $B_d(n)$ as $\operatorname {GL}(V_n)$ -modules

In this subsection, for simplicity, we write $V=V_n$ , $B_{d,k}=B_{d,k}(n)$ and $B_{d,k}^c=B_{d,k}^c(n)$ .

Let N be a nonnegative integer and $\lambda \vdash N$ . Recall from Section 5.1 that $S^\lambda $ denotes the Specht module, which is an irreducible representation of $\mathfrak {S}_N$ corresponding to $\lambda $ . Let $V_{\lambda }=\mathbb {S}_{\lambda }V$ denote the image of V under the Schur functor $\mathbb {S}_{\lambda }$ . Note that $V_{\lambda }$ is a simple $\operatorname {GL}(V)$ -module if $n\geq r(\lambda )$ and that $V_{\lambda }=0$ if $n<r(\lambda )$ , where $r(\lambda )$ is the number of rows of $\lambda $ .

We use the Littlewood–Richardson rule, plethysms and results by Bar-Natan [Reference Bar-Natan4] to compute the irreducible decompositions of the $\operatorname {GL}(V)$ -modules $B_d$ .

Proposition 7.4 Bar-Natan [Reference Bar-Natan4]

As $\mathfrak {S}_{2d-k}$ -modules, we have isomorphisms

$$ \begin{align*}D^c_{1,0}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{2,1}\cong S^{(1^3)},\quad D^c_{2,2}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{3,2}\cong S^{(2^2)},\quad D^c_{3,3}\cong S^{(1^3)},\quad D^c_{3,4}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{4,3}\cong S^{(3,1^2)},\quad D^c_{4,4}\cong S^{(4)}\oplus S^{(2^2)} ,\quad D^c_{4,5}\cong S^{(1^3)} ,\quad D^c_{4,6}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{5,4}\cong S^{(4,2)}\oplus S^{(2^3)}\oplus S^{(3,1^3)},\quad D^c_{5,5}\cong ({S^{(3,1^2)}})^{\oplus 2},\end{align*} $$

$$ \begin{align*}D^c_{5,6}\cong S^{(4)}\oplus ({S^{(2^2)}})^{\oplus 2},\quad D^c_{5,7}\cong ({S^{(1^3)}})^{\oplus 2},\quad D^c_{5,8}\cong ({S^{(2)}})^{\oplus 2}.\end{align*} $$

Lemma 7.5. We have the following isomorphisms of the $\operatorname {GL}(V)$ -modules:

$$ \begin{align*}B^c_{1,0}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{2,1}\cong V_{(1^3)},\quad B^c_{2,2}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{3,2}\cong V_{(2^2)},\quad B^c_{3,3}\cong V_{(1^3)},\quad B^c_{3,4}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{4,3}\cong V_{(3,1^2)},\quad B^c_{4,4}\cong V_{(4)}\oplus V_{(2^2)} ,\quad B^c_{4,5}\cong V_{(1^3)},\quad B^c_{4,6}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{5,4}\cong V_{(4,2)}\oplus V_{(2^3)}\oplus V_{(3,1^3)},\quad B^c_{5,5}\cong (V_{(3,1^2)})^{\oplus 2},\end{align*} $$

$$ \begin{align*}B^c_{5,6}\cong V_{(4)}\oplus (V_{(2^2)})^{\oplus 2},\quad B^c_{5,7}\cong (V_{(1^3)})^{\oplus 2},\quad B^c_{5,8}\cong (V_{(2)})^{\oplus 2}.\end{align*} $$

Proof. By using Theorem 7.2, Lemma 7.5 and plethysm, we have

$$ \begin{align*}B_{3,0}\cong\operatorname{Sym}^3(B^c_{1,0})\cong\mathbb{S}_{(3)}(\mathbb{S}_{(2)} V) \cong V_{(6)}\oplus V_{(4,2)}\oplus V_{(2^3)}. \end{align*} $$

By using Theorem 7.2, Lemma 7.5 and the Littlewood–Richardson rule, we have

$$ \begin{align*}B_{3,1}\cong B^c_{2,1}\otimes B^c_{1,0}\cong V_{(1^3)}\otimes V_{(2)}\cong V_{(3,1^2)}\oplus V_{(2,1^3)}, \end{align*} $$

and

$$ \begin{align*}B_{3,2}\cong B^c_{3,2}\oplus(B^c_{2,2}\otimes B^c_{1,0})\cong V_{(2^2)}\oplus (V_{(4)}\oplus V_{(3,1)} \oplus V_{(2^2)}). \end{align*} $$

The other isomorphisms of (1) follow from Lemma 7.5.

The irreducible decompositions (2) and (3) follow in a similar way.

We need the irreducible decompositions of $B_{d,0}$ and $B_{d,1}$ to study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ . For $\lambda =(\lambda _1,\cdots ,\lambda _r)\vdash N$ , let $2\lambda $ denote the partition $(2\lambda _1,\cdots ,2\lambda _r)$ of $2N$ .