A solvable extended logarithm of the Johnson homomorphism

Takefumi Nosaka111 E-mail address<EMAIL_ADDRESS>
###### Abstract

We suggest an extension of a certain logarithm of the total Johnson map in
terms of solvable Lie groups. Here, the domain of the map is extended to a
subset consisting of exponential solvable elements in the mapping class group
of a surface.

Keywords
mapping class group, exponential solvable Lie group, knot

Subject Codes
14J50, 22E25, 16N40, 58K15, 22E25

##  1 Introduction

Let $\Sigma_{g,1}$ be an orientable compact surface of genus $g>0$ with one
boundary component and $\mathcal{M}_{g,1}$ be the mapping class group of
$\Sigma_{g,1}$ relative to the boundary. Since $\mathcal{M}_{g,1}$ acts on the
homology $H_{1}(\Sigma_{g,1};{\mathbb{Z}})\cong{\mathbb{Z}}^{2g}$, we have a
homomorphism $\mathcal{P}:\mathcal{M}_{g,1}\rightarrow GL_{2g}({\mathbb{Z}})$;
the kernel is called the Torelli group $\mathcal{I}_{g,1}$. Let $\mathcal{L}$
be the free Lie algebra of rank $2g$ over ${\mathbb{Q}}$.

The Johnson homomorphism [Jo, Mo1] whose domain is the Torelli group
$\mathcal{I}_{g,1}$ has been a source of insight into $\mathcal{M}_{g,1}$ in
terms of nilpotent Lie algebras and is extended to the total Johnson map [Ka].
To describe this map, we prepare the lower central series
$\mathcal{L}=\mathcal{L}_{1}\supset\mathcal{L}_{2}\supset\cdots$, denote the
inverse limit $\lim_{\infty\leftarrow k}\mathcal{L}/\mathcal{L}_{k}$ by
$\mathcal{L}^{\rm nil}$. We also take the automorphism group, ${\rm Aut}_{\rm
Lie}(\mathcal{L}^{\rm nil})$ of the Lie algebra $\mathcal{L}^{\rm nil}$, and a
certain subgroup ${\rm Aut}_{\rm 0}(\mathcal{L}^{\rm nil})$, which preserves a
symplectic structure; see (1) for details. Then, the total Johnson map222More
explicitly, as mentioned in Section 4, we need to choose a symplectic Magnus
expansion. is a map from $\mathcal{M}_{g,1}$ to the subgroup ${\rm Aut}_{\rm
0}(\mathcal{L}^{\rm nil})$. Meanwhile, this ${\rm Aut}_{\rm
0}(\mathcal{L}^{\rm nil})$ has a Lie algebra isomorphic to the set,
$\mathrm{Der}_{\omega}(\mathcal{L}^{\rm nil})$, consisting of derivations of
$\mathcal{L}^{\rm nil}$ which annihilate a symplectic form $\omega$. The set
$\mathrm{Der}_{\omega}(\mathcal{L}^{\rm nil})$ has been widely studied as a
Lie algebra of the Lie wor(l)d in the sense of Kontsevich [Kon]. Since
$\mathrm{Der}_{\omega}(\mathcal{L}^{\rm nil})$ canonically injects a
completion of the Goldman Lie algebra of $\Sigma_{g,1}$ (see [KK1, KK2] for
details), if we can define a logarithm from ${\rm Aut}_{\rm
0}(\mathcal{L}^{\rm nil})$ to $\mathrm{Der}_{\omega}(\mathcal{L}^{\rm nil})$,
then the total Johnson map serves as a bridge from $\mathcal{M}_{g,1}$ to the
Goldman Lie algebra.

However, the known domains of the logarithm have ambiguous aspects. In [KK1],
some logarithms are constructed from the images of $\mathcal{I}_{g,1}$ and the
Dehn twists in $\mathcal{M}_{g,1}$. More generally, the paper [MT] gives an
extension of the domain in ${\rm Aut}_{\rm Lie}(\mathcal{L}^{\rm nil})$ and
defines a generalization of (the logarithm of) Dehn twists. However, any
existing element $\phi\in\mathcal{M}_{g,1}$ which defines a logarithm has the
property that every eigenvalue of $\mathcal{P}(\phi)$ is one.

In this paper, we suggest an extension of the logarithm in terms of solvable
Lie groups over ${\mathbb{C}}$. Here, for a matrix $A\in GL_{n}({\mathbb{C}})$
with eigenvalues $\lambda_{1},\dots,\lambda_{n}\in{\mathbb{C}}^{\times}$, we
define $\mathrm{Eig}(A)$ to be the multiplicative subgroup of
${\mathbb{C}}^{\times}$ generated by
$\lambda_{1},\dots,\lambda_{n},\bar{\lambda_{1}},\dots,\bar{\lambda_{n}}$; we
say that $A$ is exponential solvable if
$\mathrm{Eig}(A)\cap\\{z\in{\mathbb{C}}\mid 1=|z|\\}=\\{1\\}$. Our main
theorem (Theorem 2.5) states that if $\phi\in\mathcal{M}_{g,1}$ is such that
$\mathcal{P}(\phi)$ is an exponential solvable matrix, we can define a
logarithm of $\phi$. More generally, we discuss the existence of logarithms of
exponential solvable elements, $\phi$, of ${\rm Aut}_{\rm
Lie}(\mathcal{L}^{\rm nil})$ (see Theorem 2.2); we show these logarithms to be
an extension of the ones in [KK1, MT]; see Theorem 2.6 and (3).

While there have been many nilpotent approaches to $\mathcal{M}_{g,1}$, this
paper suggests an approach from the solvability viewpoint. The point of
constructing logarithms is that, starting from any solvable exponential
$\phi\in{\rm Aut}_{\rm Lie}(\mathcal{L}^{\rm nil})$, we can find a solvable
Lie group such that the exponential map $\mathrm{Exp}$ is diffeomorphic and
the image contains $\phi$; accordingly, we can define the logarithm of $\phi$
to be $\mathrm{Exp}^{-1}(\phi)$; see Sections 5–6 for the details.
Furthermore, we suggest a procedure for computing $\mathrm{Exp}^{-1}(\phi)$
concretely in terms of the Baker-Campbell-Hausdorff (BCH) formula; see Section
3. In Section 4, we discuss little applications of the logarithms to
topological invariants.

Conventional notation. Let ${\mathbb{K}}$ be one of the fields
${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$. By $\mathcal{L}$, we mean the free
Lie algebra of rank $n\in\mathbb{N}$ over ${\mathbb{Q}}$.

##  2 Main theorems and applications

The purpose of this section is to state the main theorems (the proofs are
given in Section 6). For this, we first introduce base matrices of an
invertible $(n\times n)$-matrix $A$ over ${\mathbb{C}}$. Consider the
exponential map $\exp:\mathrm{Mat}(n\times n;{\mathbb{C}})\rightarrow{\rm
GL}_{n}(,{\mathbb{C}})$ that sends $B$ to $\sum_{i=0}^{\infty}B^{i}/i!$. In
this paper, we call an $(n\times n)$-matrix $B$ satisfying $\exp(B)=A$ a base
matrix of $A$.

######  Example 2.1.

If we fix a branch of the logarithm function on ${\mathbb{C}}^{\times}$, we
can uniquely choose a base matrix for every $A\in{\rm GL}_{n}({\mathbb{C}})$
as follows. Let $\ln:{\mathbb{C}}^{\times}\rightarrow{\mathbb{C}}$ be the map
which takes $z$ to $\ln(|z|)+\sqrt{-1}\mathrm{arg}(z)$, where
$-\pi<\mathrm{arg}(z)\leq\pi$.

First, we suppose the case where $g\in{\rm GL}_{n}({\mathbb{C}})$ exists such
that $g^{-1}Ag=J_{\lambda}(n)$, where
$J_{\lambda}(\ell)\in\mathrm{Mat}(\ell\times\ell,{\mathbb{C}})$ is the Jordan
block of size $\ell$ and eigenvalue $\lambda$. Then, we define

$\ln(A):=\ln(\lambda)I_{\ell}+\sum_{i:1\leq
i<\ell}(-1)^{i-1}\frac{(\lambda^{-1}A-I_{\ell})^{i}}{i}.$

In general, if $g\in{\rm GL}(n,{\mathbb{C}})$ exists such that
$g^{-1}Ag=J_{\lambda_{1}}(\ell_{1})\oplus\cdots\oplus
J_{\lambda_{m}}(\ell_{m})$ by Jordan decomposition, we define $\ln(A)$ to be
$g\bigl{(}\ln(J_{\lambda_{1}}(\ell_{1}))\oplus\cdots\oplus\ln(J_{\lambda_{m}}(\ell_{m}))\bigr{)}g^{-1}$.
Accordingly, we can easily check that $\exp(\ln(A))=A$, i.e., $\ln(A)$ is a
base matrix of $A$.

Furthermore, we need some terminology for the derivations. Let $\mathfrak{g}$
be a real Lie algebra of finite dimension and $\mathrm{Der}(\mathfrak{g})$ be
the Lie algebra consisting of derivations on $\mathfrak{g}$, that is,

$\mathrm{Der}(\mathfrak{g})=\\{f\in\mathop{\mathrm{Hom}}\nolimits(\mathfrak{g},\mathfrak{g})\mid
f([x,y])=[f(x),y]+[x,f(y)]\ \ \mathrm{for\ any\ }x,y\in\mathfrak{g}\\}.$

In addition, let us define ${\rm Aut}_{\rm Lie}(\mathfrak{g})$ to be the group
consisting of Lie algebra automorphisms of $\mathfrak{g}$. It is a Lie group,
since it is a closed subgroup of ${\rm GL}(\mathfrak{g})$. Then, as is
classically known (see [SW, Proposition 7.3]), the Lie algebra of ${\rm
Aut}_{\rm Lie}(\mathfrak{g})$ coincides with $\mathrm{Der}(\mathfrak{g})$, and
the exponential map
$\exp:\mathrm{Der}(\mathfrak{g})\rightarrow\mathrm{Aut}_{\rm
Lie}(\mathfrak{g})$ is given by $D\mapsto\sum_{j=0}^{\infty}D^{j}/j!$. Let
$\mathfrak{g}_{\rm ab}$ be the abelianization of $\mathfrak{g}$, i.e.,
$\mathfrak{g}_{\rm ab}=\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$. If
$\mathfrak{g}$ is nilpotent, we define projections

$p_{k}:\mathrm{Aut}_{\rm Lie}(\mathfrak{g})\rightarrow\mathrm{Aut}_{\rm
Lie}(\mathfrak{g}_{\rm ab})={\rm GL}(\mathfrak{g}_{\rm ab}),\ \ \ \
p_{k}:\mathrm{Der}(\mathfrak{g})\rightarrow\mathrm{Der}(\mathfrak{g}_{\rm
ab})=\mathrm{End}(\mathfrak{g}_{\rm ab})$

to be the induced maps from the projection
$\mathfrak{g}\rightarrow\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$.

The main theorems are for the case in which $\mathfrak{g}$ is the
complexification of a free nilpotent Lie algebra. More precisely,

######  Theorem 2.2.

Let $\mathcal{L}/\mathcal{L}_{k}$ be the $k$-th nilpotent quotient of the free
Lie algebra $\mathcal{L}$ of finite rank over ${\mathbb{Q}}$ and
$\mathfrak{g}$ be the complexification
$\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}}$. For
$\Phi\in\mathrm{Aut}_{\rm Lie}(\mathfrak{g})$, we choose a base matrix,
$\phi$, of $p_{k}(\Phi)$. Suppose that
$p_{k}(\Phi)\in\mathrm{Aut}(\mathfrak{g}_{\rm ab})$ is exponential solvable,
that is, $\mathrm{Eig}(p_{k}(\Phi))\cap\\{z\in{\mathbb{C}}\mid
1=|z|\\}=\\{1\\}$ (see the introduction for the definition of
$\mathrm{Eig}(p_{k}(\Phi))$).

Then, there is a unique derivation $D_{\Phi}\in\mathrm{Der}(\mathfrak{g})$
such that $\exp(D_{\Phi})=\Phi$ and $p_{k}(D_{\Phi})=\phi$.

######  Remark 2.3.

The assumption of exponential solvability is necessary. For example, if
$p_{k}(\Phi)=\mu_{m}I_{n}$ for some a primitive $m$-th root of unity, we can
not subsequently apply Lemma 5.2, which plays a key role in the proof. In
particular, it seems hard to define a logarithm of the hyper-elliptic
involution in $\mathcal{M}_{g,1}$.

In terms of the base matrices in Example 2.1, we introduce the natural
logarithms of $\mathrm{Aut}_{\rm Lie}(\mathfrak{g})$:

######  Definition 2.4.

For $\Phi\in\mathrm{Aut}_{\rm Lie}(\mathfrak{g})$ such that $p_{k}(\Phi)$ is
exponential solvable, we define $\ln(\Phi)$ to be the derivation $D_{\Phi}$ in
the above theorem satisfying $p_{k}(D_{\Phi})=\ln(p_{k}(\Phi))$.

In addition, let us discuss the symplecticity of $\Phi$. Suppose $n=2g$ for
some $g\in{\mathbb{N}}$. Take a basis, $x_{1},x_{2},\dots,x_{2g}$, of
$\mathcal{L}$, and define
$\omega:=\sum_{i=1}^{g}[x_{2i-1},x_{2i}]\in\mathcal{L}/\mathcal{L}_{k}$. As in
[GL] and [Mo2, Definition 3.4], we define two groups by setting

${\rm
Aut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})^{\prime}:=\\{\Phi\in{\rm
Aut}_{\rm Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})\ |\
\Phi(\omega)=\omega\\},$ ${\rm
Aut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}}):=q_{k+1}({\rm
Aut}_{0}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{K}})^{\prime}),$ (1)

where $q_{k+1}$ is the projection ${\rm Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{K}})\rightarrow{\rm
Aut}_{\rm Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$. In parallel,
we define the Lie subalgebra,
$\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$, to be
$\\{D\in\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})\mid
D(\omega)=0\\}$.

######  Theorem 2.5.

Let $\mathfrak{g}$ be the complexification
$\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}}$ as in Theorem 2.2. Take
$\Phi\in\mathrm{Aut}_{\rm Lie}(\mathfrak{g})$. If we can choose an exponential
solvable base matrix, $\phi$, of $p_{k}(\Phi)$, and if $\Phi$ lies in
$\mathrm{Aut}_{0}(\mathfrak{g})$, then the derivation $D_{\Phi}$ claimed in
Theorem 2.2 lies in the subalgebra $\mathrm{Der}_{\omega}(\mathfrak{g}).$

Furthermore, let us discuss the formal Maclaurin expansion and convergence.
For $\Phi\in\mathrm{Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$, consider the formal sum
$\mathrm{Log}(\Phi):=-\sum_{i=1}^{\infty}(\mathrm{id}-\Phi)^{i}/i$. As in
[KK2, Sections 4 and 7], let us define $\mathcal{M}^{\circ}_{k,{\mathbb{K}}}$
to be the subset of $\mathrm{Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$ consisting of $\Phi$’s
such that $\mathrm{Log}(\Phi)$ converges in
$\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k})\otimes{\mathbb{K}}$. We further
define $\mathcal{M}^{\circ}_{\infty,{\mathbb{K}}}$ to be the limit
$\displaystyle{\lim_{\infty\leftarrow
k}}\mathcal{M}^{\circ}_{k,{\mathbb{K}}}\subset\mathrm{Aut}_{\rm
Lie}(\displaystyle{\lim_{\infty\leftarrow
k}}\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$For example, as shown in
[KK1], if $\Phi$ is an element derived from the Torelli group or Dehn twists
in $\mathcal{M}_{g,1}$, $\Phi$ lies in
$\mathcal{M}^{\circ}_{\infty,{\mathbb{Q}}}$; see also [MT, Sections 4 and 8]
for a generalization of these examples; however, as mentioned in Section 1,
all of the existing examples of $\Phi\in\mathcal{M}^{\circ}_{k,{\mathbb{K}}}$
are such that every eigenvalue of $p_{k}(\Phi)\in{\rm GL}_{n}({\mathbb{C}})$
is one (see [MT, Section 4.3]), and the set
$\mathcal{M}^{\circ}_{{\mathbb{K}}}$ is considered to be far from being
explicitly determined.

Now, let us make a generalization of these results:

######  Theorem 2.6.

If $\Phi\in\mathrm{Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$ satisfies that every
eigenvalue of $p_{k}(\Phi)$ is one, then $\Phi$ lies in
$\mathcal{M}^{\circ}_{k,{\mathbb{K}}}$. Moreover, if
${\mathbb{K}}={\mathbb{C}}$, equality $\mathrm{Log}(\Phi)=\mathrm{In}(\Phi)$
holds in $\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$. This
statement holds true even considering the inverse limit according to
$k\rightarrow\infty.$

Finally, we now give a similar theorem in terms of Hopf algebras. According to
[Kon], this discussion is regarded in the associative context. Let $\hat{T}$
be the ring ${\mathbb{K}}[\\![X_{1},\dots,X_{n}]\\!]$ of formal power series
with non-commutative variables $X_{1},\dots,X_{n}$, and $\hat{T}_{k}$ be the
ideal of $\hat{T}$ generated by formal power series of degree $\geq k$. Let
$\hat{T}_{\infty}$ be $\\{0\\}$. For $k\in\mathbb{N}\cup\\{\infty\\}$, let us
consider the coproduct map
$\hat{\Delta}:\hat{T}/\hat{T}_{k}\rightarrow\hat{T}/\hat{T}_{k}\otimes\hat{T}/\hat{T}_{k}$
defined by $\hat{\Delta}(X_{i})=1\otimes X_{i}+X_{i}\otimes 1$, where
$\otimes$ with $k=\infty$ is defined to be the complete tensor product as in
[Q, Appendix A.1]. Take the following two automorphism groups:

$\displaystyle{\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})$ $\displaystyle:=$
$\displaystyle\\{\mathrm{A\ }{\mathbb{K}}\textrm{-algebra \ automorphism \
}U\textrm{ of }\hat{T}/\hat{T}_{k}\mid\
U(\hat{T}_{p})\subset\hat{T}_{p}\mathrm{\ for\ any\ }p\in{\mathbb{N}}\\},$
$\displaystyle{\rm Aut}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$ $\displaystyle:=$
$\displaystyle\\{U\in{\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})\mid U\mathrm{\
preserves\ }\hat{\Delta}\\}.$ (2)

As is well-known, the set of primitive elements of $\hat{T}/\hat{T}_{k}$,

$\mathcal{P}(\hat{T}/\hat{T}_{k}):=\\{x\in\hat{T}/\hat{T}_{k}\mid\hat{\Delta}(x)=1\otimes
x+x\otimes 1\\},$

has a Lie algebra structure with $[X,Y]=XY-YX$ and is isomorphic to
$\mathcal{L}/\mathcal{L}_{k}$. According to [Q, Theorem A.3.3], the
restrictions of Hopf automorphisms of $\hat{T}/\hat{T}_{k}$ give rise to a
group isomorphism ${\rm Aut}_{\rm Hopf}(\hat{T}/\hat{T}_{k})\cong{\rm
Aut}_{\rm Lie}(\mathcal{L}/\mathcal{L}_{k})$. Thus, this ${\rm Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k})$ is regarded as a subgroup opf ${\rm
Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})$.

######  Theorem 2.7.

Let $\mathrm{Der}(\hat{T}/\hat{T}_{k}\otimes{\mathbb{C}})$ be the set of all
derivations of $\hat{T}/\hat{T}_{k}\otimes{\mathbb{C}}$, and $p_{k}:{\rm
Aut}_{\rm alg}(\hat{T}/\hat{T}_{k}\otimes{\mathbb{C}})\rightarrow{\rm
Aut}_{\rm alg}(\hat{T}/\hat{T}_{2}\otimes{\mathbb{C}})={\rm
GL}_{n}({\mathbb{C}})$ be the projection. For $\Phi\in{\rm Aut}_{\rm
alg}(\hat{T}/\hat{T}_{k})$, we choose a base matrix, $\phi$, of $p_{k}(\Phi)$.
If $p_{k}(\Phi)\in{\rm GL}_{n}({\mathbb{C}})$ is exponential solvable, then
there is a unique derivation
$D_{\Phi}\in\mathrm{Der}(\hat{T}/\hat{T}_{k}\otimes{\mathbb{C}})$ such that
$\exp(D_{\Phi})=\Phi$ and $p_{k}(D_{\Phi})=\phi$.

The above theorems can be summarized as a commutative diagram,

$\textstyle{\ln:\\{\Phi\in{\rm Aut}_{\rm
alg}(\hat{T}/\hat{T}_{k}\otimes{\mathbb{C}})\mid p_{k}(\Phi)\textrm{ is
exponential
solvable}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\mathrm{Der}(\hat{T}/\hat{T}_{k}\otimes{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\ln:\\{\Phi\in\mathrm{Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})\mid p_{k}(\Phi)\textrm{
is exponential
solvable}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\ln:\\{\Phi\in\mathrm{Aut}_{\rm
0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})\mid p_{k}(\Phi)\textrm{ is
exponential
solvable}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{g,1}\cup\\{\mathrm{Dehn\
twists}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Log}}$$\textstyle{\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}}).}$
(3)

##  3 Computations of the natural logarithm $\mathrm{In}(\Phi)$

This section shows a procedure for computing the natural logarithm
$\mathrm{In}(\Phi)$ from the viewpoints of generalized Magnus expansions and
rational homotopy theory [Q]. Recall the isomorphism ${\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\cong{\rm Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k})$ from the previous section. Thus, to
investigate the above groups ${\rm Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k})$ and ${\rm
Aut}_{\omega}(\mathcal{L}/\mathcal{L}_{k})$, we may focus on ${\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})$ hereafter.

As is done in [Ka, Section 1], we first observe the decompositions (4) below.
Recall the lemma:

######  Lemma 3.1 ([Ka, Lemma 1.2]).

Take $k\in\mathbb{N}\cup\\{\infty\\}$. A ${\mathbb{K}}$-algebra endomorphism
$U$ of $\hat{T}/\hat{T}_{k}$ lies in ${\rm Aut}_{\rm
alg}(\hat{T}/\hat{T}_{k})$ if and only if $U(\hat{T}_{p})\subset\hat{T}_{p}$
for each $p\in{\mathbb{N}}$ and the projection of $U$ on
$\hat{T}_{1}/\hat{T}_{2}=H_{{\mathbb{K}}}$ is an isomorphism.

Let $H_{{\mathbb{K}}}$ be $\hat{T}_{1}/\hat{T}_{2}\cong{\mathbb{K}}^{n}$ as a
vector space. For $U\in{\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})$, we denote
the projection of $U$ on $\hat{T}_{1}/\hat{T}_{2}$ by $p_{k}(U)\in{\rm
GL}_{n}({\mathbb{K}})$ and have the following group homomorphisms,

$p_{k}:{\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})\rightarrow{\rm
GL}_{n}({\mathbb{K}}),\ \ \ \ \ {\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\rightarrow{\rm GL}_{n}({\mathbb{K}}).$

We denote the kernels by $\mathrm{IA}_{\rm alg}(\hat{T}/\hat{T}_{k})$ and
$\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$. We will show splittings of the
homomorphisms. For $A\in{\rm GL}_{n}({\mathbb{K}})$ and $j\in{\mathbb{N}}$,
consider the tensor representation, i.e., $A(z_{j}):=A^{\otimes j}z_{j}$ for
$z_{j}\in H_{{\mathbb{K}}}^{\otimes j}$. Identifying $\hat{T}/\hat{T}_{k}$
with $\prod_{j=0}^{k-1}H_{{\mathbb{K}}}^{\otimes j}$, the tensor
representation yields a homomorphism $\mathfrak{s}:{\rm
GL}_{n}({\mathbb{K}})\rightarrow GL(\hat{T}/\hat{T}_{k})$. We can easily
verify that it lies in ${\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})$ and
preserves $\hat{\Delta}$. Namely, $\mathfrak{s}$ is a splitting of $p_{k}$.
Hence, we have the semi-direct products,

${\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})\cong\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k})\rtimes GL(H_{\mathbb{K}}),\ \ \ \ \ {\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\cong\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\rtimes GL(H_{\mathbb{K}}).$ (4)

To gain an understanding of the subgroup $\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k})$, let us examine the bijection (5). Every
${\mathbb{K}}$-linear homomorphism
$\alpha:H_{{\mathbb{K}}}\rightarrow\hat{T}_{2}/\hat{T}_{k}$ can be uniquely
extended to an algebra homomorphism
$\tilde{\alpha}:\hat{T}/\hat{T}_{k}\rightarrow\hat{T}_{2}/\hat{T}_{k}$, which
is an isomorphism, by Lemma 3.1. Thus, we have a bijection,

$E:\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{2}/\hat{T}_{k})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k});\ \ \alpha\longmapsto\tilde{\alpha}.$ (5)

Moreover, the preimage of $\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$
implies

$E^{-1}(\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{k}))=\\{f\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{2}/\hat{T}_{k})\mid
f(x)\in\mathcal{P}(\hat{T}/\hat{T}_{k})\ \mathrm{for\ any\ }x\in
H_{\mathbb{K}}\\}.$ (6)

From viewpoints of (4) and (5), the group structure of ${\rm Aut}_{\rm
alg}(\hat{T}/\hat{T}_{k})$ can be described as follows (The description for
$k\leq 4$ is in [Ka, Lemma 1.4]). Let $\mathcal{P}_{m}$ be the set of
partitions of $m$. For $i_{1},\dots,i_{\ell}\in\mathbb{Z}_{\geq 1}$ with
$\sum_{j=1}^{\ell}i_{j}=m$, we denote the associated partition by
$[i_{1}|i_{2}|\cdots|i_{\ell}]\in\mathcal{P}_{m}$. In addition, for
$u\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{2}/\hat{T}_{k})$
and ${\rm GL}(H_{{\mathbb{K}}})$, we define $((u,A))$ to be $\tilde{u}\circ
A\in{\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})$. We define $u_{m}$ to be the
element in
$\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{m}/\hat{T}_{m+1})$
that is determined by

$((u,A))a=Aa+\sum_{m:1\leq m<k}u_{m}(Aa)\in\hat{T}/\hat{T}_{k}$

for any $a\in H_{{\mathbb{K}}}.$ By a straightforward computation, we can show
the following:

######  Lemma 3.2 (cf. [Ka, Lemma 1.4]).

Suppose $((w,C))=((u,A))((v,B))\in{\rm Aut}_{\rm alg}(\hat{T}/\hat{T}_{k})$
for $(u,A)$, $(v,B),$ and $(w,C)\in\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k})\rtimes GL(H_{\mathbb{K}})$. Then, $C=AB$ and

$w_{m}(a)=u_{m}(a)+\sum_{[i_{1}|i_{2}|\cdots|i_{\ell}]\in\mathcal{P}_{m}}(u_{i_{1}}\otimes\cdots\otimes
u_{i_{\ell}})A^{\otimes\ell}\bigl{(}v_{\ell}(A^{-1}a)\bigr{)}$

hold for any $a\in H_{{\mathbb{K}}}$ and $m\geq 2$. Here,
$u_{1}=\mathrm{id}_{H_{{\mathbb{K}}}}$ and $v_{1}=0$.

######  Example 3.3.

By abusing the notation, we will denote the operation $a\mapsto
A^{\otimes\ell}(v_{\ell}(A^{-1}a))$ by $Av_{\ell}$ and give concrete
descriptions of $w_{m}$ for $m\leq 4$:

$w_{2}=u_{2}+Av_{2},\ \ \ w_{3}=u_{3}+(u_{2}\otimes 1+1\otimes
u_{2})Av_{2}+v_{3},$ $w_{4}=u_{4}+(u_{3}\otimes 1+1\otimes u_{3}+u_{2}\otimes
u_{2})Av_{2}+(u_{2}\otimes 1\otimes 1+1\otimes u_{2}\otimes 1+1\otimes
1\otimes u_{2})Av_{3}+Av_{4}.$

Finally, for ${\mathbb{K}}={\mathbb{C}}$, we can show a procedure for
computing the logarithm $\ln(\Phi)$ for $\Phi\in{\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\cong{\rm Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$ such that $p_{k}(\Phi)$
is exponential solvable. Following the semi-direct products (4), we decompose
$\Phi$ as $(\mathrm{I}\Phi,p_{k}(\Phi))\in\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\rtimes{\rm GL}(H_{\mathbb{C}})$. As mentioned in
Sections 5 and 6, there uniquely exists
$Y\in\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$ satisfying
$\exp(Y)=(\mathrm{I}\Phi,1)$. Letting $X\in\mathfrak{gl}_{n}({\mathbb{C}})$ be
$\ln(p_{k}(\Phi))$, we formally obtain the following from the BCH formula:

$\ln(\Phi)=\ln((\mathrm{I}\Phi,1)(1,p_{k}(\Phi)))=\log((\exp(Y),1)(1,\exp(X)))=$
$={\displaystyle
X+Y+{\frac{1}{2}}[X,Y]+{\frac{1}{12}}[X,[X,Y]]-{\frac{1}{12}}[Y,[X,Y]]+\cdots}.$
(7)

######  Lemma 3.4 (See §6 for the proof).

The BCH-formula (7) converges in
$\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$.

To conclude, if we find concretely a $Y$ satisfying
$\exp(Y)=(\mathrm{I}\Phi,1)$ and a decomposition
$\Phi=(\mathrm{I}\Phi,p_{k}(\Phi))$ with small $k$, we can sometimes compute
$\ln(\Phi)$ via the BCH formula; see [KK1, Section 7] and [MT, Sections 5 and
8] for a description of computing such $Y$’s. For example, let us give an
concrete computation for $k=3$, where we denote $[X,Y]$ by
$\mathrm{ad}(X)(Y)$.

######  Example 3.5.

First, suppose that $k=3$. Then, any term in (7) which contains two instances
of $Y$ vanishes. Thus, we can apply a reduction of the BCH formula (see [Reu,
Section 3.3]) to $\mathrm{Im}(\Phi)$, as
$X+\mathrm{ad}(\frac{X}{1-e^{X}})(Y)$; more precisely, if we define the
Bernoulli numbers $b_{2n}$ by the Taylor expansion
$x/(1-e^{x})=1+x/2+\sum_{n=1}^{\infty}\frac{1}{(2n)!}b_{2n}x^{2n}$, the
$\mathrm{ad}(\frac{X}{1-e^{X}})(Y)$ is defined by
$Y+\frac{1}{2}[X,Y]+\sum_{n=1}^{\infty}\frac{1}{(2n)!}b_{2n}\mathrm{ad}(X)^{2n}(Y).$
For example, if $X$ is a diagonal matrix, it is not so hard to compute
$\mathrm{ad}(\frac{X}{1-e^{X}})(Y)$.

Concerning the case $k\geq 4$, if a term in the BCH formula contains a bracket
of $Y$ $(k-2)$ times, the term vanishes. Since we can find explicit
descriptions of order $O(Y^{k})$ in the BCH formula (see, e.g., [ML, Section
4] or [Reu, Section 3.3]), we can find an explicit formula for $\ln(\Phi)$, as
in Example 3.5.

##  4 Motivation for studying the Johnson map and topological invariants

In this section, we explain our motivation behind defining an extension of the
logarithm. Throughout this section, we will fix $k\in\mathbb{Z}_{\geq
2}\cup\\{\infty\\}$, and the free group, $F$, with basis
$x_{1},x_{2},\dots,x_{n}$,

Let us begin by introducing Magnus expansions. Take an invertible matrix
$A=\\{a_{ij}\\}_{1\leq i,j\leq n}\in GL_{n}({\mathbb{K}})$. For $m\geq 0$, we
define $(\hat{T}/\hat{T}_{k})^{\times}$ to be the multiplicative group
consisting of invertible elements of $\hat{T}/\hat{T}_{k}$. Inspired by [Ka,
KK1], we define a Magnus expansion (over $A$) to be a group homomorphism
$\theta:F\rightarrow(\hat{T}/\hat{T}_{k})^{\times}$ satisfying
$\theta(x_{i})=1+\sum_{j=1}^{n}a_{ij}X_{j}$ modulo $\hat{T}_{2}$ for any
$i\leq n$. If $A$ is the identity matrix $I_{n}$, this definition is the same
as the generalized Magnus expansion [Ka, KK1]. A Magnus expansion $\theta$ is
said to be group-like if $\hat{\Delta}(\theta(x))=x\otimes x$ for any $x\in
F$. For example, the Magnus expansion $\theta_{\exp}$ defined by
$\theta_{\exp}(x_{i})=1+\sum_{j=1}^{\infty}X_{i}^{j}/j!$ is group-like.
Furthermore, we define two sets

$\Theta_{n}^{\rm grp}:=\\{\mathrm{A\ group}\textrm{-}\mathrm{like\ Magnus\
expansion\ }\theta:F\rightarrow(\hat{T}/\hat{T}_{k})^{\times}\mathrm{\ over\
}A\mid A\in GL_{n}({\mathbb{K}})\\}.$

For $\theta\in\Theta_{n}^{\rm grp}$ and $U\in{\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})$, we define $U\cdot\theta$ by $U\circ\theta$. From
the definitions, $U\cdot\theta$ lies in $\Theta_{n}^{\rm grp}$. This
correspondence defines an action of ${\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})$ on $\Theta_{n}^{\rm grp}$. Similarly to [Ka,
Theorem 1.3 (2)], we can prove the following:

######  Proposition 4.1.

The action of ${\rm Aut}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$ on $\Theta_{n}^{\rm
grp}$ is free and transitive.

The proof is almost the same as the original one, so we will omit it here. As
a result, we have a bijection between ${\rm Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})$ and $\Theta_{n}^{\rm grp}$. According to Lemma
3.2, we can describe the group structure on $\Theta_{n}^{\rm grp}$ via the
bijectivity, where the identity $\in\Theta_{n}^{\rm grp}$ is the above
$\theta_{\exp}$.

Next, let us briefly mention symplectic expansions, which were first
introduced in [Ma]. Let $n$ be $2g$, and $\zeta\in F$ be
$[x_{1},x_{2}]\cdots[x_{2g-1},x_{2g}]$, and $\omega\in\hat{T}/\hat{T}_{k}$ be
$\sum_{i=1}^{g}X_{2i-1}X_{2i}-X_{2i}X_{2i-1}$. A symplectic expansion is a
group-like expansion $\theta:F\rightarrow(\hat{T}/\hat{T}_{k})^{\times}$
satisfying $\theta(\zeta)=\exp(\omega)=\sum_{m=0}^{\infty}\omega^{m}/m!$. Let
$\Theta_{n}^{\rm sym}$ be a subset composed of all symplectic expansions, and
${\rm Aut}_{\rm\omega}(\hat{T}/\hat{T}_{k})$ be the subgroup of ${\rm
Aut}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$ of automorphisms preserving $\omega$,
which is isomorphic to the subgroup ${\rm
Aut}_{\omega}(\mathcal{L}/\mathcal{L}_{k})$ in (1). As shown in [Ma],
$\Theta_{n}^{\rm sym}$ is not empty. As an extension of [KK1, Proposition
2.8.1], if $k=\infty$, we can show that the restricted action of ${\rm
Aut}_{\rm\omega}(\hat{T}/\hat{T}_{k})$ on $\Theta_{n}^{\rm sym}$ is also free
and transitive.

Furthermore, let us review the total Johnson map from [KK1, Section 2.5]. Let
${\rm Ker}(\epsilon)\subset{\mathbb{K}}[F]$ be the augmentation ideal, and
$\widehat{{\mathbb{K}}}[F]$ the completion algebra, i.e.,
$\widehat{{\mathbb{K}}}[F]:=\lim_{\infty\leftarrow j}{\mathbb{K}}[F]/({\rm
Ker}(\epsilon))^{j}$. As in [Ka, Theorem 1.3], any Magnus expansion $\theta$
gives rise to a filtered algebra isomorphism
$\theta:\widehat{{\mathbb{K}}}[F]\cong\widehat{T}$. Any mapping class
$\varphi\in\mathcal{M}_{g,1}$ can be regarded as being in $\mathrm{Aut}(F)$
via the natural action
$\mathcal{M}_{g,1}\curvearrowright\pi_{1}(\Sigma_{g,1})=F$; thus, we uniquely
have $T^{\theta}(\varphi)\in{\rm Aut}_{\rm alg}(\widehat{T})$ such that
$T^{\theta}(\varphi)\circ\theta=\theta\circ\varphi$. The map
$T^{\theta}:\mathcal{M}_{g,1}\rightarrow{\rm Aut}_{\rm alg}(\widehat{T})$ is
also called the total Johnson map. It is known (see, e.g., [KK1, KK2]) that
this map is injective, and if $\theta$ is group-like (resp. symplectic), then
the image of $T^{\theta}$ is contained in ${\rm Aut}_{\rm Hopf}(\hat{T})$
(resp. ${\rm Aut}_{\rm\omega}(\hat{T})$); the composite of the restriction of
$T^{\theta}$ on $\mathcal{I}_{g,1}\cup\\{\textrm{Dehn twists}\\}$ and the
logarithm from $\mathrm{IA}_{\rm Hopf}(\hat{T})$ are studied in [KK1, KK2, Ma,
MT] for the case of ${\mathbb{K}}={\mathbb{Q}}$ and $\theta$ is over
$A=I_{n}$.

In contrast, thanks to Theorem 2.2, when ${\mathbb{K}}={\mathbb{C}}$, we can
define the logarithms from the set,

$\mathrm{SE}_{k}:=\\{\phi\in\mathrm{Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\mid\mathrm{Eig}(p_{k}(\phi))\cap\\{z\in{\mathbb{C}}\mid
1=|z|\\}=1\\}.$

To conclude, we have succeeded in giving an extension of the composite
$\mathrm{Log}\circ T^{\theta}$ from the preimage
$(T^{\theta})^{-1}(\mathrm{SE}_{k})\subset\mathcal{M}_{g,1}$. It might be an
interesting exercise to determine the union $\cup_{\theta\in\Theta_{n}^{\rm
grp}}\cup_{k\geq
1}^{\infty}(T^{\theta})^{-1}(\mathrm{SE}_{k})\subset\mathcal{M}_{g,1}$. For
example, if this inclusion $\subset$ is surjective, we can define a logarithm
for every element of $\mathcal{M}_{g,1}$.

Now let us discuss some small applications to topological invariants. The
paper [GL] defines a class of homology cylinders over $\Sigma_{g,1}$, for
which we will not give a definition of here, and a certain monoid
homomorphism,

$\sigma_{k}:\\{M:\textrm{A homology cobordism over
}\Sigma_{g,1}\\}\longrightarrow{\rm
Aut}_{0}(\mathcal{L}/\mathcal{L}_{k})\cong{\rm
Aut}_{\omega}(\hat{T}/\hat{T}_{k}),$

where $\mathrm{rk}\mathcal{L}=2g$. As an analogy, the author [No] defines a
knot invariant as the map,

$\sigma_{k}^{\prime}:\\{\textrm{a knot }K\textrm{ in }S^{3}\textrm{ with
}\mathrm{deg}\Delta_{K}=2g\\}\longrightarrow\mathrm{Out}_{0}(\mathcal{L}/\mathcal{L}_{k})/{\rm
conj},$

where $\mathrm{deg}\Delta_{K}$means the degree of the Alexander polynomial of
$K$ and the symbol $``{\rm/conj}"$ means the set of the conjugacy classes. The
groups ${\rm Aut}_{\omega}(\hat{T}/\hat{T}_{k})$ and
$\mathrm{Out}_{0}(\mathcal{L}/\mathcal{L}_{k})$ seem far from being
computable; however, if $\sigma_{k}(M)$ lies in $\mathrm{SE}_{k}$, the
composite $\ln\circ\sigma_{k}(M)$ is valued in the vector space
$\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$. On
the other hand, concerning another ${\sigma}_{k}^{\prime}$, notice the
equality,

$X-e^{-Y}Xe^{Y}=X-\sum_{j:j\geq
1}[[\cdots[X,\underbrace{Y]\cdots],Y}_{j\textrm{-times}}]/j!\in\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$
(8)

for any $X,Y\in\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$
(see, e.g., [MT, Appendix (A.0.2)]). In conclusion, if we can define
$\ln\circ\sigma_{k}^{\prime}(K)$, the invariant is valued in the quotient
linear space
$\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})/\mathrm{Inn}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$
modulo the formula (8) for any
$X,Y\in\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$.
Here, $\mathrm{Inn}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$
is the submodule consisting of inner derivations killing $\omega$.
Additionally, we remark that the quotient space modulo (8) is non-trivial. In
fact, the space surjects on the abelianization of
$\mathrm{Der}_{\omega}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$ which
is related to the rational cohomology of $\mathrm{Out}(F)$; see the main
theorem of [Kon] and [CKV, Section 8]. For example, if $g\geq 6$ or $g=1$, the
paper [MSS] explicitly gives 1-cocycles of
$\mathrm{Der}_{\omega}(\lim_{\infty\leftarrow
k}\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$ in terms of the Enomoto-
Satoh trace.

As in the Kontsevich knot invariant and the LMO invariant of homology
3-spheres, if we can define the logarithms of some invariants, we should be
able to discover interesting properties and relations between invariants from
the viewpoint of the logarithms. It seems reasonable to hope that the above
composites $\ln\circ\sigma_{k}$ and $\ln\circ\sigma_{k}^{\prime}$ have similar
properties.

##  5 Two lemmas

Here, we turn our attention to the proofs of the main theorems. As a
preliminary, let us prepare two lemmas. Proposition 5.1 below is inspired by
[Mo1, §2] and [Mo2]. We will use the terminology in Section 3. Let
$m\in\mathbb{N}.$ Notice from Lemma 3.1 that the canonical projection
$q_{k}:\mathrm{Aut}_{\bullet}(\hat{T}/\hat{T}_{m+1})\rightarrow\mathrm{Aut}_{\bullet}(\hat{T}/\hat{T}_{m})$
is surjective. Recall from (5) that any ${\mathbb{K}}$-homomorphism
$\alpha:H_{{\mathbb{K}}}\rightarrow\hat{T}_{m}/\hat{T}_{m+1}$ such that
$p_{k}\circ\alpha:H_{{\mathbb{K}}}\rightarrow H_{{\mathbb{K}}}$ is isomorphic
can be uniquely extended to an algebra isomorphism
$\tilde{\alpha}:\hat{T}/\hat{T}_{m+1}\rightarrow\hat{T}/\hat{T}_{m+1}$; thus,
the kernel of $q_{k}$ is
$\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{m}/\hat{T}_{m+1})$.
Notice that $\mathrm{Aut}_{\bullet}(\hat{T}/\hat{T}_{m})$ is a closed subgroup
of the matrix Lie group $\mathrm{GL}(\hat{T}/\hat{T}_{m+1})$ and $q_{k}$ is
continuous; hence, if ${\mathbb{K}}={\mathbb{R}}$ or
${\mathbb{K}}={\mathbb{C}}$, then
$\mathrm{Aut}_{\bullet}(\hat{T}/\hat{T}_{m})$ is a Lie group and $q_{k}$ is
smooth. In summary, we have the Lie group extensions,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{m}/\hat{T}_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{{\rm
Aut}_{\rm
alg}(\hat{T}/\hat{T}_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{k}}$$\textstyle{{\rm
Aut}_{\rm
alg}(\hat{T}/\hat{T}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\mathcal{P}(\hat{T}_{m}/\hat{T}_{m+1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{{\rm
Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{k}}$$\textstyle{{\rm
Aut}_{\rm
Hopf}(\hat{T}/\hat{T}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$
(9)

The sequence (3) over ${\mathbb{Q}}$ follows from Proposition 2.3 in [Mo1] and
is not central; however, to see a centrality, we only need recall the
subgroups in (4); we have

######  Lemma 5.1.

The horizontal sequences of the restriction of the diagram (9),

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{m}/\hat{T}_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{k}}$$\textstyle{\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\mathcal{P}(\hat{T}_{m}/\hat{T}_{m+1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{k}}$$\textstyle{\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$
(10)

are central extensions. In particular,
$\mathrm{IA}_{\rm\bullet}(\hat{T}/\hat{T}_{m})$ is a nilpotent Lie group and
is contractible.

###### Proof.

We will show the centrality of only the upper sequence. For
$f\in\mathop{\mathrm{Hom}}\nolimits(H_{{\mathbb{K}}},\hat{T}_{m}/\hat{T}_{m+1})$
and $\alpha\in\mathrm{IA}_{\rm alg}(\hat{T}/\hat{T}_{m+1})$, by (4), it
suffices to show the identity $(\alpha\circ
E(f))^{\sim}=(E(f)\circ\alpha)^{\sim}$. Let
$r_{k}:\hat{T}/\hat{T}_{m+1}\rightarrow\hat{T}_{m}/\hat{T}_{m+1}$ be the
projection. Then,
$r_{k}((\alpha\circ\iota(f))^{\sim})=r_{k}(\alpha)^{\sim}+r_{k}(\iota(f))^{\sim}=r_{k}((\iota(f)\circ\alpha)^{\sim})$
by the definition of $\mathrm{IA}_{\rm alg}(\hat{T}/\hat{T}_{m+1})$. Since
$q_{k}(\alpha\circ E(f))=q_{k}(\alpha)=q_{k}(E(f)\circ\alpha)$ by the
definition of $f$, we have the required identity. ∎

Next, let us change our focus to be on exponential solvable Lie groups. Let
$G$ be a real solvable Lie group with Lie algebra $\mathfrak{g}$. If the
exponential map $\mathfrak{g}\rightarrow G$ is diffeomorphic, $G$ is said to
be exponential solvable. For example, every simply-connected nilpotent Lie
group is exponential solvable. As an application, let us consider the
situation that the additive Lie group ${\mathbb{R}}$ smoothly acts on a Lie
group $N$ with Lie algebras $\mathfrak{n}$. Then, the action defines the semi-
direct product $N\rtimes{\mathbb{R}}$ and induces a representation
$\tau:{\mathbb{R}}\rightarrow\mathrm{GL}(\mathfrak{n})$. For
$x\in\mathfrak{n}$, let $T(x)\subset{\mathbb{R}}$ be the subgroup of $\tau$
fixing $x$. We shall cite the following:

######  Lemma 5.2 (A special case of [MM, Theorem 5]).

Let $N$ be a simply-connected nilpotent Lie group and $G=N\rtimes{\mathbb{R}}$
be the semidirect product associated with $\tau$. Then, $G$ is exponential
solvable if and only if $T(x)=\\{0\\}$ or $T(x)={\mathbb{R}}$ for each
$x\in\mathfrak{n}$.

######  Remark 5.3.

The Jordan decomposition theorem implies that the condition in Lemma 5.2 holds
if no eigenvalue of $\tau(1)\in\mathrm{GL}(\mathfrak{n})$ lies in
$\\{b\in{\mathbb{C}}^{\times}\mid b\neq 1,|b|=1\\}$. This is equivalent to
saying that no eigenvalue of the differential
$d\tau(\bullet)\in\mathrm{End}(\mathfrak{n}\otimes{{\mathbb{C}}})$ lies in
$\\{\sqrt{-1}a\mid a\in{\mathbb{R}}^{\times}\\}$ (cf. the concept of weights
on exponential solvable Lie groups; see, e.g., [MM]).

##  6 Proofs of Theorems 2.2, 2.5, 2.6, and 2.7

Before giving the proofs, we will briefly examine the decomposition (11)
below. For $\lambda\in{\mathbb{C}}\setminus\\{0\\}$ and $\ell\in{\mathbb{N}}$,
let $J_{\lambda}(\ell)\in\mathrm{Mat}(\ell\times\ell;{\mathbb{C}})$ be the
Jordan block of size $\ell$ and eigenvalue $\lambda$. As is known (see, e.g.,
[MV, Theorem 2]), the tensor product of $J_{\lambda}(\ell)$ and $J_{\mu}(n)$
in $\mathrm{Mat}(n\ell\times n\ell;{\mathbb{C}})$ has the following
decomposition:

$J_{\lambda}(\ell)\otimes_{{\mathbb{C}}}J_{\mu}(n)\cong\bigoplus_{w:1\leq
w\leq\mathrm{min}(\ell,n)}J_{\lambda\mu}(\ell+n-2w-1).$ (11)

In addition, let us observe the action of ${\rm GL}(H_{{\mathbb{K}}})$ on the
kernel $\mathrm{IA}_{\bullet}(\hat{T}/\hat{T}_{k})$ in (4). By Lemma 3.1, the
Lie algebra of $\mathrm{IA}_{\rm alg}(\hat{T}/\hat{T}_{k})$ is linearly
isomorphic to
$\oplus_{j=2}^{k-1}\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{K}}}(H_{{\mathbb{K}}},\hat{T}_{j}/\hat{T}_{j+1})$.
Here, we consider the action of ${\rm GL}(H_{{\mathbb{K}}})$ on
$\mathop{\mathrm{Hom}}\nolimits(H_{{\mathbb{K}}},\hat{T}_{j}/\hat{T}_{j+1})$
defined by

$(Af)(u):=A^{\otimes j}f(A^{-1}u)\ \ \ \ \ \ \ (A\in{\rm
GL}(H_{{\mathbb{K}}}),\ u\in H_{{\mathbb{K}}},\
f\in\mathop{\mathrm{Hom}}\nolimits(H_{{\mathbb{K}}},\hat{T}_{j}/\hat{T}_{j+1})).$
(12)

From Lemma 3.2 and the semi-direct products (4), the sum of this action (12)
coincides with the action of ${\rm GL}(H_{{\mathbb{K}}})$ on $\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k})$, although this coincidence was shown in the case
${\mathbb{K}}={\mathbb{Q}}$ (see, e.g., [Mo1, Mo2]). Hence, the restricted
action of ${\rm GL}(H_{{\mathbb{K}}})$ on $\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})=\oplus_{j=1}^{k-1}\mathop{\mathrm{Hom}}\nolimits(H_{{\mathbb{K}}},\mathcal{P}(\hat{T}_{j}/\hat{T}_{j+1}))$
can be regarded as a subrepresentation of
$\oplus_{j=2}^{k-1}H_{{\mathbb{K}}}^{*}\otimes H^{\otimes j}_{{\mathbb{K}}}$.

###### Proof of Theorem 2.2.

Let $\Phi\in{\rm Aut}_{\rm Lie}(\mathcal{L}/\mathcal{L}_{k})\cong{\rm
Aut}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$, and let ${\mathbb{K}}$ be
${\mathbb{C}}$. If $p_{k}(\Phi)$ is the identity matrix, then $\phi$ is the
zero matrix and $\Phi$ lies in $\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{m})$.
Since any contractible nilpotent Lie group is exponential solvable as
mentioned in Section 5, so is $\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{m})$,
by Lemma 5.1. Hence, it is enough to define $D_{\Phi}$ to be
$\exp^{-1}(\Phi)$.

Next, we suppose that $p_{k}(\Phi)$ is not the identity. Let $K$ be the
1-dimensional Lie subgroup of $\mathrm{GL}(H_{{\mathbb{C}}})$ generated by
$p_{k}(\Phi)$, that is, $K=\langle\exp(\phi)\rangle$. From the definition of
the exponential solvability of $p_{k}(\Phi)$, $K$ is diffeomorphic to
${\mathbb{R}}$ in $\mathrm{GL}(H_{{\mathbb{C}}}).$ The restricted action from
the semi-direct product (4) is regarded as a representation
$\mathfrak{s}:K\rightarrow\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\subset{\rm GL}(\hat{T}/\hat{T}_{k})$, which the
above discussion shows is a subrepresentation of ${\mathbb{R}}$ on
$\oplus_{j=2}^{k-1}H_{{\mathbb{C}}}^{*}\otimes H^{\otimes j}_{{\mathbb{C}}}$.
Notice from the decomposition (11) that any eigenvalue of
$\mathfrak{s}(\exp(\phi))$ lies in $\mathrm{Eig}(p_{k}(\Phi))$, and,
therefore, does not meet the set $\\{b\in{\mathbb{C}}^{\times}\mid b\neq
1,|b|=1\\}$ under the assumption of $p_{k}(\Phi)$. From Remark 5.3, the action
of $K$ on the Lie algebra of $\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$
satisfies the condition in Lemma 5.2; thus, $\mathrm{IA}_{\rm
Hopf}(\hat{T}/\hat{T}_{k})\rtimes K$ is an exponential solvable Lie group and
contains $\Phi$. Hence, if we define $D_{\Phi}$ to be $\exp^{-1}(\Phi)$, then
$D_{\Phi}$ satisfies the required conditions $\exp(D_{\Phi})=\Phi$ and
$p_{k}(D_{\Phi})=\phi$ by construction. Finally, uniqueness follows from the
bijectivity of $\exp$. ∎

###### Proof of Theorem 2.7.

If we replace ${\rm Aut}_{\rm Hopf}$ by ${\rm Aut}_{\rm alg}$ in the above
proof, the same discussion runs. Thus, we omit the details. ∎

Next, to prove Theorem 2.5, we review some of the results from [Mo2]. Let
$\mathrm{rk}(\mathcal{L})=2g$, and
$\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$ be ${\rm
Ker}(p_{k})\cap\mathrm{Aut}_{\rm
Lie}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$. Consider the kernel of
the bracketing:

$\mathfrak{h}_{g,1}^{{\mathbb{K}}}(k):=\mathop{\mathrm{Ker}}\nolimits([,]:H_{{\mathbb{K}}}\otimes_{{\mathbb{Q}}}(\mathcal{L}_{k}/\mathcal{L}_{k+1})\longrightarrow(\mathcal{L}_{k+1}/\mathcal{L}_{k+2})\otimes_{{\mathbb{Q}}}{\mathbb{K}}),$
(13)

Under the identification of
$\mathop{\mathrm{Hom}}\nolimits(H_{{\mathbb{K}}},\mathcal{L}_{k}/\mathcal{L}_{k+1}\otimes{\mathbb{K}})$
and $H_{{\mathbb{K}}}\otimes_{{\mathbb{Q}}}\mathcal{L}_{k}/\mathcal{L}_{k+1}$
by the Poincaré duality $H_{{\mathbb{K}}}\cong H_{{\mathbb{K}}}^{*}$, as shown
in [Mo2, (3)], if ${\mathbb{K}}={\mathbb{Q}}$, then the intersection
$\mathrm{Im}(E)\cap{\rm
SpAut}_{0}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{K}})$ is equal to the
$\mathfrak{h}_{g,1}^{{\mathbb{K}}}(k)$. Further, equality holds over
${\mathbb{K}}={\mathbb{C}}$, since
$\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{Q}})\subset\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{R}})$
is dense, and
$\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{R}})\subset\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k+1}\otimes{\mathbb{C}})$
is a canonical complexification. Meanwhile, any
$f\in\mathop{\mathrm{Hom}}\nolimits(H_{{\mathbb{K}}},\mathcal{L}_{k}\otimes{\mathbb{K}})\cong
H_{{\mathbb{K}}}\otimes(\mathcal{L}_{k}\otimes{\mathbb{K}})$ can be uniquely
extended to a derivation $D_{f}$ in
$\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$; we have an
injection
$\oplus_{j=0}^{k-1}\mathfrak{h}_{g,1}^{{\mathbb{K}}}(j)\hookrightarrow\mathrm{Der}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$
that sends $f$ to $D_{f}$, whose image is a Lie subalgebra. According to [Mo2,
Theorem 3.3], if ${\mathbb{K}}={\mathbb{Q}}$, the Lie algebras of ${\rm
Aut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$ and ${\rm
Aut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})\cap\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{K}})$
are isomorphic to the subalgebras
$\oplus_{j=0}^{k-1}\mathfrak{h}_{g,1}^{{\mathbb{K}}}(j)$ and
$\oplus_{j=1}^{k-1}\mathfrak{h}_{g,1}^{{\mathbb{K}}}(j)$, respectively. The
original statement is true only over ${\mathbb{Q}}$; however, we can similarly
verify that the isomorphisms hold over ${\mathbb{R}}$ and ${\mathbb{C}}$.

###### Proof of Theorem 2.5.

Since $\mathfrak{h}_{g,1}^{{\mathbb{C}}}(k)=\mathrm{Im}(E)\cap{\rm
SpAut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$ as above, the
extension (10) means that
$\mathrm{IA}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})\cap{\rm
SpAut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$ is contractible.
Since it is nilpotent, it is an exponential solvable Lie group. Hence, the
same discussion as in the proof of Theorem 2.2 applies if we replace
$\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$ by
$(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})\cap{\rm
SpAut}_{0}(\mathcal{L}/\mathcal{L}_{k}\otimes{\mathbb{C}})$. ∎

It remains to give the proofs of Lemma 3.4 and Theorem 2.6:

###### Proof of Lemma 3.4.

Let $m\in\mathbb{N}$. We may suppose that $p_{k}(\Phi)$ is not the identity.
As in the above proof, consider the exponential solvable Lie group
$\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{k})\rtimes K$. In general, it is
known (see, e.g., [Ro, Lemma 1.1 or Sections 1.3 and 1.9]) that the BCH
formula in any exponential solvable Lie group $G$ converges for any element of
the Lie algebra $\mathfrak{g}$. Hence, the BCH formula (7) is in the Lie
algebra of $\mathrm{IA}_{\rm Hopf}(\hat{T}/\hat{T}_{k})\rtimes K$ converges
for any $X,Y$. ∎

###### Proof of Theorem 2.6.

Let $\Phi\in{\rm Aut}_{\rm Lie}(\mathcal{L}/\mathcal{L}_{k})\cong{\rm
Aut}_{\rm Hopf}(\hat{T}/\hat{T}_{k})$ such that every eigenvalue of
$p_{k}(\Phi)$ is one. If we can show the existence of $N_{k}\in{\mathbb{N}}$
satisfying $(\Phi-\mathrm{id})^{N_{k}}=0$ with ${\mathbb{K}}={\mathbb{C}}$,
then the logarithm $\mathrm{Log}$ converges, while
$\ln(\Phi)=\mathrm{Log}(\Phi)$ immediately follows from the uniqueness in
Theorem 2.2. Here, the statement in the case $k=\infty$ is due to the inverse
limit of $\ln(\Phi)=\mathrm{Log}(\Phi)$ according to $k\rightarrow\infty.$

As for the proof of the existence of $N_{k}$, let
$\phi\in\mathfrak{gl}(\hat{T}/\hat{T}_{2})$ be
$\mathrm{Log}(p_{k}(\Phi))=-\sum_{i=1}^{n-1}(p_{k}(\Phi)-I_{n})^{i}/i$. Let
$\mathfrak{IA}\subset\mathfrak{gl}(\hat{T}/\hat{T}_{k})$ and
$\mathfrak{k}=\langle\phi\rangle$ be the Lie algebras of the nilpotent Lie
group $\mathrm{IA}_{\rm alg}(\hat{T}/\hat{T}_{k})$ and of $K=\langle
p_{k}(\Phi)\rangle$, respectively. We claim that any
$(Y,\phi)\in\mathfrak{IA}\rtimes\mathfrak{k}$ ensures $n_{Y}\in\mathbb{N}$
such that $(Y,\phi)^{n_{Y}}v=0$ for any $v\in\hat{T}/\hat{T}_{k}$. By the
decomposition (11), if
$d\geq\mathop{\mathrm{dim}}\nolimits\hat{T}/\hat{T}_{k}$, then
$(0,\phi)^{d}(\hat{T}/\hat{T}_{k})=0$. Thus, by the Poincaré-Birkhoff-Witt
theorem, we have that $X\in\mathfrak{IA}$ satisfies $(Y,\phi)^{d}v=Xv$ for any
$v\in\hat{T}/\hat{T}_{k}.$ Notice from Engel’s theorem on the nilpotent
$\mathfrak{IA}$ (or Lemma 6.1) that any $Z\in\mathfrak{IA}$ admits
$n_{Z}\in\mathbb{N}$ such that $Z^{n_{Z}}(\hat{T}/\hat{T}_{k})=0$; hence, to
complete our proof of the claim, we may define $n_{Y}$ to be $n_{Z}d$. Then,
from this claim and Lemma 6.1 below, we can choose an appropriate basis of
$\hat{T}/\hat{T}_{k}$ such that the semi-direct product $\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k})\rtimes K$ is a Lie subgroup of
$U(\hat{T}/\hat{T}_{k})$, where $U(\hat{T}/\hat{T}_{k})$ is the subgroup of
${\rm GL}(\hat{T}/\hat{T}_{k})$ consisting of upper right triangular matrices
with only eigenvalue one. Since $\Phi\in\mathrm{IA}_{\rm
alg}(\hat{T}/\hat{T}_{k})\rtimes K\subset U(\hat{T}/\hat{T}_{k})$, we have
$(\Phi-\mathrm{id})^{\mathop{\mathrm{dim}}\nolimits\hat{T}/\hat{T}_{k}}(\hat{T}/\hat{T}_{k})=0$,
as required. ∎

######  Lemma 6.1 (A corollary of Engel’s theorem. See, e.g., [SW, Corollary
11.11]).

Let $V$ be a real vector space with
$1\leq\mathop{\mathrm{dim}}\nolimits_{{\mathbb{R}}}V<\infty$, and $L$ be a Lie
subalgebra of $\mathfrak{gl}(V)$. If any $X\in L$ admits $n_{X}\in\mathbb{N}$
satisfying $X^{n_{X}}V=0$, then we can choose a basis
$\\{v_{i}\\}_{i=1}^{\mathop{\mathrm{dim}}\nolimits V}$ of $V$ such that $L$ is
contained in $\mathfrak{n}(V)$. Here, $\mathfrak{n}(V)$ is the Lie subalgebra
$\\{(n_{ij})_{1\leq i,j\leq\mathop{\mathrm{dim}}\nolimits
V}\in\mathfrak{gl}(V)=\mathrm{Mat}(\mathop{\mathrm{dim}}\nolimits
V\times\mathop{\mathrm{dim}}\nolimits V;{\mathbb{R}})\mid n_{ij}=0\ \ (i\geq
j)\\}$.

###  6.1 Acknowledgments

The author sincerely expresses his gratitude to Nariya Kawazumi, Yusuke Kuno,
and Masatoshi Sato for giving him valuable comments. The work was partially
supported by JSPS KAKENHI, Grant Number 00646903.

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Department of Mathematics, Tokyo Institute of Technology 2-12-1 Ookayama,
Meguro-ku Tokyo 152-8551 Japan