# Derivations, extensions, and rigidity of subalgebras of the Witt algebra

Lucas Buzaglo

###### Abstract.

Let $\Bbbk$ be an algebraically closed field of characteristic 0. We study
some cohomological properties of Lie subalgebras of the Witt algebra
$W=\operatorname{Der}(\Bbbk[t,t^{-1}])$ and the one-sided Witt algebra
$W_{\geq-1}=\operatorname{Der}(\Bbbk[t])$. In the first part of the paper, we
consider finite codimension subalgebras of $W_{\geq-1}$. We compute
derivations and one-dimensional extensions of such subalgebras. These
correspond to $\operatorname{Ext}_{U(L)}^{1}(M,L)$, where $L$ is a subalgebra
of $W_{\geq-1}$ and $M$ is a one-dimensional representation of $L$. We find
that these subalgebras exhibit a kind of rigidity: their derivations and
extensions are controlled by the full one-sided Witt algebra. As an
application of these computations, we prove that any isomorphism between
finite codimension subalgebras of $W_{\geq-1}$ extends to an automorphism of
$W_{\geq-1}$.

The second part of the paper is devoted to explaining the observed rigidity.
We define a notion of “completely non-split extension” and prove that
$W_{\geq-1}$ is the universal completely non-split extension of any of its
subalgebras of finite codimension. In some sense, this means that even when
studying subalgebras of $W_{\geq-1}$ as abstract Lie algebras, they remember
that they are contained in $W_{\geq-1}$. We also consider subalgebras of
infinite codimension, explaining the similarities and differences between the
finite and infinite codimension situations.

Almost all of the results above are also true for subalgebras of the Witt
algebra. We summarise results for $W$ at the end of the paper.

###### Key words and phrases:

Subalgebras of the Witt algebra, submodule-subalgebra, derivation, one-
dimensional extension, Lie algebra cohomology

###### 2020 Mathematics Subject Classification:

17B40, 17B56, 17B66 (Primary), 17B65, 17B68 (Secondary)

## Introduction

Throughout, we let $\Bbbk$ be an algebraically closed field of characteristic
0. All vector spaces are $\Bbbk$-vector spaces.

Since its inception in the late 1960s [GF68, GF69], the study of cohomology of
infinite-dimensional Lie algebras has had a rich history. The main focus has
been on the Witt algebra and related Lie algebras [Gon73, FF80, DZ98]. Another
class of Lie algebras that has been of particular interest in this context are
_Krichever–Novikov algebras_ [Mil97, Wag99, Wag12] (for the definition and
general theory of Krichever–Novikov algebras, see [Sch14]). Other examples can
be found in [FL96, FM06] and in Fuks’s book on the cohomology of infinite-
dimensional Lie algebras [Fuk86]. The computation of these cohomology groups
is an interesting and challenging problem with connections to various areas of
mathematics and physics, including representation theory and deformation
theory.

In this paper, we compute the first cohomology of some infinite-dimensional
Lie algebras with coefficients in the adjoint representation, which is closely
related to computing derivations of these Lie algebras: for a Lie algebra $L$,
we have

$H^{1}(L;L)\cong\operatorname{Der}(L)/\operatorname{Inn}(L),$

where $\operatorname{Der}(L)$ is the space of derivations of $L$ and
$\operatorname{Inn}(L)$ is the space of inner derivations of $L$. The study of
derivations of Lie algebras dates back to the pioneering works of Jacobson,
Hochschild, Leger, Dixmier, and others [Jac37, Hoc42, Leg53, DL57], and
continues to be an active area of research to this day (see, for example,
[GJP11, FSW13, MSS13, SSM17]).

Let $W_{\geq-1}=\operatorname{Der}(\Bbbk[t])$ be the _one-sided Witt algebra_
and let $W=\operatorname{Der}(\Bbbk[t,t^{-1}])$ be the _Witt algebra_ (or
_centerless Virasoro algebra_). These are infinite-dimensional Lie algebras
which are naturally modules over $\Bbbk[t]$ and $\Bbbk[t,t^{-1}]$,
respectively. We are interested in infinite-dimensional subalgebras of
$W_{\geq-1}$ and $W$. Of particular interest are those subalgebras of
$W_{\geq-1}$ and $W$ which are also $\Bbbk[t]$\- or
$\Bbbk[t,t^{-1}]$-submodules, which we call _submodule-subalgebras_. They are
denoted by $W_{\geq-1}(f)\coloneqq fW_{\geq-1}$ and $W(g)\coloneqq gW$ for
$f\in\Bbbk[t]$ and $g\in\Bbbk[t,t^{-1}]$. All subalgebras of $W_{\geq-1}$ of
finite codimension are “essentially” submodule-subalgebras: if $L$ is such a
subalgebra, then there exists $f\in\Bbbk[t]$ such that $W_{\geq-1}(f)\subseteq
L\subseteq W_{\geq-1}(\operatorname{rad}(f))$, where
$\operatorname{rad}(f)=\prod_{\xi\in V(f)}(t-\xi)$ [PS23, Proposition 3.2.7].
Furthermore, by [BB], any infinite-dimensional subalgebra of $W_{\geq-1}$ is
isomorphic to a subalgebra of finite codimension. We can therefore extract a
lot of information about infinite-dimensional subalgebras of $W_{\geq-1}$ by
first studying submodule-subalgebras.

Of the submodule-subalgebras of $W_{\geq-1}$, those of the form $W_{\geq
n}\coloneqq W_{\geq-1}(t^{n+1})$ for $n\in\mathbb{N}$ are of particular
significance and have been extensively studied. For example, $W_{\geq 1}$ is
sometimes called the _positive Witt algebra_ , which appeared in [Fia83] in
the context of deformation theory, where the cohomology group $H^{2}(W_{\geq
1};W_{\geq 1})$ was computed. Similar results for the Lie algebras $W_{\geq
2}$ and $W_{\geq 3}$ were established in [FP01] and [Koc06].

Submodule-subalgebras of $W_{\geq-1}$ and $W$ form a large and interesting
class of infinite-dimensional Lie algebras of linear growth. More generally,
one could study submodule-subalgebras of $\operatorname{Der}(A)$, where $A$ is
an associative $\Bbbk$-algebra. These first appeared in [Gra78], and
subsequently in [Gra93] and [Sie96]. In particular, Siebert showed that if
$L_{1}$ and $L_{2}$ are submodule-subalgebras of
$\operatorname{Der}(\Bbbk[X_{1}])$ and $\operatorname{Der}(\Bbbk[X_{2}])$,
respectively, where $X_{1}$ and $X_{2}$ are irreducible affine varieties, then
$X_{1}$ and $X_{2}$ are isomorphic if $L_{1}$ and $L_{2}$ are isomorphic,
provided $L_{1}$ and $L_{2}$ satisfy certain additional properties [Sie96,
Theorem 3].

We start off by computing derivations of infinite-dimensional subalgebras of
$W_{\geq-1}$.

######  Theorem 0.1 (Theorems 2.2 and 6.7).

Let $L$ be an infinite-dimensional subalgebra of $W_{\geq-1}$. Then

$\operatorname{Der}(L)=\\{\operatorname{ad}_{w}\mid w\in W_{\geq-1}\text{ such
that }\operatorname{ad}_{w}(L)\subseteq L\\}.$

In other words, derivations of $L$ are restrictions of derivations of
$W_{\geq-1}$. Consequently, $H^{1}(L;L)\cong N_{W_{\geq-1}}(L)/L$, where
$N_{W_{\geq-1}}(L)$ is the normaliser of $L$ in $W_{\geq-1}$.

In particular,

$\operatorname{Der}(W_{\geq-1}(f))=\\{\operatorname{ad}_{w}\mid w\in
W_{\geq-1}(\operatorname{rad}(f))\\}\cong W_{\geq-1}(\operatorname{rad}(f))$

for $f\in\Bbbk[t]$, where $\operatorname{rad}(f)=\prod_{\xi\in V(f)}(t-\xi)$.
As a result,

$\dim H^{1}(W_{\geq-1}(f);W_{\geq-1}(f))=\deg(f)-\deg(\operatorname{rad}(f)).$

For a Lie algebra $L$, we have
$H^{1}(L;L)=\operatorname{Ext}_{U(L)}^{1}(\Bbbk,L)$, where $\Bbbk$ is the one-
dimensional trivial representation of $L$. We can generalise Theorem 0.1 by
computing

$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f)),$

where $f\in\Bbbk[t]$ and $M$ is an arbitrary one-dimensional representation of
$W_{\geq-1}(f)$. An element of $\operatorname{Ext}_{U(L)}^{1}(M,L)$ can be
viewed as a non-split short exact sequence of $L$-representations

$0\to L\to\overline{L}\to M\to 0.$

We study such short exact sequences in Section 3 for $L$ a subalgebra of
$W_{\geq-1}$ of finite codimension. In particular, we prove that
$\overline{L}$ can be canonically embedded in $W_{\geq-1}$ (see Proposition
3.7).

The above results suggest that subalgebras of $W_{\geq-1}$ of finite
codimension are “rigid”, in the sense that they inherit their properties from
$W_{\geq-1}$. We explain this by showing that $W_{\geq-1}$ can be
intrinsically built from any of its subalgebras of finite codimension as the
universal _completely non-split extension_ , which is an abstract Lie-
algebraic property (see Definition 5.1).

######  Theorem 0.2 (Theorem 5.3).

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then
$W_{\geq-1}$ is the universal completely non-split extension of $L$, in the
following sense: $W_{\geq-1}$ is a completely non-split extension of $L$, and
if $\overline{L}$ is another completely non-split extension of $L$, then
$\overline{L}$ can be uniquely embedded in $W_{\geq-1}$ such that the diagram

${L}$${W_{\geq-1}}$${\overline{L}}$

commutes.

Given Siebert’s result, the rigidity noted above, and the close relationship
between subalgebras of $W_{\geq-1}$ of finite codimension and submodule-
subalgebras, it is natural to ask whether any isomorphism between subalgebras
of $W_{\geq-1}$ of finite codimension extends to an automorphism of
$W_{\geq-1}$. If we knew of this universal property in advance, this would be
immediate. In fact, we do prove this on the way to proving the universal
property.

######  Theorem 0.3 (Theorem 4.1).

Let $L_{1}$ and $L_{2}$ be subalgebras of $W_{\geq-1}$ of finite codimension
and suppose there is an isomorphism of Lie algebras $\varphi\colon L_{1}\to
L_{2}$. Then $\varphi$ extends to an automorphism of $W_{\geq-1}$.

The idea of the proof of Theorem 0.3 is as follows: let $\varphi\colon
L_{1}\to L_{2}$ be an isomorphism between proper subalgebras of $W_{\geq-1}$
of finite codimension. We show that there exist subalgebras $\overline{L}_{i}$
of $W_{\geq-1}$ with $L_{i}\subseteq\overline{L}_{i}\subseteq W_{\geq-1}$ and
$\dim(\overline{L}_{i}/L_{i})=1$ for $i=1,2$ such that $\varphi$ extends to an
isomorphism $\overline{\varphi}\colon\overline{L}_{1}\to\overline{L}_{2}$. The
proof then follows by induction on $\operatorname{codim}_{W_{\geq-1}}(L_{1})$.

In order to prove the existence of the isomorphism
$\overline{\varphi}\colon\overline{L}_{1}\to\overline{L}_{2}$, we apply our
study of non-split short exact sequences of the form

$0\to L\to\overline{L}\to\overline{L}/L\to 0$

where $L$ is a subalgebra of $W_{\geq-1}$ of finite codimension and
$\dim(\overline{L}/L)=1$. In other words, Theorem 0.3 follows from the
computation of $\operatorname{Ext}_{U(L)}^{1}(M,L)$, where $M$ is a one-
dimensional representation of $L$.

We now explain the structure and proofs of the paper in more detail. There are
two main goals: the first is to compute
$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f))$, where $f$ and
$M$ are as above, and the second is to provide an explanation for the observed
rigidity.

Sections 1 and 2 are devoted to computing derivations of $W_{\geq-1}(f)$ for
$f\in\Bbbk[t]$, which corresponds to the case where $M=\Bbbk$ is the trivial
representation. This is because

$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(\Bbbk,W_{\geq-1}(f))=H^{1}(W_{\geq-1}(f);W_{\geq-1}(f))\cong\operatorname{Der}(W_{\geq-1}(f))/\operatorname{Inn}(W_{\geq-1}(f))$

is the first cohomology of $W_{\geq-1}(f)$ with coefficients in the adjoint
representation.

It is easy to show that all derivations of $W_{\geq-1}$ are inner. Given the
rigidity of the situation, it is natural to ask if derivations of submodule-
subalgebras of $W_{\geq-1}$ also arise from the adjoint action of elements of
$W_{\geq-1}$, that is, if

$\operatorname{Der}(W_{\geq-1}(f))=\\{\operatorname{ad}_{w}\mid w\in
W_{\geq-1}\text{ and }\operatorname{ad}_{w}(W_{\geq-1}(f))\subseteq
W_{\geq-1}(f)\\},$

where $f\in\Bbbk[t]$. It is easy to see that
$\operatorname{ad}_{w}(W_{\geq-1}(f))\subseteq W_{\geq-1}(f)$ if and only if
$w\in W_{\geq-1}(\operatorname{rad}(f))$, where
$\operatorname{rad}(f)=\prod\\{t-\xi\mid f(\xi)=0\\}$. Therefore, the question
is equivalent to asking whether
$\operatorname{Der}(W_{\geq-1}(f))=\\{\operatorname{ad}_{w}\mid w\in
W_{\geq-1}(\operatorname{rad}(f))\\}$, which is precisely what Theorem 0.1
says.

As the Lie algebra $W_{\geq-1}$ is graded, the proof of Theorem 0.1 starts by
considering derivations of graded submodule-subalgebras, that is, Lie algebras
of the form $W_{\geq n}$ for $n\in\mathbb{N}$. For this situation, we can use
results from [Far88]. In particular, the space of derivations of a graded Lie
algebra is itself graded, which greatly simplifies our computations. However,
computing derivations of ungraded submodule-subalgebras requires more work.
Since submodule-subalgebras of $W_{\geq-1}$ are filtered, this is achieved by
an associated graded argument, allowing us to compute derivations of ungraded
submodule-subalgebras from derivations of graded ones.

In Section 3, we move on to computing
$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f))$, where
$f\in\Bbbk[t]$ and $M$ is an arbitrary one-dimensional representation of
$W_{\geq-1}(f)$. A key observation is that the derived subalgebra of
$W_{\geq-1}(f)$ is $W_{\geq-1}(f^{2})$, so the action of $W_{\geq-1}(f^{2})$
on any one-dimensional representation of $W_{\geq-1}(f)$ is trivial.
Consequently, there are many one-dimensional representations of
$W_{\geq-1}(f)$, corresponding to the representations of the abelian Lie
algebra $W_{\geq-1}(f)/W_{\geq-1}(f^{2})$. In other words, the one-dimensional
representations of $W_{\geq-1}(f)$ are parametrised by $\mathbb{A}^{\deg(f)}$.
However, the only one-dimensional modules $M$ which give rise to non-split
extensions

$0\to W_{\geq-1}(f)\to L\to M\to 0$

are of the form $M=\Bbbk$ or $M=W_{\geq-1}/W_{\geq-1}(t-\xi)$ for $\xi\in
V(f)$ a root of order 1. The $U(W_{\geq-1}(f))$-module structure on
$W_{\geq-1}/W_{\geq-1}(t-\xi)$ is induced from $W_{\geq-1}(f)$ acting by its
adjoint action on both $W_{\geq-1}$ and $W_{\geq-1}(t-\xi)$, noting that
$W_{\geq-1}(f)$ is a Lie subalgebra of both $W_{\geq-1}$ and
$W_{\geq-1}(t-\xi)$.

######  Theorem 0.4 (Theorem 3.3).

Let $f\in\Bbbk[t]$. If $M\cong W_{\geq-1}/W_{\geq-1}(t-\xi)$ for some
$\xi\in\Bbbk$ a root of $f$ of multiplicity 1, then

$\dim(\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(W_{\geq-1}/W_{\geq-1}(t-\xi),W_{\geq-1}(f)))=1.$

In this case, the unique non-split extension of $W_{\geq-1}/W_{\geq-1}(t-\xi)$
by $W_{\geq-1}(f)$ is $W_{\geq-1}(\frac{f}{t-\xi})$. If $M\cong\Bbbk$ is
trivial, we have

$\dim(\operatorname{Ext}_{U(W_{\geq-1}(f)}^{1}(\Bbbk,W_{\geq-1}(f)))=\deg(f)-\deg(\operatorname{rad}(f)).$

Otherwise, $\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f))=0$.

Although it is not immediately obvious, the proof is closely related to the
computation of derivations of submodule-subalgebras: we consider a short exact
sequence of $U(W_{\geq-1}(f))$-modules

$0\to W_{\geq-1}(f)\to L\to M\to 0$

as a sequence of $U(W_{\geq-1}(f^{2}))$-modules. Since $M$ is trivial as a
representation of $W_{\geq-1}(f^{2})$, the short exact sequence gives an
element of

$\operatorname{Ext}_{U(W_{\geq-1}(f^{2}))}^{1}(\Bbbk,W_{\geq-1}(f))\cong\operatorname{Der}(W_{\geq-1}(f^{2}),W_{\geq-1}(f))/\operatorname{Inn}(W_{\geq-1}(f^{2}),W_{\geq-1}(f)),$

which can be computed using the techniques from Theorem 0.1.

In Section 5, we prove Theorem 0.2, providing an explanation for the
aforementioned rigidity. We first show that $W_{\geq-1}$ is a completely non-
split extension of any of its subalgebras of finite codimension, which we
achieve by proving that any such subalgebra of $W_{\geq-1}$ has a one-
dimensional extension contained in $W_{\geq-1}$. We then prove the universal
property using results from Section 3 about one-dimensional extensions of
subalgebras of $W_{\geq-1}$.

We then briefly consider subalgebras of $W_{\geq-1}$ of infinite codimension.
Any infinite-dimensional subalgebra of $W_{\geq-1}$ is isomorphic to a
subalgebra of $W_{\geq-1}$ of finite codimension [BB], so our results in
finite codimension can be translated to infinite codimension. However, there
are also some key differences when it comes to the universal property, which
we highlight at the end of Section 5.

We conclude the paper by establishing analogous results to Theorems 0.1–0.4
for submodule-subalgebras of the Witt algebra. The proofs for submodule-
subalgebras of $W_{\geq-1}$ go through for $W$ with the obvious changes. In
fact, the situation for the Witt algebra is slightly easier: the associated
graded algebra to $W(f)$ is $W$, where $f\in\Bbbk[t,t^{-1}]$. Consequently, in
order to use the associated graded argument to compute derivations of $W(f)$,
we only need to consider derivations of $W$. Thanks to results from [Far88],
it is easy to see that $\operatorname{Der}(W)=\operatorname{Inn}(W)$.

Acknowledgements: This work was done as part of the author’s PhD research at
the University of Edinburgh.

Proposition 4.13 is part of a collaboration with Jason Bell carried out during
the author’s visit to the University of Waterloo. We are grateful to Prof.
Bell for allowing us to include it.

##  1\. Derivations of graded submodule-subalgebras

We begin by recalling the notions of derivations of algebras, and defining the
Lie algebras of interest in this paper.

######  Definition 1.1.

A _derivation_ of a $\Bbbk$-algebra $A$ is a $\Bbbk$-linear map $d\colon A\to
A$ such that

$d(ab)=ad(b)+d(a)b$

for $a,b\in A$. We write $\operatorname{Der}(A)$ for the set of derivations of
$A$.

We write $W_{\geq-1}=\operatorname{Der}(\Bbbk[t])=\Bbbk[t]\partial$ and
$W=\operatorname{Der}(\Bbbk[t,t^{-1}])=\Bbbk[t,t^{-1}]\partial$ for the _one-
sided Witt algebra_ and the _Witt algebra_ , respectively, where
$\partial=\frac{d}{dt}$. The Witt algebra is sometimes called the _centerless
Virasoro algebra_.

We let $e_{n}=t^{n+1}\partial\in W$ for $n\in\mathbb{Z}$. The Lie bracket is
given by

$[f\partial,g\partial]=(fg^{\prime}-f^{\prime}g)\partial,\quad[e_{n},e_{m}]=(m-n)e_{n+m},$

where $f,g\in\Bbbk[t,t^{-1}]$ and $n,m\in\mathbb{Z}$. Note that $W_{\geq-1}$
is spanned by $\\{e_{n}\mid n\geq-1\\}$.

For $f\in\Bbbk[t]$, we write $W_{\geq-1}(f)=fW_{\geq-1}=f\Bbbk[t]\partial$
under the obvious notation. It is clear that $W_{\geq-1}(f)$ is a subalgebra
of $W_{\geq-1}$, as well as a $\Bbbk[t]$-submodule. We call $W_{\geq-1}(f)$ a
_submodule-subalgebra_ of $W_{\geq-1}$. We write $W_{\geq n}$ for
$W_{\geq-1}(t^{n+1})$.

We will also consider the Lie algebra
$\operatorname{Der}(\Bbbk(t))=\Bbbk(t)\partial$, and view $W_{\geq-1}$ and $W$
as subalgebras of $\Bbbk(t)\partial$.

Our main goal in the first part of the paper is to compute
$\operatorname{Ext}^{1}_{U(W_{\geq-1}(f))}(M,W_{\geq-1}(f))$, where
$f\in\Bbbk[t]\setminus\\{0\\}$ and $M$ is a one-dimensional
$U(W_{\geq-1}(f))$-module. We devote the first two sections to studying the
easiest case, where $M=\Bbbk$ is the trivial one-dimensional representation of
$W_{\geq-1}(f)$. In this case,
$\operatorname{Ext}^{1}_{U(W_{\geq-1}(f))}(\Bbbk,W_{\geq-1}(f))$ is a well-
known object.

######  Definition 1.2.

For a Lie algebra $L$ and a representation $M$ of $L$, the _cohomology of $L$
with values in $M$_, denoted $H^{n}(L;M)$ for $n\in\mathbb{N}$, is defined as

$H^{n}(L;M)=\operatorname{Ext}^{n}_{U(L)}(\Bbbk,M).$

We are therefore interested in computing $H^{1}(W_{\geq-1}(f);W_{\geq-1}(f))$
for $f\in\Bbbk[t]\setminus\\{0\\}$. It is a standard fact that, for a Lie
algebra $L$ and a representation $M$ of $L$, the cohomology group $H^{1}(L;M)$
is isomorphic to the quotient
$\operatorname{Der}(L,M)/\operatorname{Inn}(L,M)$, defined below.

######  Definition 1.3.

A _derivation_ of a Lie algebra $L$ with values in an $L$-module $M$ is a
linear map $d\colon L\to M$ such that

$d([x,y])=x\cdot d(y)-y\cdot d(x)$

for $x,y\in L$. We write $\operatorname{Der}(L,M)$ for the set of derivations
$L$ with values in $M$, and simply write $\operatorname{Der}(L)$ instead of
$\operatorname{Der}(L,L)$. A derivation $d\in\operatorname{Der}(L,M)$ is
_inner_ if there exists $m\in M$ such that $d(x)=x\cdot m$. We write
$\operatorname{Inn}(L,M)$ for the set of inner derivations of $L$ with values
in $M$. As before, we write $\operatorname{Inn}(L)$ instead of
$\operatorname{Inn}(L,L)$.

If $L=\bigoplus_{n\in\mathbb{Z}}L_{n}$ is a $\mathbb{Z}$-graded Lie algebra
and $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ is a $\mathbb{Z}$-graded
representation of $L$, then for $k\in\mathbb{Z}$ we write

$\operatorname{Der}(L,M)_{k}=\\{d\in\operatorname{Der}(L,M)\mid
d(L_{n})\subseteq M_{n+k}\text{ for all }n\in\mathbb{Z}\\}.$

If $d\in\operatorname{Der}(L,M)_{k}$, we say that $d$ is a _graded derivation
of degree $k$_.

If $L$ is a Lie algebra and $M$ is a representation of $L$, we can construct
elements of $\operatorname{Ext}_{U(L)}^{1}(\Bbbk,M)$ from derivations of $L$
with values in $M$ as follows: given a derivation
$d\in\operatorname{Der}(L,M)$, consider the vector space $X=M\oplus\Bbbk d$
with $L$-action given by the usual action of $L$ on $M$ and $w\cdot d=-d(w)$
for $w\in L$. Note that $X/M$ is trivial as a representation of $L$, so we get
a short exact sequence of representations of $L$ given by

(1.3.1) $0\to M\to X\to\Bbbk\to 0.$

Furthermore, we easily see that (1.3.1) splits if and only if
$d\in\operatorname{Inn}(L,M)$.

We therefore see that computing
$H^{1}(W_{\geq-1}(f);W_{\geq-1}(f))=\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(\Bbbk,W_{\geq-1}(f))$
is equivalent to computing derivations of $W_{\geq-1}(f)$. Thanks to the
following result of Farnsteiner, studying derivations of graded Lie algebras
is easier than ungraded ones, since we can exploit the graded structure of the
space of derivations.

######  Proposition 1.4 ([Far88, Proposition 1.1]).

If $L=\bigoplus_{n\in\mathbb{Z}}L_{n}$ is a $\mathbb{Z}$-graded finitely
generated Lie algebra and $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ is a
$\mathbb{Z}$-graded representation of $L$, then

$\operatorname{Der}(L,M)=\bigoplus_{n\in\mathbb{Z}}\operatorname{Der}(L,M)_{n}.$

The Lie algebras $W$ and $W_{\geq-1}$ are $\mathbb{Z}$-graded, with $e_{n}$
having degree $n$. However, submodule-subalgebras of $W_{\geq-1}$ are not
graded in general: the graded ones are $W_{\geq n}$ for $n\geq-1$. Still,
ungraded submodule-subalgebras are filtered, with associated graded algebras
isomorphic to subalgebras of the form $W_{\geq n}$. It is therefore natural to
begin our study of derivations of submodule-subalgebras of $W_{\geq-1}$ by
considering the graded ones, and later attempt to extract as much information
as we can for general submodule-subalgebras via an associated graded argument.

The goal for this section is to compute $\operatorname{Der}(W_{\geq n},W)$ for
$n\in\mathbb{N}$, where we view $W$ as a representation of $W_{\geq-1}$ under
the adjoint action. We start with the cases $n=-1,0,1$. The computation of
derivations of $W_{\geq-1}$ was already a well-known folklore result, but we
still give a proof here. However, derivations of $W_{\geq n}$ for arbitrary
$n$ were not known before.

######  Proposition 1.5.

We have $\operatorname{Der}(W_{\geq-1},W)=\operatorname{Inn}(W_{\geq-1},W)$.
Consequently,
$\operatorname{Der}(W_{\geq-1})=\operatorname{Inn}(W_{\geq-1})\cong
W_{\geq-1}$ and $H^{1}(W_{\geq-1};W_{\geq-1})=0$.

We will use a theorem of Farnsteiner to prove Proposition 1.5.

######  Theorem 1.6 ([Far88, Proposition 1.2]).

Let $L=\bigoplus_{n\in\mathbb{Z}}L_{n}$ be a $\mathbb{Z}$-graded finitely
generated Lie algebra and let $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ be a graded
representation of $L$ such that

  1. (1)

$H^{1}(L_{0};M_{n})=0$ for all $n\in\mathbb{Z}\setminus\\{0\\}$.

  2. (2)

$\operatorname{Hom}_{L_{0}}(L_{n},M_{m})=0$ for $n\neq m$.

Then
$\operatorname{Der}(L,M)=\operatorname{Der}(L,M)_{0}+\operatorname{Inn}(L,M)$.

In order to compute $\operatorname{Der}(W_{\geq-1},W)$, we show that
$\operatorname{Der}(W_{\geq-1},W)_{0}\subseteq\operatorname{Inn}(W_{\geq-1},W)$.
It will then be clear that
$\operatorname{Der}(W_{\geq-1},W)=\operatorname{Inn}(W_{\geq-1},W)$, provided
the conditions of Theorem 1.6 are satisfied.

######  Lemma 1.7.

Let $n\geq-1$. We have $\operatorname{Der}(W_{\geq
n},W)_{0}=\Bbbk\operatorname{ad}_{e_{0}}\subseteq\operatorname{Inn}(W_{\geq-1},W)$.

###### Proof.

Let $d\in\operatorname{Der}(W_{\geq-1},W)_{0}$. Then for all $n\geq-1$ there
exists $\lambda_{n}\in\Bbbk$ such that $d(e_{n})=\lambda_{n}e_{n}$. We have

$d([e_{n},e_{m}])=[e_{n},d(e_{m})]+[d(e_{n}),e_{m}]=(\lambda_{n}+\lambda_{m})[e_{n},e_{m}]=(m-n)(\lambda_{n}+\lambda_{m})e_{n+m}.$

On the other hand,

$d([e_{n},e_{m}])=(m-n)d(e_{n+m})=(m-n)\lambda_{n+m}e_{n+m}.$

Combining the above, we conclude that $\lambda_{n}+\lambda_{m}=\lambda_{n+m}$
for all $n\neq m$. Now, $\lambda_{0}+\lambda_{1}=\lambda_{1}$, so
$\lambda_{0}=0$. Furthermore, $\lambda_{-1}+\lambda_{1}=\lambda_{0}=0$, so
$\lambda_{-1}=-\lambda_{1}$.

Now consider $2\lambda_{1}+\lambda_{3}$:

$2\lambda_{1}+\lambda_{3}=\lambda_{1}+\lambda_{4}=\lambda_{5}=\lambda_{2}+\lambda_{3}.$

Hence, $\lambda_{2}=2\lambda_{1}$. Inductively, we deduce that
$n\lambda_{1}=\lambda_{n}$ for all $n\geq-1$. Letting $\lambda=\lambda_{1}$,
it follows that
$d=\lambda\operatorname{ad}_{e_{0}}\in\operatorname{Inn}(W_{\geq-1},W)$. ∎

###### Proof of Proposition 1.5.

By Lemma 1.7, it suffices to check the conditions for Theorem 1.6 are
satisfied when $L=W_{\geq-1}$ and $M=W$.

First, we must show that

$H^{1}(\Bbbk e_{0};\Bbbk e_{n})\cong\operatorname{Der}(\Bbbk e_{0},\Bbbk
e_{n})/\operatorname{Inn}(\Bbbk e_{0},\Bbbk e_{n})=0$

for all $n\in\mathbb{Z}\setminus\\{0\\}$. Let
$n\in\mathbb{Z}\setminus\\{0\\}$. We have

$\operatorname{Inn}(\Bbbk e_{0},\Bbbk
e_{n})=\Bbbk\operatorname{ad}_{e_{n}}\subseteq\operatorname{Der}(\Bbbk
e_{0},\Bbbk e_{n})\subseteq\operatorname{Hom}_{\Bbbk}(\Bbbk e_{0},\Bbbk
e_{n}).$

By a dimension count, $H^{1}(\Bbbk e_{0};\Bbbk e_{n})=0$.

To end the proof, we can easily see that $\operatorname{Hom}_{\Bbbk
e_{0}}(\Bbbk e_{n},\Bbbk e_{m})=0$ for $n\neq m$, where
$n\in\mathbb{Z}_{\geq-1}$ and $m\in\mathbb{Z}$, since $e_{n}$ and $e_{m}$ have
different eigenvalues with respect to the action of $e_{0}$. ∎

The computation of derivations of $W_{\geq 0}$ is almost identical to
derivations of $W_{\geq-1}$. We therefore omit the proof.

######  Proposition 1.8.

We have $\operatorname{Der}(W_{\geq 0},W)=\operatorname{Inn}(W_{\geq 0},W)$.
Consequently, $\operatorname{Der}(W_{\geq 0})=\operatorname{Inn}(W_{\geq
0})\cong W_{\geq 0}$ and $H^{1}(W_{\geq 0};W_{\geq 0})=0$. ∎

Proposition 1.8 shows that $\operatorname{Ext}_{U(W_{\geq
0})}^{1}(\Bbbk,W_{\geq 0})=0$. However, $W_{\geq 0}$ does have nontrivial
extensions by one-dimensional representations: consider the short exact
sequence

(1.8.1) $0\to W_{\geq 0}\to W_{\geq-1}\to W_{\geq-1}/W_{\geq 0}\to 0.$

This is because the $U(W_{\geq 0})$-module $W_{\geq-1}/W_{\geq 0}$ is not the
trivial one-dimensional module, since $e_{0}$ acts as $-\operatorname{id}$. In
fact, we will see later that $\dim(\operatorname{Ext}_{U(W_{\geq
0})}^{1}(W_{\geq-1}/W_{\geq 0},W_{\geq 0}))=1$ and that (1.8.1) is the unique
nontrivial one-dimensional extension of $W_{\geq 0}$ (see Theorem 3.3).

Computing $\operatorname{Der}(W_{\geq 1},W)$ requires more work, since
$e_{0}\not\in W_{\geq 1}$, so we cannot use Theorem 1.6. Instead, we can use
the graded structure of $\operatorname{Der}(W_{\geq 1})$ from Proposition 1.4.

######  Proposition 1.9.

We have $\operatorname{Der}(W_{\geq 1},W)=\operatorname{Inn}(W_{\geq 1},W)$.
Consequently, $\operatorname{Der}(W_{\geq 1})=\\{\operatorname{ad}_{w}\mid
w\in W_{\geq 0}\\}\cong W_{\geq 0}$ and $\dim(H^{1}(W_{\geq 1};W_{\geq
1}))=1$.

###### Proof.

By Proposition 1.4, it suffices to prove that $\operatorname{Der}(W_{\geq
1},W)_{n}\subseteq\operatorname{Inn}(W_{\geq 1},W)$ for $n\in\mathbb{Z}$.

The same proof as in Lemma 1.7 works to show that $\operatorname{Der}(W_{\geq
1},W)_{0}=\Bbbk\operatorname{ad}_{e_{0}}$.

We start by considering $\operatorname{Der}(W_{\geq 1},W)_{1}$. Note that,
since $W_{\geq 1}$ is generated by $e_{1}$ and $e_{2}$ as a Lie algebra, any
derivation $d\in\operatorname{Der}(W_{\geq 1},W)$ is uniquely determined by
$d(e_{1})$ and $d(e_{2})$. Let $d\in\operatorname{Der}(W_{\geq 1},W)_{1}$, so
$d(e_{1})=\lambda e_{2}$ and $d(e_{2})=\mu e_{3}$ for some
$\lambda,\mu\in\Bbbk$. Let
$d^{\prime}=d-\mu\operatorname{ad}_{e_{1}}\in\operatorname{Der}(W_{\geq
1},W)$, so that $d^{\prime}(e_{1})=\lambda e_{2}$ and $d^{\prime}(e_{2})=0$.
We have

$\displaystyle d^{\prime}(e_{3})$
$\displaystyle=d^{\prime}([e_{1},e_{2}])=[e_{1},d^{\prime}(e_{2})]+[d^{\prime}(e_{1}),e_{2}]=\lambda[e_{2},e_{2}]=0,$
$\displaystyle d^{\prime}(e_{4})$
$\displaystyle=\frac{1}{2}d^{\prime}([e_{1},e_{3}])=\frac{1}{2}([e_{1},d^{\prime}(e_{3})]+[d^{\prime}(e_{1}),e_{3}])=\frac{1}{2}\lambda[e_{2},e_{3}]=\frac{1}{2}\lambda
e_{5},$ $\displaystyle d^{\prime}(e_{5})$
$\displaystyle=\frac{1}{3}d^{\prime}([e_{1},e_{4}])=\frac{1}{3}([e_{1},d^{\prime}(e_{4})]+[d^{\prime}(e_{1}),e_{4}])=\frac{1}{3}(\frac{1}{2}\lambda[e_{1},e_{5}]+\lambda[e_{2},e_{4}])=\frac{4}{3}\lambda
e_{6}.$

On the other hand,

$d^{\prime}(e_{5})=d^{\prime}([e_{2},e_{3}])=[e_{2},d^{\prime}(e_{3})]+[d^{\prime}(e_{2}),e_{3}]=0.$

Therefore, $\lambda=0$, so $d^{\prime}=0$. It follows that
$d=\mu\operatorname{ad}_{e_{1}}$, so $\operatorname{Der}(W_{\geq
1},W)_{1}\subseteq\operatorname{Inn}(W_{\geq 1},W)$.

Now let $d\in\operatorname{Der}(W_{\geq 1},W)_{2}$, so $d(e_{1})=\lambda
e_{3}$ and $d(e_{2})=\mu e_{4}$ for some $\lambda,\mu\in\Bbbk$. Consider the
subalgebra $V_{2}$ of $W_{\geq-1}$ spanned by $e_{2},e_{4},e_{6},\ldots$.
Certainly, $d$ restricts to a derivation of $V_{2}$. Note that $V_{2}$ is
isomorphic to $W_{\geq 1}$ via the map $e_{2n}\mapsto 2e_{n}$. Under this
isomorphism, $d$ induces a derivation $\widetilde{d}$ of $W_{\geq 1}$ of
degree 1. By the above computation of $\operatorname{Der}(W_{\geq 1},W)_{1}$,
we know that $\widetilde{d}$ is a scalar multiple of
$\operatorname{ad}_{e_{1}}$, and therefore
${\left.\kern-1.2ptd\vphantom{\big{|}}\right|_{V_{2}}}$ is a scalar multiple
of $\operatorname{ad}_{e_{2}}$. In particular, we see that $d(e_{2})=0$, so
that $\mu=0$. It follows that $d=-\lambda\operatorname{ad}_{e_{2}}$, so
$\operatorname{Der}(W_{\geq 1},W)_{2}\subseteq\operatorname{Inn}(W_{\geq
1},W)$.

Finally, let $n\in\mathbb{Z}\setminus\\{0,1,2\\}$ and consider
$d\in\operatorname{Der}(W_{\geq 1},W)_{n}$. Then $d(e_{1})=\lambda e_{n+1}$
and $d(e_{2})=\mu e_{n+2}$ for some $\lambda,\mu\in\Bbbk$. Taking Lie brackets
up to $e_{5}$ and using the Leibniz rule for $d$ as above, we get

$(\lambda-\mu)n^{3}-\lambda n^{2}+(4\lambda-5\mu)n-12\lambda+6\mu=0.$

Let $\alpha=\frac{1}{n-1}\lambda$ and $\beta=\frac{1}{n-2}\mu$. Substituting
$\alpha$ and $\beta$ into the equation above,

$(n^{4}-2n^{3}+5n^{2}-16n+12)(\alpha-\beta)=0.$

The quartic equation in $n$ does not vanish for $n\geq 3$ (its only integer
solutions are $n=1,2$), so we conclude that $\alpha=\beta$. Therefore,
$d(e_{1})=\alpha(n-1)e_{n+1}$ and $d(e_{2})=\alpha(n-2)e_{n+2}$. It follows
that $d=-\alpha\operatorname{ad}_{e_{n}}$, so $\operatorname{Der}(W_{\geq
1},W)_{n}\subseteq\operatorname{Inn}(W_{\geq 1},W)$ for all $n\geq 3$. This
finishes the proof. ∎

We now move on to computing $\operatorname{Der}(W_{\geq n},W)$ for $n\geq 2$.

######  Proposition 1.10.

For $n\in\mathbb{N}$, we have $\operatorname{Der}(W_{\geq
n},W)=\operatorname{Inn}(W_{\geq n},W)$. Consequently,
$\operatorname{Der}(W_{\geq n})=\\{\operatorname{ad}_{w}\mid w\in W_{\geq
0}\\}\cong W_{\geq 0}$ and $\dim(H^{1}(W_{\geq n};W_{\geq n}))=n$.

In order to prove Proposition 1.10, it suffices to compute graded derivations
of $W_{\geq n}$, by Proposition 1.4. The idea of the proof is similar to that
of Proposition 1.5. In that proof, we used the fact that we can take brackets
up to $e_{5}$ in two different ways, which puts restrictions on the space of
derivations of $W_{\geq 1}$. In other words, we exploited the relation

$[e_{1},[e_{1}[e_{1},e_{2}]]]+6[e_{2},[e_{1},e_{2}]]=0$

in $W_{\geq 1}$. For the proof of Proposition 1.10, we first consider
derivations of $W_{\geq n}$ of degree 0 and then find a relation in $W_{\geq
n}$. It is not difficult to show that any derivation of $W_{\geq n}$ of degree
0 must be a multiple of $\operatorname{ad}_{e_{0}}$.

######  Lemma 1.11.

Letting $n\in\mathbb{N}$, we have $\operatorname{Der}(W_{\geq
n},W)_{0}=\Bbbk\operatorname{ad}_{e_{0}}$.

###### Proof.

The result has already been shown for $n=0,1$, so we may assume that $n\geq
2$.

Let $d\in\operatorname{Der}(W_{\geq n},W)_{0}$. Then for all $k\geq n$ there
exists $\lambda_{k}\in\Bbbk$ such that $d(e_{k})=\lambda_{k}e_{k}$. For all
$\ell\geq n$, we have

$d([e_{k},e_{\ell}])=[e_{k},d(e_{\ell})]+[d(e_{k}),e_{\ell}]=(\lambda_{k}+\lambda_{\ell})[e_{k},e_{\ell}]=(\ell-k)(\lambda_{k}+\lambda_{\ell})e_{k+\ell}.$

On the other hand,

$d([e_{k},e_{\ell}])=(\ell-k)d(e_{k+\ell})=(\ell-k)\lambda_{k+\ell}e_{k+\ell}.$

Combining the above, we conclude that
$\lambda_{k}+\lambda_{\ell}=\lambda_{k+\ell}$ for all $k\neq\ell$.

Consider $2\lambda_{k}$ for some $k\geq n$. Let $\ell>2k$. Then

$2\lambda_{k}+\lambda_{\ell}=\lambda_{k}+\lambda_{k+\ell}=\lambda_{2k+\ell}=\lambda_{2k}+\lambda_{\ell}.$

Therefore, $2\lambda_{k}=\lambda_{2k}$. Inductively, we deduce that
$m\lambda_{k}=\lambda_{mk}$ for all $m\in\mathbb{N}$.

Now let $\mu=\lambda_{n}+\frac{\lambda_{n}}{n}$. Then

$n\mu=\lambda_{n}+n\lambda_{n}=\lambda_{n}+\lambda_{n^{2}}=\lambda_{n(n+1)}=n\lambda_{n+1},$

so $\mu=\lambda_{n+1}$. Now, we have
$\lambda_{n}+\frac{\lambda_{n}}{n}=\lambda_{n+1}$, so

$\frac{\lambda_{n}}{n}=\lambda_{n+1}-\lambda_{n},$

and thus $\lambda_{n}=n(\lambda_{n+1}-\lambda_{n})$. Let
$\lambda=\lambda_{n+1}-\lambda_{n}$. We claim that
$d=\lambda\operatorname{ad}_{e_{0}}$.

Then, for $m\geq n$,

$\lambda_{m}+\lambda=\lambda_{m}+\lambda_{n+1}-\lambda_{n}=\lambda_{n+m+1}-\lambda_{n}.$

But we also know that $\lambda_{n+m+1}=\lambda_{m+1}+\lambda_{n}$, so

$\lambda_{m}+\lambda=\lambda_{m+1}+\lambda_{n}-\lambda_{n}=\lambda_{m+1}.$

Since $\lambda_{n}=n\lambda$ and $\lambda_{m+1}=\lambda_{m}+\lambda$ for all
$m\geq n$, it follows by induction that $\lambda_{m}=m\lambda$ for all $m\geq
n$, proving the claim. ∎

We are nearly ready to prove Proposition 1.10: we only need to find a relation
in $W_{\geq n}$.

######  Lemma 1.12.

Let $n,m\in\mathbb{Z}$. We have the following relation on $e_{n}$ and $e_{m}$:

$nm[[e_{n},e_{m}],[[e_{n},e_{m}],[[e_{n},e_{m}],[e_{m},[e_{n},[e_{n},e_{m}]]]]]]\\\
+3(m-n)(n+m)[[e_{m},[e_{n},[e_{n},e_{m}]]],[[e_{m},[e_{n},[e_{n},e_{m}]]],[e_{n},e_{m}]]]=0.$

###### Proof.

We have

(1.12.1) $\displaystyle[e_{n},e_{m}]$ $\displaystyle=(m-n)e_{n+m},$
$\displaystyle[e_{n},[e_{n},e_{m}]]=(m-n)[e_{n},e_{n+m}]$
$\displaystyle=(m-n)me_{2n+m},$ (1.12.2)
$\displaystyle[e_{m},[e_{n},[e_{n},e_{m}]]]=(m-n)m[e_{m},e_{2n+m}]$
$\displaystyle=2(m-n)nme_{2n+2m}.$

We can easily find a relation between $e_{n+m}$ and $e_{2n+2m}$:

(1.12.3) $\displaystyle[e_{n+m},[e_{n+m},[e_{n+m},e_{2n+2m}]]]$
$\displaystyle=6(n+m)^{3}e_{5n+5m},$ (1.12.4)
$\displaystyle[e_{2n+2m},[e_{2n+2m},e_{n+m}]]$
$\displaystyle=-(n+m)^{2}e_{5n+5m}.$

Therefore,
$[e_{n+m},[e_{n+m},[e_{n+m},e_{2n+2m}]]]+6(n+m)[e_{2n+2m},[e_{2n+2m},e_{n+m}]]=0$.
We can now find a relation between $e_{n}$ and $e_{m}$ by substituting
(1.12.1) and (1.12.2) into (1.12.3) and (1.12.4):

$\displaystyle[[e_{n},e_{m}],[[e_{n},e_{m}],[[e_{n},e_{m}],[e_{m},[e_{n},[e_{n},e_{m}]]]]]]$
$\displaystyle=12nm(m-n)^{4}(n+m)^{3}e_{5n+5m},$
$\displaystyle[[e_{m},[e_{n},[e_{n},e_{m}]]],[[e_{m},[e_{n},[e_{n},e_{m}]]],[e_{n},e_{m}]]]$
$\displaystyle=-4n^{2}m^{2}(m-n)^{3}(n+m)^{2}e_{5n+5m}.$

The lemma follows immediately. ∎

We can now prove Proposition 1.10.

###### Proof of Proposition 1.10.

The result has already been shown for $n=0,1$, so fix $n\geq 2$. By Lemma
1.11, we already know that $\operatorname{Der}(W_{\geq
n},W)_{0}=\Bbbk\operatorname{ad}_{e_{0}}$. So, let
$k\in\mathbb{Z}\setminus\\{0\\}$ and consider $d\in\operatorname{Der}(W_{\geq
n},W)_{k}$. For $m\geq n$, write $d(e_{m})=\lambda_{m}e_{m+k}$, where
$\lambda_{m}\in\Bbbk$. We claim that $d=\alpha\operatorname{ad}_{e_{k}}$ for
some $\alpha\in\Bbbk$, or in other words, that $\lambda_{m}=(m-k)\alpha$ for
all $m\geq n$.

Letting $r,s\geq n$ with $r\neq s$, Lemma 1.12 implies that

$d\Bigl{(}rs[[e_{r},e_{s}],[[e_{r},e_{s}],[[e_{r},e_{s}],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+3(s-r)(r+s)[[e_{s},[e_{r},[e_{r},e_{s}]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[e_{r},e_{s}]]]\Bigr{)}=0.$

Therefore, applying the Leibniz rule repeatedly,

(1.12.5)
$\begin{split}rs\Bigl{(}&[[d(e_{r}),e_{s}],[[e_{r},e_{s}],[[e_{r},e_{s}],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},d(e_{s})],[[e_{r},e_{s}],[[e_{r},e_{s}],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[d(e_{r}),e_{s}],[[e_{r},e_{s}],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},d(e_{s})],[[e_{r},e_{s}],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},e_{s}],[[d(e_{r}),e_{s}],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},e_{s}],[[e_{r},d(e_{s})],[e_{s},[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},e_{s}],[[e_{r},e_{s}],[d(e_{s}),[e_{r},[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},e_{s}],[[e_{r},e_{s}],[e_{s},[d(e_{r}),[e_{r},e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},e_{s}],[[e_{r},e_{s}],[e_{s},[e_{r},[d(e_{r}),e_{s}]]]]]]\\\
+&[[e_{r},e_{s}],[[e_{r},e_{s}],[[e_{r},e_{s}],[e_{s},[e_{r},[e_{r},d(e_{s})]]]]]]\Bigr{)}\\\
+3(s-r)(r+s)\Bigl{(}&[[d(e_{s}),[e_{r},[e_{r},e_{s}]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[d(e_{r}),[e_{r},e_{s}]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[d(e_{r}),e_{s}]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},d(e_{s})]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},e_{s}]]],[[d(e_{s}),[e_{r},[e_{r},e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},e_{s}]]],[[e_{s},[d(e_{r}),[e_{r},e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},e_{s}]]],[[e_{s},[e_{r},[d(e_{r}),e_{s}]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},e_{s}]]],[[e_{s},[e_{r},[e_{r},d(e_{s})]]],[e_{r},e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},e_{s}]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[d(e_{r}),e_{s}]]]\\\
+&[[e_{s},[e_{r},[e_{r},e_{s}]]],[[e_{s},[e_{r},[e_{r},e_{s}]]],[e_{r},d(e_{s})]]]\Bigr{)}=0.\end{split}$

Substituting $d(e_{r})=\lambda_{r}e_{r+k}$ and $d(e_{s})=\lambda_{s}e_{s+k}$
into (1.12.5) and simplifying,

$(k^{2}+ks+6s^{2}+kr+12rs+6r^{2})(k^{2}+ks-2s^{2}+kr-2r^{2})\\\
\times((s-k)\lambda_{r}-(r-k)\lambda_{s})k(s-r)^{3}rse_{5r+5s+k}=0.$

Now, $k(s-r)^{3}rs\neq 0$, since $r,s,k\neq 0$ and $r\neq s$. Hence,

$(k^{2}+ks+6s^{2}+kr+12rs+6r^{2})(k^{2}+ks-2s^{2}+kr-2r^{2})((s-k)\lambda_{r}-(r-k)\lambda_{s})=0,$

for all $r,s\geq n$ with $r\neq s$. For brevity, let

$H_{k}(r,s)=(k^{2}+ks+6s^{2}+kr+12rs+6r^{2})(k^{2}+ks-2s^{2}+kr-2r^{2}).$

It follows that

(1.12.6) $(s-k)\lambda_{r}=(r-k)\lambda_{s}$

for all $r,s\geq n$ such that $H_{k}(r,s)\neq 0$.

Fix $m\geq n$ such that $m\neq k$. Letting $\alpha=\frac{\lambda_{m}}{m-k}$,
we get $\lambda_{\ell}=(\ell-k)\alpha$ for all $\ell\geq n$ such that
$H_{k}(m,\ell)\neq 0$, by (1.12.6).

Now suppose $\ell\geq n$ such that $H_{k}(m,\ell)=0$. We want to show that we
still have $\lambda_{\ell}=(\ell-k)\alpha$ in this case. Since $H_{k}(m,r)$ is
non-constant in $r$, we can choose $r\geq n$ such that $r\neq k$,
$H_{k}(m,r)\neq 0$ and $H_{k}(\ell,r)\neq 0$. Since $H_{k}(m,r)\neq 0$, we
already know that $\lambda_{r}=(r-k)\alpha$. Furthermore, by (1.12.6), we get

$(r-k)\lambda_{\ell}=(\ell-k)\lambda_{r}=(\ell-k)(r-k)\alpha,$

since $H_{k}(\ell,r)\neq 0$. It follows that $\lambda_{\ell}=(\ell-k)\alpha$
for all $\ell\geq n$, even if $H_{k}(m,\ell)=0$. Therefore,
$d=\alpha\operatorname{ad}_{e_{k}}$, as claimed. ∎

##  2\. Derivations of general submodule-subalgebras

We now consider derivations of general submodule-subalgebras of $W_{\geq-1}$.
It is standard that $\operatorname{Ext}$ groups of a filtered object are
related to $\operatorname{Ext}$ groups of the associated graded object (see,
for example, [Bjö79, Proposition 2.6.10]). We will see that this is also the
case in our situation: by an associated graded argument, we will be able to
extract a lot of information about derivations of ungraded submodule-
subalgebras from the results of Section 1.

We begin with some definitions.

######  Definition 2.1.

For an element $w=\sum_{i=m}^{n}\lambda_{i}e_{i}\in W$, where
$n,m\in\mathbb{Z}$ with $n>m$, and $\lambda_{i}\in\Bbbk$ with $\lambda_{n}\neq
0$, we call $n$ the _degree_ of $w$ and write $\deg(w)=n$. Furthermore, we
call $\lambda_{n}e_{n}$ the _leading term_ of $w$ and write
$\operatorname{LT}(w)=\lambda_{n}e_{n}$. Note that if $\deg(u)\neq\deg(v)$, we
have $\deg([u,v])=\deg(u)+\deg(v)$, where $u,v\in W$.

For $f=\alpha\prod_{i=0}^{n}(t-\lambda_{i})^{k_{i}}\in\Bbbk[t]$, where
$\alpha,\lambda_{i}\in\Bbbk$, $\lambda_{i}\neq\lambda_{j}$ for $i\neq j$ and
$k_{i}\geq 1$ for all $i$, the _radical_ of $f$ is
$\operatorname{rad}(f)=\prod_{i=0}^{n}(t-\lambda_{i})$. If $k_{i}=1$ for all
$i$, we say $f$ is _reduced_. The _vanishing locus_ of $f$ is
$V(f)=\\{\lambda_{1},\ldots,\lambda_{n}\\}$.

Let $f\in\Bbbk(t)\setminus\\{0\\}$ and write $f=\frac{p}{q}$, where
$p,q\in\Bbbk[t]\setminus\\{0\\}$. Let $\xi\in\Bbbk$ and write

$p=(t-\xi)^{k}r,\quad q=(t-\xi)^{\ell}s,$

where $k,\ell\in\mathbb{N}$ and $r,s\in\Bbbk[t]$ do not vanish at $\xi$. The
_order of vanishing of $f$ at $\xi$_ is $\operatorname{ord}_{\xi}(f)=k-\ell$.
If $\operatorname{ord}_{\xi}(f)<0$, we say $f$ has a _pole of order
$-\operatorname{ord}_{\xi}(f)$_. By convention,
$\operatorname{ord}_{\xi}(0)=\infty$ for all $\xi\in\Bbbk$. Furthermore, for
the derivation $f\partial\in\operatorname{Der}(\Bbbk(t))$, we simply write
$\operatorname{ord}_{\xi}(f\partial)=\operatorname{ord}_{\xi}(f)$.

The goal of this section is to compute $\operatorname{Der}(W_{\geq-1}(f))$ for
$f\in\Bbbk[t]\setminus\\{0\\}$, thus generalising Proposition 1.10 to
arbitrary submodule-subalgebras of $W_{\geq-1}$.

######  Theorem 2.2.

Let $f\in\Bbbk[t]\setminus\\{0\\}$. We have
$\operatorname{Der}(W_{\geq-1}(f))=\\{\operatorname{ad}_{w}\mid w\in
W_{\geq-1}(\operatorname{rad}(f))\\}$. Consequently,
$\operatorname{Der}(W_{\geq-1}(f))\cong W_{\geq-1}(\operatorname{rad}(f))$,
and

$\dim H^{1}(W_{\geq-1}(f);W_{\geq-1}(f))=\deg(f)-\deg(\operatorname{rad}(f)).$

######  Remark 2.3.

A _complete_ Lie algebra is a Lie algebra with trivial center whose
derivations are all inner. Theorem 2.2 implies that $W_{\geq-1}(f)$ is
complete if and only if $f\in\Bbbk[t]$ is reduced.

In order to understand derivations of $W_{\geq-1}(f)$, we will construct
associated graded derivations, which are graded derivations of $W_{\geq n}$,
where $n\coloneqq\deg(f\partial)$. This will essentially be done by only
considering the leading terms of elements of $W_{\geq-1}(f)$. The following
result is the first step in the construction of associated graded derivations.

######  Lemma 2.4.

Let $f\in\Bbbk[t]\setminus\\{0\\}$. For
$d\in\operatorname{Der}(W_{\geq-1}(f),W)$, the set

$\\{\deg(d(x))-\deg(x)\mid x\in W_{\geq-1}(f)\\}$

is bounded above.

###### Proof.

Let $d\in\operatorname{Der}(W_{\geq-1}(f),W)\setminus\\{0\\}$. Let $n=\deg(f)$
and $x_{i}=t^{i-1}f\partial$, so that $x_{1},\ldots,x_{n}$ generate
$W_{\geq-1}(f)$ as a Lie algebra. Let

$N=\max\\{\deg(d(x_{i}))-\deg(x_{i})\mid i\in\\{1,\ldots,n\\}\\}.$

We claim that $\deg(d(x))-\deg(x)\leq N$ for all $x\in W_{\geq-1}(f)$. Note
that

$d([x,y])=[x,d(y)]+[d(x),y].$

Therefore, provided $\deg(d(x))\leq\deg(x)+N$ and $\deg(d(y))\leq\deg(y)+N$,
we see that $\deg(d([x,y]))\leq\deg([x,y])+N$. The claim now follows by an
easy induction on $\deg(x)$. ∎

Lemma 2.4 gives us a notion of degree for derivations
$d\in\operatorname{Der}(W_{\geq-1}(f),W)$.

######  Definition 2.5.

Let $f\in\Bbbk[t]\setminus\\{0\\}$. For
$d\in\operatorname{Der}(W_{\geq-1}(f),W)\setminus\\{0\\}$, the _degree_ of $d$
is

$\deg(d)=\max\\{\deg(d(x))-\deg(x)\mid x\in W_{\geq-1}(f)\\}.$

By convention, the degree of the zero derivation is $-\infty$.

We now consider leading terms of elements $x\in W_{\geq-1}(f)$ such that
$\deg(d(x))=\deg(x)+\deg(d)$.

######  Lemma 2.6.

Let $f\in\Bbbk[t]\setminus\\{0\\}$, $d\in\operatorname{Der}(W_{\geq-1}(f),W)$,
and $x\in W_{\geq-1}(f)$ such that $\deg(d(x))=\deg(x)+\deg(d)$. Then
$\deg(d(y))=\deg(y)+\deg(d)$ for all $y\in W_{\geq-1}(f)$ such that
$\deg(y)=\deg(x)$. Furthermore, if $\operatorname{LT}(x)=\operatorname{LT}(y)$
then $\operatorname{LT}(d(x))=\operatorname{LT}(d(y))$.

###### Proof.

Let $y\in W_{\geq-1}(f)$ such that $\deg(y)=\deg(x)$. Rescaling $y$ if
necessary, we may assume that $\operatorname{LT}(y)=\operatorname{LT}(x)$. Let
$z=x-y$, so $\deg(z)<\deg(x)$ and $d(y)=d(x)-d(z)$. By definition of
$\deg(d)$, we know that

$\deg(d(z))\leq\deg(z)+\deg(d)<\deg(x)+\deg(d)=\deg(d(x)).$

It follows that
$\operatorname{LT}(d(y))=\operatorname{LT}(d(x)-d(z))=\operatorname{LT}(d(x))$
and that

$\deg(d(y))=\deg(d(x))=\deg(x)+\deg(d)=\deg(y)+\deg(d),$

as required. ∎

Given Lemma 2.6, we can now make the following definition.

######  Definition 2.7.

Let $f\in\Bbbk[t]\setminus\\{0\\}$, $d\in\operatorname{Der}(W_{\geq-1}(f),W)$,
and $n=\deg(f\partial)$. We say that $k\in\mathbb{Z}_{\geq n}$ is _$d$
-compatible_ if $\deg(d(x))=\deg(x)+\deg(d)$ for some (and therefore all)
$x\in W_{\geq-1}(f)$ such that $\deg(x)=k$.

######  Example 2.8.

Let $f=t^{2}+1$ and let
$d=\operatorname{ad}_{e_{5}+e_{3}}\in\operatorname{Der}(W_{\geq-1}(f))$. Then
$\deg(d)=5$. Therefore, 5 is not $d$-compatible, while every
$n\in\mathbb{Z}_{\geq 1}\setminus\\{5\\}$ is $d$-compatible.

We now complete the construction of associated graded derivations.

######  Proposition 2.9.

Let $f\in\Bbbk[t]\setminus\\{0\\}$ and
$d\in\operatorname{Der}(W_{\geq-1}(f),W)$. Define a $\Bbbk$-linear map
$\operatorname{gr}(d)\colon W_{\geq n}\to W$ by

$\operatorname{gr}(d)(e_{k})=\begin{cases}\operatorname{LT}(d(x)),&\text{where
}x\in W_{\geq-1}(f)\text{ and }\operatorname{LT}(x)=e_{k},\text{ if }k\text{
is }d\text{-compatible,}\\\ 0,&\text{if }k\text{ is not
}d\text{-compatible,}\end{cases}$

for $k\in\mathbb{Z}_{\geq n}$. Then
$\operatorname{gr}(d)=\lambda\operatorname{ad}_{e_{N}}\in\operatorname{Der}(W_{\geq
n},W)$ for some $\lambda\in\Bbbk$ and $N\in\mathbb{Z}$.

###### Proof.

First of all, by Lemma 2.6, $\operatorname{gr}(d)$ is well-defined.

If we prove that $\operatorname{gr}(d)\in\operatorname{Der}(W_{\geq n},W)$,
then the result will follow from Proposition 1.10. Therefore, it suffices to
show that $\operatorname{gr}(d)$ is a derivation. In other words, it is enough
to show that

$\operatorname{gr}(d)([e_{k},e_{\ell}])=[e_{k},\operatorname{gr}(d)(e_{\ell})]+[\operatorname{gr}(d)(e_{k}),e_{\ell}],$

for all $k,\ell\in\mathbb{Z}_{\geq n}$. Let $k,\ell\in\mathbb{Z}_{\geq n}$ be
distinct, and let $u,v\in W_{\geq-1}(f)$ such that
$\operatorname{LT}(u)=e_{k}$ and $\operatorname{LT}(v)=e_{\ell}$. We have

(2.9.1) $d([u,v])=[u,d(v)]+[d(u),v].$

The rest of the proof now follows by a straightforward case-by-case analysis,
depending on whether $k$ and $\ell$ are $d$-compatible. We explicitly show the
case where $k$ and $\ell$ are both $d$-compatible and leave the rest of the
cases to the reader. Let $N=\deg(d)$. Since $\deg(d(u))=k+N$, we have
$\deg([d(u),v])=k+\ell+N$ if $k+N\neq\ell$, but $\deg([d(u),v])<k+\ell+N$ if
$k+N=\ell$. We will therefore need to divide the proof into two cases.

Case 1: Either $k+N=\ell$ or $\ell+N=k$.

Without loss of generality, say $k+N=\ell$ and $\ell+N\neq k$. Then $k+\ell$
is $d$-compatible (which can be seen by looking at the degrees in (2.9.1)),
and
$[\operatorname{gr}(d)(e_{k}),e_{\ell}]=[\operatorname{LT}(d(x)),e_{\ell}]=0$,
since $\deg(d(x))=k+N=\ell$. Furthermore,

$\operatorname{gr}(d)([e_{k},e_{\ell}])=\operatorname{LT}(d([u,v]))=\operatorname{LT}([u,d(v)])=[\operatorname{LT}(u),\operatorname{LT}(d(v))]=[e_{k},\operatorname{gr}(d)(e_{\ell})].$

Therefore,
$\operatorname{gr}(d)([e_{k},e_{\ell}])=[e_{k},\operatorname{gr}(d)(e_{\ell})]+[\operatorname{gr}(d)(e_{k}),e_{\ell}]$.

Case 2: $k+N\neq\ell$ and $\ell+N\neq k$.

We have

$\displaystyle[e_{k},\operatorname{gr}(d)(e_{\ell})]+[\operatorname{gr}(d)(e_{k}),e_{\ell}]$
$\displaystyle=[\operatorname{LT}(u),\operatorname{LT}(d(v))]+[\operatorname{LT}(d(u)),\operatorname{LT}(v)]$
$\displaystyle=\operatorname{LT}([u,d(v)])+\operatorname{LT}([d(u),v]).$

Suppose $\operatorname{LT}([u,d(v)])+\operatorname{LT}([d(u),v])\neq 0$. Then
$k+\ell$ is $d$-compatible and

$[e_{k},\operatorname{gr}(d)(e_{\ell})]+[\operatorname{gr}(d)(e_{k}),e_{\ell}]=\operatorname{LT}([u,d(v)]+[d(u),v])=\operatorname{LT}(d([u,v]))=\operatorname{gr}(d)([e_{k},e_{\ell}]).$

Otherwise, $\operatorname{LT}([u,d(v)])+\operatorname{LT}([d(u),v])=0$, in
which case $k+\ell$ is not $d$-compatible, so

$\operatorname{gr}(d)([e_{k},e_{\ell}])=0=[e_{k},\operatorname{gr}(d)(e_{\ell})]+[\operatorname{gr}(d)(e_{k}),e_{\ell}].$

This completes the proof. ∎

We now study $\operatorname{Der}(W_{\geq-1}(f),W_{\geq-1})$. This is not
strictly necessary to prove Theorem 2.2, which only considers
$\operatorname{Der}(W_{\geq-1}(f))$, but it is not much more difficult than
considering $\operatorname{Der}(W_{\geq-1}(f))$, and will be useful for the
next section.

######  Proposition 2.10.

Let $f\in\Bbbk[t]\setminus\\{0\\}$ and
$d\in\operatorname{Der}(W_{\geq-1}(f),W_{\geq-1})$. Then
$d=\operatorname{ad}_{\frac{h}{f}\partial}$ for some $h\in\Bbbk[t]$.

###### Proof.

Consider $\operatorname{gr}(d)\in\operatorname{Der}(W_{\geq n},W_{\geq-1})$.
By Proposition 2.9,
$\operatorname{gr}(d)=\lambda_{0}\operatorname{ad}_{e_{k_{0}}}$ for some
$\lambda_{0}\in\Bbbk$, $k_{0}\in\mathbb{Z}$. Now let
$d_{1}=d-\lambda_{0}\operatorname{ad}_{e_{k_{0}}}\in\operatorname{Der}(W_{\geq-1}(f),W)$
and note that $\deg(d_{1})<\deg(d)$. Similarly to before,
$\operatorname{gr}(d_{1})=\lambda_{1}\operatorname{ad}_{e_{k_{1}}}$ for some
$\lambda_{1}\in\Bbbk$ and $k_{1}\in\mathbb{Z}$, so we can consider
$d_{2}=d_{1}-\lambda_{1}\operatorname{ad}_{e_{k_{1}}}\in\operatorname{Der}(W_{\geq-1}(f),W)$
with $\deg(d_{2})<\deg(d_{1})$. Continuing inductively, we get
$d_{1},d_{2},d_{3},\ldots\in\operatorname{Der}(W_{\geq-1}(f),W)$ such that
$\deg(d)>\deg(d_{1})>\deg(d_{2})>\deg(d_{3})>\ldots$, and

$d=d_{m+1}+\sum_{i=0}^{m}\lambda_{i}\operatorname{ad}_{e_{k_{i}}},$

for all $m\in\mathbb{N}$, where we set $\lambda_{i}=0$ if $d_{i}=0$. Let
$g=\sum_{i=0}^{\infty}\lambda_{i}t^{k_{i}+1}\in\Bbbk((t^{-1}))$ and consider
$\operatorname{ad}_{g\partial}\in\operatorname{Der}(\Bbbk((t^{-1}))\partial)$,
so that
$d={\left.\kern-1.2pt\operatorname{ad}_{g\partial}\vphantom{\big{|}}\right|_{W_{\geq-1}(f)}}$.
In other words, $[g\partial,hf\partial]\in W_{\geq-1}$ for all $h\in\Bbbk[t]$.
Hence, $[g\partial,f\partial]\in W_{\geq-1}$, so
$f^{\prime}g-fg^{\prime}\in\Bbbk[t]$, and thus

$tf^{\prime}g-tfg^{\prime}\in\Bbbk[t].$

Similarly, $[g\partial,tf\partial]\in W_{\geq-1}$, so

$fg+tf^{\prime}g-tfg^{\prime}\in\Bbbk[t].$

Combining the above, we conclude that $h\coloneqq fg\in\Bbbk[t]$, so
$g=\frac{h}{f}\in\Bbbk(t)$. ∎

If $d\in\operatorname{Der}(W_{\geq-1}(f))$, then Proposition 2.10 tells us
that $d=\operatorname{ad}_{g\partial}$ for some $g\in\Bbbk(t)$. For the proof
of Theorem 2.2, we need to show that $g\in\operatorname{rad}(f)\Bbbk[t]$, in
other words, that $g\in\Bbbk[t]$ and that $g$ vanishes at all the roots of
$f$. In order to achieve this, we will consider orders of vanishing of
$[f\partial,g\partial]$ at roots of $f$. The following easy lemma computes
$\operatorname{ord}_{\xi}([f\partial,g\partial])$ for $\xi\in\Bbbk$.

######  Lemma 2.11.

Let $\xi\in\Bbbk$, $f\in\Bbbk[t]\setminus\\{0\\}$, and
$g\in\Bbbk(t)\setminus\\{0\\}$. Then

$\operatorname{ord}_{\xi}([f\partial,g\partial])=\operatorname{ord}_{\xi}(f)+\operatorname{ord}_{\xi}(g)-1$

if $\operatorname{ord}_{\xi}(g)\neq\operatorname{ord}_{\xi}(f)$ and
$\operatorname{ord}_{\xi}([f\partial,g\partial])\geq
2\operatorname{ord}_{\xi}(f)$ if
$\operatorname{ord}_{\xi}(g)=\operatorname{ord}_{\xi}(f)$.

###### Proof.

Write $f=(t-\xi)^{k}p$ and $g=(t-\xi)^{\ell}q$, where
$p\in\Bbbk[t],q\in\Bbbk(t)$ and
$\operatorname{ord}_{\xi}(p)=\operatorname{ord}_{\xi}(q)=0$, so that
$k=\operatorname{ord}_{\xi}(f)$ and $\ell=\operatorname{ord}_{\xi}(g)$.
Differentiating $f$ and $g$, we get

$f^{\prime}=(t-\xi)^{k-1}(kp+(t-\xi)p^{\prime}),\quad
g^{\prime}=(t-\xi)^{\ell-1}(\ell q+(t-\xi)q^{\prime}).$

Therefore,

$[f\partial,g\partial]=(fg^{\prime}-f^{\prime}g)\partial=(t-\xi)^{k+\ell-1}((\ell-k)pq+(t-\xi)(pq^{\prime}-p^{\prime}q)).$

Let $r=(\ell-k)pq+(t-\xi)(pq^{\prime}-p^{\prime}q)$. It is clear that
$\operatorname{ord}_{\xi}(r)=0$ if $k\neq\ell$. Therefore,

$\operatorname{ord}_{\xi}(fg^{\prime}-f^{\prime}g)=\operatorname{ord}_{\xi}((t-\xi)^{k+\ell-1}r)=k+\ell-1$

if $k\neq\ell$. If $k=\ell$, then $\operatorname{ord}_{\xi}(r)\geq 1$, so

$\operatorname{ord}_{\xi}(fg^{\prime}-f^{\prime}g)=\operatorname{ord}_{\xi}((t-\xi)^{2k-1}r)\geq
2k,$

which concludes the proof. ∎

We are now ready to prove Theorem 2.2.

###### Proof of Theorem 2.2.

If $w\in W_{\geq-1}(\operatorname{rad}(f))$, then by Lemma 2.11, it follows
that $\operatorname{ad}_{w}\in\operatorname{Der}(W_{\geq-1}(f))$.

Let $d\in\operatorname{Der}(W_{\geq-1}(f))$. By Proposition 2.10, we know that
$d=\operatorname{ad}_{\frac{h}{f}\partial}$ for some $h\in\Bbbk[t]$. Let
$g=\frac{h}{f}$. Since $[g\partial,f\partial]\in W_{\geq-1}(f)$, $f$ divides
$f^{\prime}g-fg^{\prime}$. Therefore,

$\operatorname{ord}_{\xi}(f^{\prime}g-fg^{\prime})\geq\operatorname{ord}_{\xi}(f)$

for all $\xi\in V(f)$. Assume, for a contradiction, that
$\operatorname{ord}_{\xi}(g)\leq 0$ for some $\xi\in V(f)$. Then, by Lemma
2.11,

$\operatorname{ord}_{\xi}(f)\leq\operatorname{ord}_{\xi}(f^{\prime}g-fg^{\prime})=\operatorname{ord}_{\xi}(f)+\operatorname{ord}_{\xi}(g)-1\leq\operatorname{ord}_{\xi}(f)-1,$

a contradiction. Therefore, $\operatorname{ord}_{\xi}(g)\geq 1$ for all
$\xi\in V(f)$. It follows that $g\in\Bbbk[t]$ (since $g$ can only have poles
at roots of $f$), and that $\operatorname{rad}(f)$ divides $g$, so
$g\partial\in W_{\geq-1}(\operatorname{rad}(f))$. ∎

##  3\. Extensions of submodule-subalgebras

Having computed $\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f))$
for $M=\Bbbk$, we move on to the computation of the $\operatorname{Ext}$ group
when $M$ is an arbitrary one-dimensional representation of $W_{\geq-1}(f)$. We
may view an element of
$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f))$ as a non-split
short exact sequence of $U(W_{\geq-1}(f))$-modules

(3.0.1) $0\to W_{\geq-1}(f)\to X\to M\to 0.$

Although it is not immediately obvious, this computation is closely related to
the results from previous sections on derivations. This is because $M$ is
trivial as a representation of
$W_{\geq-1}(f^{2})=[W_{\geq-1}(f),W_{\geq-1}(f)]$ (see Lemma 3.4). Therefore,
regarding (3.0.1) as a sequence of $U(W_{\geq-1}(f^{2}))$-modules, we get an
element of

$\operatorname{Ext}_{U(W_{\geq-1}(f^{2}))}^{1}(\Bbbk,W_{\geq-1}(f))\cong\operatorname{Der}(W_{\geq-1}(f^{2}),W_{\geq-1}(f))/\operatorname{Inn}(W_{\geq-1}(f^{2}),W_{\geq-1}(f))$

(recall the isomorphism
$\operatorname{Ext}_{U(L)}^{1}(\Bbbk,M)\cong\operatorname{Der}(L,M)/\operatorname{Inn}(L,M)$
from the beginning of Section 1). We can use Proposition 2.10 to compute
$\operatorname{Der}(W_{\geq-1}(f^{2}),W_{\geq-1}(f))$, which then allows us to
determine the structure of $X$ as a representation of $W_{\geq-1}(f^{2})$.
This will give a lot of information into the structure of $X$ as a
representation of $W_{\geq-1}(f)$.

First of all, it is straightforward to show that $X$ has the structure of a
Lie algebra in a natural way.

######  Lemma 3.1.

Let $0\to L\to\overline{L}\xrightarrow{\pi}M\to 0$ be a short exact sequence
of $U(L)$-modules, where $L$ is a Lie algebra and $\dim(M)=1$. Writing
$\overline{L}=L\oplus\Bbbk x$ as a vector space (where $\pi(\Bbbk x)=M$),
there is a unique Lie bracket on $\overline{L}$ extending the action of $L$ on
$\overline{L}$. This is given by $[u,v]_{\overline{L}}=[u,v]_{L}$ and
$[u,x]_{\overline{L}}=u\cdot x$ for $u,v\in L$.

###### Proof.

It suffices to check that

(3.1.1) $[u,[v,x]]+[v,[x,u]]+[x,[u,v]]=0$

for $u,v\in L$. We have

$[[u,v],x]=[u,v]\cdot x=u\cdot(v\cdot x)-v\cdot(u\cdot x),$

since $\overline{L}$ is a $U(L)$-module. Therefore,

$[[u,v],x]=[u,[v,x]]-[v,[u,x]],$

so (3.1.1) holds. ∎

Lemma 3.1 says that studying the group
$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(W_{\geq-1}(f),M)$, where $M$ is a
one-dimensional $U(W_{\geq-1}(f))$-module, is essentially the same as studying
Lie algebras $\overline{L}$ containing $W_{\geq-1}(f)$ such that
$\dim(\overline{L}/W_{\geq-1}(f))=1$. Thus, we make the following definition.

######  Definition 3.2.

Let $L$ be a Lie algebra. A _one-dimensional extension_ of $L$ is a Lie
algebra $\overline{L}$ containing $L$ such that $\dim(\overline{L}/L)=1$ and
the short exact sequence of $U(L)$-modules

$0\to L\to\overline{L}\to\overline{L}/L\to 0$

is non-split.

We therefore consider one-dimensional extensions of $W_{\geq-1}(f)$, where
$f\in\Bbbk[t]\setminus\\{0\\}$. There are two types of “obvious” one-
dimensional extensions of $W_{\geq-1}(f)$. Firstly, suppose $f$ is non-
reduced, so that there exists $w\in W_{\geq-1}(\operatorname{rad}(f))\setminus
W_{\geq-1}(f)$. Since $W_{\geq-1}(f)$ is an ideal of
$W_{\geq-1}(\operatorname{rad}(f))$, it follows that $\overline{L}\coloneqq
W_{\geq-1}(f)\oplus\Bbbk w$ is a one-dimensional extension of $W_{\geq-1}(f)$.
Note that $\overline{L}/W_{\geq-1}(f)$ is trivial as a representation of
$W_{\geq-1}(f)$, since $W_{\geq-1}(f)$ is an ideal of $\overline{L}$. Hence,
this construction corresponds to an element of
$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(\Bbbk,W_{\geq-1}(f))$.

The other “obvious” one-dimensional extensions come from removing non-repeated
roots of $f$. In other words, if $\xi\in V(f)$ with
$\operatorname{ord}_{\xi}(f)=1$, then $W_{\geq-1}(\frac{f}{t-\xi})$ is a one-
dimensional extension of $W_{\geq-1}(f)$ which is not contained in
$W_{\geq-1}(\operatorname{rad}(f))$. In this case, the quotient
$W_{\geq-1}(\frac{f}{t-\xi})/W_{\geq-1}(f)$ is isomorphic to
$W_{\geq-1}/W_{\geq-1}(t-\xi)$ as a $U(W_{\geq-1}(f))$-module, where the
module structure on $W_{\geq-1}/W_{\geq-1}(t-\xi)$ is induced from
$W_{\geq-1}(f)$ acting on $W_{\geq-1}$ and $W_{\geq-1}(t-\xi)$, noting that
$W_{\geq-1}(f)$ is a subalgebra of both $W_{\geq-1}$ and $W_{\geq-1}(t-\xi)$.
Therefore, this corresponds to an element of

$\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(W_{\geq-1}/W_{\geq-1}(t-\xi),W_{\geq-1}(f)).$

The main goal of this section is to prove that all one-dimensional extensions
of $W_{\geq-1}(f)$ are one of these two types.

######  Theorem 3.3.

Let $f\in\Bbbk[t]\setminus\\{0\\}$ and let $M$ be a one-dimensional
$U(W_{\geq-1}(f))$-module. If $M\cong W_{\geq-1}/W_{\geq-1}(t-\xi)$ for some
$\xi\in V(f)$ with $\operatorname{ord}_{\xi}(f)=1$, then

$\dim(\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(W_{\geq-1}/W_{\geq-1}(t-\xi),W_{\geq-1}(f)))=1.$

In this case, the unique non-split extension of $W_{\geq-1}/W_{\geq-1}(t-\xi)$
by $W_{\geq-1}(f)$ is $W_{\geq-1}(\frac{f}{t-\xi})$. If $M\cong\Bbbk$ is
trivial, we have

$\dim(\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(\Bbbk,W_{\geq-1}(f)))=\deg(f)-\deg(\operatorname{rad}(f)).$

Otherwise, $\operatorname{Ext}_{U(W_{\geq-1}(f))}^{1}(M,W_{\geq-1}(f))=0$.

The first step in proving Theorem 3.3 is computing the derived subalgebra of
$W_{\geq-1}(f)$. This will be used to deduce that any one-dimensional
representation of $W_{\geq-1}(f)$ is trivial when viewed as a representation
of $W_{\geq-1}(f^{2})$.

######  Lemma 3.4.

For $f\in\Bbbk[t]$, the derived subalgebra of $W_{\geq-1}(f)$ is
$W_{\geq-1}(f^{2})$.

###### Proof.

The result follows by noting that
$[gf\partial,hf\partial]=(gh^{\prime}-g^{\prime}h)f^{2}\partial$ for
$g,h\in\Bbbk[t]$. ∎

######  Remark 3.5.

For a Lie algebra $L$, write $L_{(0)}=L$ and $L_{(n+1)}=[L,L_{(n)}]$ for the
lower central series of $L$, where $n\in\mathbb{N}$. We say that a Lie algebra
$L$ is _residually nilpotent_ if $\bigcap_{n\in\mathbb{N}}L_{(n)}=0$.

Since we computed $W_{\geq-1}(f)_{(1)}$ in Lemma 3.4, one might ask about
further terms in the lower central series of $W_{\geq-1}(f)$. It is not
difficult to show that

$W_{\geq-1}(f)_{(2)}=W_{\geq-1}(\gcd(f^{\prime},f)f^{2})$

for $f\in\Bbbk[t]$. Note that $\gcd(f^{\prime},f)=1$ if and only if $f$ is
reduced, so we have
$W_{\geq-1}(f)_{(2)}=W_{\geq-1}(f^{2})=W_{\geq-1}(f)_{(1)}$ and therefore
$W_{\geq-1}(f)_{(n)}=W_{\geq-1}(f^{2})$ for all $n\geq 1$. On the other hand,
when $f\in\Bbbk[t]$ is not reduced, one can easily show that $W_{\geq-1}(f)$
is residually nilpotent.

For the next step, we will need a result from [PS23]. Note that the result in
[PS23] considers subalgebras of $W$, but a similar result is true for
subalgebras of $W_{\geq-1}$ with a nearly identical proof.

######  Proposition 3.6 ([PS23, Proposition 3.2.7]).

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then there
exists a monic polynomial $f\in\Bbbk[t]\setminus\\{0\\}$ such that

$W_{\geq-1}(f)\subseteq L\subseteq W_{\geq-1}(\operatorname{rad}(f)).$

Furthermore, if we assume $f$ is of minimal degree, then such $f$ is unique.

The following result implies that all one-dimensional extensions of
$W_{\geq-1}(f)$ are contained in $W_{\geq-1}$. Note that the proposition
considers arbitrary subalgebras of $W_{\geq-1}$ of finite codimension, which
is more difficult than restricting to submodule-subalgebras, but will be
useful for Sections 4 and 5.

######  Proposition 3.7.

Let $L$ be a Lie subalgebra of $W_{\geq-1}$ of finite codimension, and let
$\overline{L}$ be a one-dimensional extension of $L$. Then $\overline{L}$ can
be uniquely embedded in $W_{\geq-1}$ such that the diagram

${L}$${W_{\geq-1}}$${\overline{L}}$

commutes.

###### Proof.

By Proposition 3.6, there exists $f\in\Bbbk[t]\setminus\\{0\\}$ such that

$W_{\geq-1}(f)\subseteq L\subseteq W_{\geq-1}(\operatorname{rad}(f)),$

since $L$ has finite codimension in $W_{\geq-1}$. Write
$\overline{L}=L\oplus\Bbbk x$ as a vector space. Since $\overline{L}/L$ is a
one-dimensional representation of $L$, it follows that $\overline{L}/L$ is
trivial as a representation of $[L,L]$. Hence, $[x,[L,L]]\subseteq L$. Since
$W_{\geq-1}(f)\subseteq L$, we know that $W_{\geq-1}(f^{2})\subseteq[L,L]$.
Therefore,

$[x,W_{\geq-1}(f^{2})]\subseteq L\subseteq W_{\geq-1}(\operatorname{rad}(f)).$

Hence,
${\left.\kern-1.2pt\operatorname{ad}_{x}\vphantom{\big{|}}\right|_{W_{\geq-1}(f^{2})}}\in\operatorname{Der}(W_{\geq-1}(f^{2}),W_{\geq-1}(\operatorname{rad}(f)))$.
By Proposition 2.10, we have
$\operatorname{ad}_{x}=\operatorname{ad}_{\frac{h}{f^{2}}\partial}$ on
$W_{\geq-1}(f^{2})$, for some $h\in\Bbbk[t]$. Let $g=\frac{h}{f^{2}}$, so that
$[w,x]=[w,g\partial]$ for all $w\in W_{\geq-1}(f^{2})$. Note that
$W_{\geq-1}(f)$ and $W_{\geq-1}(f^{2})$ are ideals of $L$, since $L\subseteq
W_{\geq-1}(\operatorname{rad}(f))$.

Let $u\in L\setminus\\{0\\},v\in W_{\geq-1}(f^{2})\setminus\\{0\\}$. We have

(3.7.1)
$[v,[u,x]]=[u,[v,x]]-[[u,v],x]=[u,[v,g\partial]]-[[u,v],g\partial]=[v,[u,g\partial]],$

where we used that $[u,v]\in W_{\geq-1}(f^{2})$ in the second equality.

Let $M=\overline{L}/L$ as a $U(L)$-module. We can regard the short exact
sequence

(3.7.2) $0\to L\to\overline{L}\to M\to 0$

as an element of $\operatorname{Ext}_{U(L)}^{1}(M,L)$ and consider the map

$\varphi\colon\operatorname{Ext}_{U(L)}^{1}(M,L)\to\operatorname{Ext}_{U(L)}^{1}(M,\Bbbk(t)\partial),$

where we view $L$ as a subalgebra of $\Bbbk(t)\partial$. Under the map
$\varphi$, the short exact sequence (3.7.2) gets mapped to

(3.7.3) $0\to\Bbbk(t)\partial\to X\to M\to 0,$

where $X$ is the $U(L)$-module $\Bbbk(t)\partial\oplus\Bbbk x$ with $L$ acting
by its adjoint action on $\Bbbk(t)\partial$ and the action of $L$ on $x$
agreeing with the bracket on $\overline{L}$. We claim that (3.7.3) splits, in
other words, that the image of (3.7.2) under $\varphi$ is zero. This will then
allow us to conclude that $\overline{L}$ can be embedded into
$\Bbbk(t)\partial$.

Let $y=x-g\partial\in X$. By (3.7.1), we have $[v,[u,y]]=0$ for all $u\in
L\setminus\\{0\\}$, $v\in W_{\geq-1}(f^{2})\setminus\\{0\\}$. There exist
$\Bbbk$-linear maps $a\colon L\to\Bbbk(t)\partial$ and $\lambda\colon
L\to\Bbbk$ such that

$[u,y]=a(u)+\lambda(u)x$

for all $u\in L$. We can also write

$[u,x]=b(u)+\lambda(u)x$

for all $u\in L$, where $b(u)=a(u)+[u,g\partial]\in L$. Therefore,

$0=[v,[u,y]]=[v,a(u)+\lambda(u)x]=[v,a(u)+\lambda(u)g\partial],$

where in the final equality we used that $[v,x]=[v,g\partial]$ for all $v\in
W_{\geq-1}(f^{2})$. It follows that
$a(u)+\lambda(u)g\partial\in\Bbbk(t)\partial$ is a scalar multiple of $v$,
since the centraliser of any nonzero element of $\Bbbk(t)\partial$ is one-
dimensional. But this must hold for all $v\in
W_{\geq-1}(f^{2})\setminus\\{0\\}$, so $a(u)+\lambda(u)g\partial=0$. Now, we
have

(3.7.4)
$[u,y]=a(u)+\lambda(u)x=a(u)+\lambda(u)x-(a(u)+\lambda(u)g\partial)=\lambda(u)y,$

for all $u\in L$. Letting $\overline{x}$ be the image of $x$ in $M$, the map

$\displaystyle s\colon M$ $\displaystyle\to X$ $\displaystyle\overline{x}$
$\displaystyle\mapsto y$

is a section of (3.7.3), so (3.7.3) splits, as claimed.

Suppose $g\partial\in L$. Then $y\in\overline{L}$, so we could take the
codomain of $s$ to be $\overline{L}$. In this case, $s\colon M\to\overline{L}$
is a section of (3.7.2). However, we assumed that (3.7.2) is non-split, so
this is a contradiction. Hence, we must have $g\partial\not\in L$, and
therefore $\overline{L}\cong L\oplus\Bbbk g\partial$. Without loss of
generality, we make the identification $\overline{L}=L\oplus g\partial$, so we
view $\overline{L}$ as a subalgebra of $\Bbbk(t)\partial$.

We claim that $\operatorname{ord}_{\xi}(g)\geq 0$ for all $\xi\in V(f)$. Since
$g=\frac{h}{f^{2}}$ can only have poles at roots of $f$, this will imply that
$g\in\Bbbk[t]$. In other words, this will show that $\overline{L}\subseteq
W_{\geq-1}$, which is enough to finish the proof.

Take $\xi\in V(f)$ and suppose $\operatorname{ord}_{\xi}(g)<0$. Define sets
$S=\\{\operatorname{ord}_{\xi}(w)\mid w\in L\\}$ and
$\overline{S}=\\{\operatorname{ord}_{\xi}(w)\mid w\in\overline{L}\\}$. Let
$n_{0}=\min(S)$, and let $w_{0}\in L$ such that
$\operatorname{ord}_{\xi}(w_{0})=n_{0}$. If $w=u+\alpha
g\partial\in\overline{L}$, where $u\in L$ and $\alpha\in\Bbbk$, then either
$\operatorname{ord}_{\xi}(w)=\operatorname{ord}_{\xi}(g)$ if $\alpha\neq 0$,
or $\operatorname{ord}_{\xi}(w)\geq\operatorname{ord}_{\xi}(w_{0})$ if
$\alpha=0$. Therefore, $\overline{S}=S\cup\\{\operatorname{ord}_{\xi}(g)\\}$.
Furthermore, since $W_{\geq-1}(f)\subseteq L$, we know that $S$ contains
$\mathbb{Z}_{\geq\operatorname{ord}_{\xi}(f)}$.

Since $[w_{0},g\partial]\in\overline{L}$, we know that
$\operatorname{ord}_{\xi}([w_{0},g\partial])\in\overline{S}=S\cup\\{\operatorname{ord}_{\xi}(g)\\}$.
By Lemma 2.11, we have

$\operatorname{ord}_{\xi}([w_{0},g\partial])=\operatorname{ord}_{\xi}(w_{0})+\operatorname{ord}_{\xi}(g)-1<n_{0}.$

By minimality of $n_{0}$, $\operatorname{ord}_{\xi}([w_{0},g\partial])\not\in
S$, so we must have that
$\operatorname{ord}_{\xi}([w_{0},g\partial])=\operatorname{ord}_{\xi}(g)$.
Therefore, $n_{0}=\operatorname{ord}_{\xi}(w_{0})=1$.

Let $n=1-\operatorname{ord}_{\xi}(g)\in\mathbb{N}$. Then
$\operatorname{ord}_{\xi}([w,g\partial])=\operatorname{ord}_{\xi}(w)-n$ for
all $w\in L$, by Lemma 2.11. Let $m\in S$ be such that $m\geq 2$, and let
$w\in L$ such that $\operatorname{ord}_{\xi}(w)=m$. Then

$\operatorname{ord}_{\xi}([w,g\partial])=m-n\geq
2-n=\operatorname{ord}_{\xi}(g)+1.$

Since
$\operatorname{ord}_{\xi}([w,g\partial])\in\overline{S}=S\cup\\{\operatorname{ord}_{\xi}(g)\\}$
and $\operatorname{ord}_{\xi}([w,g\partial])\neq\operatorname{ord}_{\xi}(g)$,
we must have that $\operatorname{ord}_{\xi}([w,g\partial])\in S$. Hence,
$m-n\in S$ for all $m\in S$ such that $m\geq 2$. In particular, this means
that if $m\in S$ such that $m\geq 2$, then $m$ must be larger than $n$, since
$\min(S)=1$. Noting that $n=1-\operatorname{ord}_{\xi}(g)\geq 2$, we therefore
see that $2\not\in S$.

Applying the above inductively, we deduce that $m-kn\in S$ for all $m\in
S,k\in\mathbb{N}$ such that $m>kn$. Since $2\not\in S$, it follows that $S$
does not contain any elements which are 2 modulo $n$. However, this
contradicts $\mathbb{Z}_{\geq\operatorname{ord}_{\xi}(f)}\subseteq S$. It
follows that $\operatorname{ord}_{\xi}(g)\geq 0$ for all $\xi\in V(f)$, as
claimed. ∎

A special case of Proposition 3.7 is a description of derivations of arbitrary
subalgebras of $W_{\geq-1}$ of finite codimension.

######  Corollary 3.8.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then

$\operatorname{Der}(L)=\\{\operatorname{ad}_{w}\mid w\in W_{\geq-1}\text{ such
that }\operatorname{ad}_{w}(L)\subseteq L\\}.$

In other words, derivations of $L$ are restrictions of derivations of
$W_{\geq-1}$. Consequently, $H^{1}(L;L)\cong N_{W_{\geq-1}}(L)/L$, where
$N_{W_{\geq-1}}(L)$ is the normaliser of $L$ in $W_{\geq-1}$. ∎

Having shown that all one-dimensional extensions of $W_{\geq-1}(f)$ are
contained in $W_{\geq-1}$, we can now explicitly compute them to prove Theorem
3.3.

###### Proof of Theorem 3.3.

If $M\cong\Bbbk$ then the result follows by Theorem 2.2. So, assume $M$ is
nontrivial as a representation of $W_{\geq-1}(f)$.

Let $0\to W_{\geq-1}(f)\to\overline{L}\to M\to 0$ be a non-split short exact
sequence of representations of $W_{\geq-1}(f)$. By Proposition 3.7, we may
assume that $\overline{L}\subseteq W_{\geq-1}$. It suffices to show that
$\overline{L}=W_{\geq-1}(\frac{f}{t-\xi})$ for some $\xi\in V(f)$ with
$\operatorname{ord}_{\xi}(f)=1$.

Write $\overline{L}=W_{\geq-1}(f)\oplus\Bbbk g\partial$ as a vector space,
where $g\in\Bbbk[t]\setminus\\{0\\}$. If $g\partial\in
W_{\geq-1}(\operatorname{rad}(f))$, then $\overline{L}\subseteq
W_{\geq-1}(\operatorname{rad}(f))$. So, suppose $g\partial\not\in
W_{\geq-1}(\operatorname{rad}(f))$, in other words, there exists $\xi\in V(f)$
such that $g(\xi)\neq 0$. Note that, if $\lambda\in\Bbbk\setminus\\{0\\}$ and
$p\in\Bbbk[t]$, then $\operatorname{ord}_{\xi}(pf+\lambda g)=0$, while
$\operatorname{ord}_{\xi}(pf)\geq\operatorname{ord}_{\xi}(f)$. It follows that
if $w\in\overline{L}$, then $\operatorname{ord}_{\xi}(w)=0$ or
$\operatorname{ord}_{\xi}(w)\geq\operatorname{ord}_{\xi}(f)$. Note that
$[f\partial,g\partial]\in\overline{L}\setminus\\{0\\}$, and

$\operatorname{ord}_{\xi}([f\partial,g\partial])=\operatorname{ord}_{\xi}(f)+\operatorname{ord}_{\xi}(g)-1=\operatorname{ord}_{\xi}(f)-1,$

by Lemma 2.11. It follows that
$\operatorname{ord}_{\xi}([f\partial,g\partial])=0$, so
$\operatorname{ord}_{\xi}(f)=1$.

For $k\in\mathbb{N}$, write
$[t^{k}f\partial,g\partial]=w_{k}+\lambda_{k}g\partial$, where $w_{k}\in
W_{\geq-1}(f)$ and $\lambda_{k}\in\Bbbk$. Note that

$[t^{k}f\partial,g\partial]=(t^{k}fg^{\prime}-kt^{k-1}fg-t^{k}f^{\prime}g)\partial.$

Therefore,
$w_{k}=(t^{k}fg^{\prime}-kt^{k-1}fg-t^{k}f^{\prime}g)\partial-\lambda_{k}g\partial$.
Evaluating $w_{k}$ at $\xi$, we get

$-\xi^{k}f^{\prime}(\xi)g(\xi)-\lambda_{k}g(\xi)=0,$

since $w_{k}\in W_{\geq-1}(f)$ vanishes at $\xi$. Hence,
$\lambda_{k}=-\xi^{k}f^{\prime}(\xi)$ for all $k\in\mathbb{N}$, since
$g(\xi)\neq 0$.

The only assumptions we made on $\xi$ were that $\xi\in V(f)$ and $g(\xi)\neq
0$. There is thus only one such $\xi$, and in particular $g(\xi^{\prime})=0$
for all $\xi^{\prime}\in V(f)\setminus\\{\xi\\}$.

To end the proof, we claim that $g\partial\in W_{\geq-1}(\frac{f}{t-\xi})$, so
that $\overline{L}\subseteq W_{\geq-1}(\frac{f}{t-\xi})$. In particular, since
$W_{\geq-1}(\frac{f}{t-\xi})$ is a one-dimensional extension of
$W_{\geq-1}(f)$, this will immediately imply that
$\overline{L}=W_{\geq-1}(\frac{f}{t-\xi})$. In order to prove that
$g\partial\in W_{\geq-1}(\frac{f}{t-\xi})$, it suffices to show that
$\operatorname{ord}_{\mu}(g)\geq\operatorname{ord}_{\mu}(f)$ for all $\mu\in
V(f)\setminus\\{\xi\\}$.

We have already shown that $\operatorname{ord}_{\mu}(g)\geq 1$ for all $\mu\in
V(f)\setminus\\{\xi\\}$, so if $\operatorname{ord}_{\mu}(f)=1$ then
$\operatorname{ord}_{\mu}(g)\geq\operatorname{ord}_{\mu}(f)$. So, suppose
$\operatorname{ord}_{\mu}(f)\geq 2$ and
$\operatorname{ord}_{\mu}(g)<\operatorname{ord}_{\mu}(f)$ for some $\mu\in
V(f)\setminus\\{\xi\\}$. Recall that

$[f\partial,g\partial]=w_{0}+\lambda_{0}g\partial,$

where $w_{0}\in W_{\geq-1}(f)$ and $\lambda_{0}=-f^{\prime}(\xi)\neq 0$.
Therefore,
$\operatorname{ord}_{\mu}([f\partial,g\partial])=\operatorname{ord}_{\mu}(g)$.
On the other hand, Lemma 2.11 says that

$\operatorname{ord}_{\mu}([f\partial,g\partial])=\operatorname{ord}_{\mu}(f)+\operatorname{ord}_{\mu}(g)-1\geq\operatorname{ord}_{\mu}(g)+1,$

since $\operatorname{ord}_{\mu}(f)\geq 2$, a contradiction. Hence,
$\operatorname{ord}_{\mu}(g)\geq\operatorname{ord}_{\mu}(f)$ for all $\mu\in
V(f)\setminus\\{\xi\\}$, as claimed. ∎

The following corollary, which follows immediately from Theorem 3.3,
summarises the two types of one-dimensional extensions which $W_{\geq-1}(f)$
has.

######  Corollary 3.9.

Suppose $\overline{L}$ is a one-dimensional extension of $W_{\geq-1}(f)$,
where $f\in\Bbbk[t]\setminus\\{0\\}$. Then either $\overline{L}\subseteq
W_{\geq-1}(\operatorname{rad}(f))$ or
$\overline{L}=W_{\geq-1}(\frac{f}{t-\xi})$, where $\xi\in V(f)$ with
$\operatorname{ord}_{\xi}(f)=1$. ∎

##  4\. Isomorphisms between subalgebras of finite codimension

As an application of our results on extensions of subalgebras of $W_{\geq-1}$,
we study isomorphisms between subalgebras of $W_{\geq-1}$ of finite
codimension. We prove that any isomorphism between subalgebras of finite
codimension extends to an automorphism of $W_{\geq-1}$, which allows us to
classify submodule-subalgebras of $W_{\geq-1}$ up to isomorphism and compute
automorphism groups of submodule-subalgebras. This will be done by an
inductive argument: if we have an isomorphism $\varphi\colon L_{1}\to L_{2}$
between subalgebras of $W_{\geq-1}$, we will show that $\varphi$ extends to an
isomorphism between one-dimensional extensions of $L_{1}$ and $L_{2}$.

######  Theorem 4.1.

Let $L_{1}$ and $L_{2}$ be subalgebras of $W_{\geq-1}$ of finite codimension
and suppose there is an isomorphism of Lie algebras $\varphi\colon L_{1}\to
L_{2}$. Then $\varphi$ extends to an automorphism of $W_{\geq-1}$.

In particular, if $f\in\Bbbk[t]\setminus\\{0\\}$, then Theorem 4.1 implies
that automorphisms of $W_{\geq-1}(f)$ are restrictions of automorphisms of
$W_{\geq-1}$. This is similar to Theorem 2.2, which says that derivations of
$W_{\geq-1}(f)$ are restrictions of derivations of $W_{\geq-1}$.

Theorem 4.1 also implies that if $L_{1}$ and $L_{2}$ are isomorphic
subalgebras of $W_{\geq-1}$ of finite codimension, then
$\operatorname{codim}_{W_{\geq-1}}(L_{1})=\operatorname{codim}_{W_{\geq-1}}(L_{2})$,
which is not obvious a priori.

######  Remark 4.2.

Theorem 4.1 is not true for arbitrary infinite-dimensional subalgebras of
$W_{\geq-1}$. Taking $f\in\Bbbk[t]$ of degree at least 2, consider the Lie
algebra $L(f)$ from [Buz22] (see Notation 6.1). In particular, $L(f)$ has
infinite codimension in $W_{\geq-1}$, and there exists $h\in\Bbbk[t]$ such
that $L(f)\cong W_{\geq-1}(h)$, by [Buz22, Lemma 4.12]. However, this
isomorphism clearly does not extend to an automorphism of $W_{\geq-1}$. In
fact, it does not even extend to an endomorphism of $W_{\geq-1}$, since any
nonzero endomorphism of $W_{\geq-1}$ is an automorphism [Du04].

If we have a submodule-subalgebra $W_{\geq-1}(f)$, where $f\in\Bbbk[t]$ is
non-constant, then we know that $W_{\geq-1}(f)$ has one-dimensional extensions
(for example, we can take $W_{\geq-1}(\frac{f}{t-\xi})$, where $f(\xi)=0$).
However, it is not immediately clear whether an arbitrary subalgebra of
$W_{\geq-1}$ of finite codimension has any one-dimensional extensions. Our
next goal is to show that any such subalgebra has one-dimensional extensions.
We will requite notation for the derived series of a Lie algebra.

######  Notation 4.3.

For a Lie algebra $L$, let $D(L)=[L,L]$. We write $D^{0}(L)=L$ and

$D^{n+1}(L)=D(D^{n}(L))=[D^{n}(L),D^{n}(L)]$

for the derived series of $L$, where $n\in\mathbb{N}$.

######  Remark 4.4.

Analogously to residual nilpotence, we say a Lie algebra $L$ is _residually
solvable_ if $\bigcap_{n\in\mathbb{N}}D^{n}(L)=0$. By Lemma 3.4, we can see
that if $f\in\Bbbk[t]$ is non-constant, then $W_{\geq-1}(f)$ is residually
solvable.

Note that residual solvability is, in some sense, a weak property. For
example, a free Lie algebra is residually solvable. On the other hand,
$W_{\geq-1}$ is not residually solvable, since $D^{n}(W_{\geq-1})=W_{\geq-1}$
for all $n\in\mathbb{N}$.

In fact, we will see that $W_{\geq-1}(f)$ satisfies a stronger property
defined below, provided $f\in\Bbbk[t]$ is non-constant.

######  Definition 4.5.

We say that a Lie algebra $L$ is _strongly residually solvable_ if $L$ is
residually solvable and every finite-dimensional quotient of $L$ is solvable.

Given a subalgebra $L\subseteq W_{\geq-1}$ of finite codimension, the next
result shows that $L$ is strongly residually solvable. Lie’s theorem will then
allow us to deduce that all finite-dimensional irreducible representations of
$L$ are one-dimensional.

######  Lemma 4.6.

If $L$ is a proper subalgebra of $W_{\geq-1}$ of finite codimension then $L$
is strongly residually solvable.

###### Proof.

Let $I$ be an ideal of $L$ of finite codimension. By Proposition 3.6, there
exist $f,g\in\Bbbk[t]\setminus\\{0\\}$ such that

$W_{\geq-1}(f)\subseteq L\subseteq W_{\geq-1}(\operatorname{rad}(f)),\quad
W_{\geq-1}(g)\subseteq I\subseteq W_{\geq-1}(\operatorname{rad}(g)).$

Since $W_{\geq-1}(\operatorname{rad}(f))$ is residually solvable, it follows
that $L$ is also residually solvable. It remains to show that $L/I$ is
solvable.

Note that $W_{\geq-1}(g)\subseteq I\subseteq L\subseteq
W_{\geq-1}(\operatorname{rad}(f))$, so $\operatorname{rad}(f)$ divides
$\operatorname{rad}(g)$. Suppose
$\operatorname{rad}(g)\neq\operatorname{rad}(f)$. Then there exists $\xi\in
V(g)\setminus V(f)$. Let $w\in I\setminus\\{0\\}$ such that
$\operatorname{ord}_{\xi}(w)$ is minimal. Then $[f\partial,w]\in I$, since
$f\partial\in L$ and $I$ is an ideal of $L$. By Lemma 2.11,

$\operatorname{ord}_{\xi}([f\partial,w])=\operatorname{ord}_{\xi}(f)+\operatorname{ord}_{\xi}(w)-1=\operatorname{ord}_{\xi}(w)-1,$

contradicting the minimality of $\operatorname{ord}_{\xi}(w)$. Therefore,
$\operatorname{rad}(g)=\operatorname{rad}(f)$. In particular, this means that
there exists $n\in\mathbb{N}$ such that $g$ divides
$\operatorname{rad}(f)^{n}$, in other words, we have
$W_{\geq-1}(\operatorname{rad}(f)^{n})\subseteq W_{\geq-1}(g)$. We know that
the derived series of $W_{\geq-1}(\operatorname{rad}(f))$ is

$W_{\geq-1}(\operatorname{rad}(f))\supseteq
W_{\geq-1}(\operatorname{rad}(f)^{2})\supseteq
W_{\geq-1}(\operatorname{rad}(f)^{4})\supseteq
W_{\geq-1}(\operatorname{rad}(f)^{8})\supseteq\ldots.$

Since $L\subseteq W_{\geq-1}(\operatorname{rad}(f))$, we have
$D^{k}(L)\subseteq
D^{k}(\operatorname{rad}(f))=W_{\geq-1}(\operatorname{rad}(f)^{2^{k}})$. Let
$m\in\mathbb{N}$ such that $2^{m}\geq n$, so that
$W_{\geq-1}(\operatorname{rad}(f)^{2^{m}})\subseteq
W_{\geq-1}(\operatorname{rad}(f)^{n})\subseteq W_{\geq-1}(g)$. It follows that
$D^{m}(L)\subseteq W_{\geq-1}(g)\subseteq I$. Hence, $L/I$ is solvable. ∎

######  Corollary 4.7.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then all
finite-dimensional irreducible representations of $L$ are one-dimensional.

###### Proof.

Let $M$ be a finite-dimensional irreducible representation of $L$. Then
$\operatorname{Ann}(M)=\\{w\in L\mid w\cdot M=0\\}$ is an ideal of $L$ of
codimension at most $\dim(M)^{2}<\infty$, since $\operatorname{Ann}(M)$ is the
kernel of the map

$\displaystyle\rho\colon L$ $\displaystyle\to\mathfrak{gl}(M)$ $\displaystyle
w$ $\displaystyle\mapsto\rho_{w},$

where $\rho_{w}(m)=w\cdot m$ for $m\in M$. Therefore,
$L/\operatorname{Ann}(M)$ is solvable by Lemma 4.6. Note that $M$ is a finite-
dimensional irreducible representation of $L/\operatorname{Ann}(M)$, so $M$ is
one-dimensional by Lie’s theorem. ∎

######  Remark 4.8.

An infinite-dimensional Lie algebra whose finite-dimensional irreducible
representations are all one-dimensional is not necessarily strongly residually
solvable. For example, all finite-dimensional irreducible representations of
an infinite-dimensional simple Lie algebra $L$ are one-dimensional, but $L$ is
not strongly residually solvable.

We can now show that any subalgebra of $W_{\geq-1}$ of finite codimension has
one-dimensional extensions.

######  Proposition 4.9.

Let $L$ be a proper subalgebra of $W_{\geq-1}$ of finite codimension. Then $L$
has a one-dimensional extension $\overline{L}$ such that the short exact
sequence of $U(L)$-modules

$0\to L\to\overline{L}\to\overline{L}/L\to 0$

is non-split. Furthermore, any such extension can be uniquely embedded in
$W_{\geq-1}$ such that the diagram

${L}$${W_{\geq-1}}$${\overline{L}}$

commutes.

###### Proof.

We start by considering the case where $\dim(W_{\geq-1}/L)=1$, so that
$W_{\geq-1}$ is a one-dimensional extension of $L$. In this case, the short
exact sequence

$0\to L\to W_{\geq-1}\to W_{\geq-1}/L\to 0$

is non-split, since $W_{\geq-1}$ does not have any one-dimensional
$U(L)$-submodules.

Now suppose $\dim(W_{\geq-1}/L)\geq 2$. By Corollary 4.7, $W_{\geq-1}/L$ has a
one-dimensional $U(L)$-submodule $M$. Let $\pi\colon W_{\geq-1}\to
W_{\geq-1}/L$ be the quotient map and let $\overline{L}=\pi^{-1}(M)$. Then
$L\subseteq\overline{L}\subseteq W_{\geq-1}$ and the short exact sequence

$0\to L\to\overline{L}\to M\to 0$

is non-split for the same reason as above: $W_{\geq-1}$ does not have any one-
dimensional $U(L)$-submodules.

The final statement is a restatement of Proposition 3.7. ∎

Proposition 4.9 allows us to prove Theorem 4.1.

###### Proof of Theorem 4.1.

By Proposition 4.9, there is a one-dimensional extension
$L_{1}\subseteq\overline{L}_{1}\subseteq W_{\geq-1}$ of $L_{1}$ such that the
short exact sequence of $U(L)$-modules

$0\to L_{1}\to\overline{L}_{1}\to\overline{L}_{1}/L_{1}\to 0$

is non-split. Now,
$L_{2}\xrightarrow{\varphi^{-1}}L_{1}\hookrightarrow\overline{L}_{1}$ is also
a one-dimensional extension of $L_{2}$. Therefore, Proposition 3.7 implies
that there is an injective map
$\overline{\varphi}\colon\overline{L}_{1}\hookrightarrow W_{\geq-1}$ such that
the diagram

${L_{2}}$${W_{\geq-1}}$${\overline{L}_{1}}$$\scriptstyle{\overline{\varphi}}$

commutes. Let $\overline{L}_{2}=\overline{\varphi}(\overline{L}_{1})$. Now, we
have an isomorphism
$\overline{\varphi}\colon\overline{L}_{1}\to\overline{L}_{2}$ such that
${\left.\kern-1.2pt\overline{\varphi}\vphantom{\big{|}}\right|_{L_{1}}}=\varphi$.
Inducting on $\operatorname{codim}_{W_{\geq-1}}(L_{1})$, we conclude that
$\varphi$ extends to an automorphism of $W_{\geq-1}$. ∎

Theorem 4.1 allows us to determine when two submodule-subalgebras of
$W_{\geq-1}$ are isomorphic. We now describe the automorphism group of
$W_{\geq-1}$. The most “obvious” automorphisms of $W_{\geq-1}$ are those which
are induced by automorphisms of $\Bbbk[t]$. We introduce notation for these
automorphisms of $W_{\geq-1}$.

######  Notation 4.10.

For $n\geq-1$ and $x\in\Bbbk$, we let $e_{n}(x)=(t-x)^{n+1}\partial$. Letting
$\alpha\in\Bbbk^{*}$ and $x\in\Bbbk$, we define a linear map

$\displaystyle\rho_{x;\alpha}\colon W_{\geq-1}$ $\displaystyle\to W_{\geq-1}$
$\displaystyle e_{n}$ $\displaystyle\mapsto\alpha^{n}e_{n}(x)$

In fact, all automorphisms of $W_{\geq-1}$ are of the form $\rho_{x;\alpha}$
for some $x\in\Bbbk,\alpha\in\Bbbk^{*}$. This result appeared without proof in
[Rud86]. For a proof, see [Bav17].

######  Proposition 4.11 ([Rud86]).

We have $\operatorname{Aut}(W_{\geq-1})=\\{\rho_{x;\alpha}\mid
x\in\Bbbk,\alpha\in\Bbbk^{*}\\}\cong\Bbbk\rtimes\Bbbk^{*}$.

As an immediate consequence, we determine exactly when two submodule-
subalgebras of $W_{\geq-1}$ are isomorphic, and compute the automorphism group
of $W_{\geq-1}(f)$.

######  Corollary 4.12.

Let $f,g\in\Bbbk[t]$. Then $W_{\geq-1}(f)\cong W_{\geq-1}(g)$ if and only if
there exist $\alpha,\gamma\in\Bbbk^{*},x\in\Bbbk$ such that $f(s)=\gamma
g(t)$, where $s=\alpha(t-x)$.

Furthermore, the group of automorphisms of $W_{\geq-1}(f)$ is

$\operatorname{Aut}(W_{\geq-1}(f))=\\{\varphi\in\operatorname{Aut}(W_{\geq-1})\mid\varphi(f\partial)=\alpha
f\partial\text{ for some }\alpha\in\Bbbk^{*}\\}.$

###### Proof.

Suppose there is an isomorphism $\varphi\colon W_{\geq-1}(f)\to
W_{\geq-1}(g)$. Then Theorem 4.1 and Proposition 4.11 imply that there exist
$\alpha\in\Bbbk^{*},x\in\Bbbk$ such that
$\varphi={\left.\kern-1.2pt\rho_{x;\alpha}\vphantom{\big{|}}\right|_{W_{\geq-1}(f)}}$.
It follows that $\rho_{x;\alpha}(f\partial)$ is a scalar multiple of
$g\partial$. In other words, $f(\alpha(t-x))$ is a scalar multiple of $g$.

Conversely, suppose that there exist $\alpha\in\Bbbk^{*},x\in\Bbbk$ such that
$f(\alpha(t-x))$ is a scalar multiple of $g$. Then it is clear that
$\rho_{x;\alpha}(W_{\geq-1}(f))=W_{\geq-1}(g)$, so $W_{\geq-1}(f)\cong
W_{\geq-1}(g)$.

The computation of $\operatorname{Aut}(W_{\geq-1}(f))$ is the special case
$g=f$ of the above. ∎

Theorem 4.1 and Proposition 4.11 imply that if $L_{1}$ and $L_{2}$ are
isomorphic subalgebras of $W_{\geq-1}$ of finite codimension, then $L_{1}$ is
a submodule-subalgebra of $W_{\geq-1}$ if and only if $L_{2}$ is also a
submodule-subalgebra. This suggests that we should be able to differentiate
between the subalgebras of $W_{\geq-1}$ which are submodules and those which
are not. One possible way of distinguishing between these is given in the next
result111Proposition 4.13 is a result of a collaboration with Jason Bell..

######  Proposition 4.13.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then
$\dim(L^{\operatorname{ab}})=\operatorname{codim}_{W_{\geq-1}}(L)$ if and only
if $L$ is a $\Bbbk[t]$-submodule of $W_{\geq-1}$.

The proof of Proposition 4.13 will be split into two cases depending on what
the associated graded algebra of $L$ looks like. We will see that if
$\operatorname{gr}(L)\neq W_{\geq r}$ for any $r$, then it is not hard to show
that $\dim(L^{\operatorname{ab}})<\operatorname{codim}_{W_{\geq-1}}(L)$ by
simply looking at degrees of elements of $[L,L]$. Therefore, the difficult
case is when $\operatorname{gr}(L)=W_{\geq r}$ for some $r\in\mathbb{N}$.

First, we introduce some notation.

######  Notation 4.14.

Let $L$ be a subalgebra of $W_{\geq-1}$. We write $\deg(L)=\\{\deg(u)\mid u\in
L\\}$.

We now prove the first case of Proposition 4.13, when
$\operatorname{gr}(L)\neq W_{\geq r}$ for any $r$, in other words, when
$\deg(L)\neq\mathbb{Z}_{\geq r}$ for any $r$.

######  Lemma 4.15.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension and suppose
$\deg(L)\neq\mathbb{Z}_{\geq r}$ for any $r\geq-1$. Then
$\dim(L^{\operatorname{ab}})<\operatorname{codim}_{W_{\geq-1}}(L)$.

###### Proof.

We know that $\deg(L)\supseteq\mathbb{Z}_{\geq r}$ for some $r\in\mathbb{N}$.
Choose $r$ minimal with this property. As a vector space, we can write
$L=L_{\geq
r}\oplus\operatorname{span}\\{g_{1}\partial,\ldots,g_{k}\partial\\}$, where

$L_{\geq
r}=\operatorname{span}\\{f_{r}\partial,f_{r+1}\partial,f_{r+3}\partial,\ldots\\}$

is some subspace of $L$ spanned by elements of degrees $r$ and above,
$\deg(f_{i}\partial)=i$ for all $i$, and $g_{i}\in\mathbb{C}[t]$ are
polynomials such that
$\deg(g_{1}\partial)<\deg(g_{2}\partial)<\ldots<\deg(g_{k}\partial)<r$. By
minimality of $r$, we must have $\deg(g_{k}\partial)<r-1$, and thus
$\deg(g_{1}\partial)<r-k$.

Now, $\operatorname{codim}_{W_{\geq-1}}(L)=r-k+1$, so we want to show that
$\dim(L/[L,L])<r-k+1$. Let $n_{i}=\deg(g_{i}\partial)$. Taking brackets of
$g_{1}\partial$ with elements in $L_{\geq r}$, we can get elements of any
degree greater than or equal to $n_{1}+r$, in other words, $\mathbb{Z}_{\geq
n_{1}+r}\subseteq\deg([L,L])$. In $[L,L]$, we also have the elements
$[g_{1}\partial,g_{i}\partial]$ for $i=2,\ldots,k$. These elements have
degrees $n_{1}+n_{i}$, which are distinct integers less than $n_{1}+r$. In
other words,

$\deg([L,L])\supseteq\mathbb{Z}_{\geq
n_{1}+r}\cup\\{n_{1}+n_{2},n_{1}+n_{3},\ldots,n_{1}+n_{k}\\}.$

We therefore see that $\operatorname{codim}_{W_{\geq-1}}([L,L])\leq
n_{1}+r+1-(k-1)=n_{1}+r-k+2$, and thus

$\dim(L/[L,L])=\operatorname{codim}_{W_{\geq-1}}([L,L])-\operatorname{codim}_{W_{\geq-1}}(L)\\\
\leq n_{1}+r-k+2-(r-k+1)=n_{1}+1<r-k+1,$

as required. ∎

The remaining case is when $\operatorname{gr}(L)=W_{\geq r}$ for some
$r\in\mathbb{N}$. For this case, we will need the following easy lemma.

######  Lemma 4.16.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension such that
$\deg(L)=\mathbb{Z}_{\geq r}$ for some $r\in\mathbb{N}$ and
$\operatorname{codim}_{W_{\geq-1}}(L)=\dim(L^{\operatorname{ab}})$. Let
$g\partial$ be an element of $L$ of degree $r$. Then $[L,L]=[g\partial,L]$.

###### Proof.

Write
$L=\operatorname{span}\\{g\partial,f_{1}\partial,f_{2}\partial,\ldots\\}$,
where $\deg(f_{i}\partial)=\deg(g\partial)+i=r+i$. Then

$[g\partial,L]=\operatorname{span}\\{[g\partial,f_{1}\partial],[g\partial,f_{2}\partial],[g\partial,f_{3}\partial],\ldots\\},$

so $[g\partial,L]$ has elements of degrees $2r+1$ and above. Hence,

$\dim(L/[g\partial,L])=r+1=\operatorname{codim}_{W_{\geq-1}}(L)=\dim(L^{\operatorname{ab}}).$

Since $[g\partial,L]\subseteq[L,L]$, it follows that $[g\partial,L]=[L,L]$. ∎

We are now ready to prove Proposition 4.13.

###### Proof of Proposition 4.13.

Suppose $L$ is a submodule-subalgebra. Then $L=W_{\geq-1}(f)$ for some
$f\in\mathbb{C}[t]$. We have $[L,L]=W_{\geq-1}(f^{2})$, and therefore

$\operatorname{codim}_{W_{\geq-1}}(L)=\dim(L^{\operatorname{ab}})=\deg(f).$

Conversely, suppose $L$ is not a submodule-subalgebra. By Lemma 4.15, we may
assume that $\deg(L)=\mathbb{Z}_{\geq r}$ for some $r\in\mathbb{N}$. Let
$g\partial\in L$ be an element of degree $r$, and write

$L=\operatorname{span}\\{g\partial,g\theta_{1}\partial,g\theta_{2}\partial,\ldots\\},$

where $\theta_{i}\in\mathbb{C}(x)$. Since $g\theta_{i}\in\mathbb{C}[x]$, the
poles of $\theta_{i}$ are limited to roots of $g$. Furthermore, since $L$ is
not a submodule-subalgebra, it must be the case that at least one $\theta_{i}$
has a pole at some root $\lambda$ of $g$. Let $p_{i}$ be the order of
vanishing of $\theta_{i}$ at $\lambda$ (which is negative if $\theta_{i}$ has
a pole at $\lambda$), and let $V=\\{p_{1},p_{2},p_{3},\ldots\\}$. Relabeling
if necessary, assume that $p_{1}<p_{2}<p_{3}<\ldots$. In particular,
$p_{1}<0$, in other words, $\theta_{1}$ has a pole at $\lambda$.

Assume, for a contradiction, that
$\dim(L/[L,L])=\operatorname{codim}_{W_{\geq-1}}(L)$. Lemma 4.16 implies that

$[L,L]=[g\partial,L]=\operatorname{span}\\{g^{2}\theta_{i}^{\prime}\partial\mid
i\geq 1\\},$

since $[g\partial,g\theta_{i}\partial]=g^{2}\theta_{i}^{\prime}\partial$.
Therefore,

$[g\theta_{i}\partial,g\theta_{j}\partial]=g^{2}[\theta_{i}\partial,\theta_{j}\partial]\in\operatorname{span}\\{g^{2}\theta_{i}^{\prime}\partial\mid
i\geq 1\\}.$

It follows that
$[\theta_{i}\partial,\theta_{j}\partial]\in\operatorname{span}\\{\theta_{i}^{\prime}\partial\mid
i\geq 1\\}$ for all $i,j$.

Note that, for $i\neq j$, the order of vanishing of
$[\theta_{i}\partial,\theta_{j}\partial]$ at $\lambda$ is $p_{i}+p_{j}-1$. The
above implies that $p_{i}+p_{j}-1=p_{k}-1$ for some $k$, and therefore
$p_{i}+p_{j}=p_{k}\in V$. Hence, $V$ is closed under addition of distinct
elements. In particular, $p_{1}+p_{2}=p_{n}$ for some $n$. Since $p_{1}<0$, we
must have $p_{n}<p_{2}$, which forces $n=1$. Hence, $p_{2}=0$. Similarly,
$p_{1}+p_{3}=p_{m}$ for some $m$. Now, $p_{1}<0$ implies that $m<3$, while
$p_{3}>p_{2}=0$ implies that $m>1$. Thus, $m=2$, so $p_{1}+p_{3}=p_{2}=0$,
meaning $p_{3}=-p_{1}$.

Therefore, $[\theta_{1}\partial,\theta_{3}\partial]$ has a pole of order 1 at
$\lambda$. But
$[\theta_{1}\partial,\theta_{3}\partial]\in\operatorname{span}\\{\theta_{1}^{\prime}\partial,\theta_{2}^{\prime}\partial,\theta_{3}^{\prime}\partial,\ldots\\}$,
leading to a contradiction. This concludes the proof. ∎

##  5\. Universal property

Up until now, all our results suggest that submodule-subalgebras of
$W_{\geq-1}$ are very rigid, in the sense that their Lie algebraic properties
are controlled by $W_{\geq-1}$. Indeed, we have shown that their derivations
and automorphisms are restrictions of derivations and automorphisms of
$W_{\geq-1}$, and that their one-dimensional extensions are all contained in
$W_{\geq-1}$. Therefore, even when studying submodule-subalgebras of
$W_{\geq-1}$ as abstract Lie algebras, they still seem to remember
$W_{\geq-1}$.

This rigidity is further demonstrated in [PS23], where the authors focused on
the Poisson algebra structure of symmetric algebras of various infinite-
dimensional Lie algebras. In particular, they show that the only elements of
$W_{\geq-1}^{*}$ which vanish on a nontrivial Poisson ideal of $S(W_{\geq-1})$
are what they call _local functions_ ; these are functions given by linear
combinations of derivatives at a finite set of points. They also show that the
same result holds for $S(W_{\geq-1}(f))$, where $f\in\Bbbk[t]$: the only
functions in $W_{\geq-1}(f)^{*}$ which vanish on a nontrivial Poisson ideal of
$S(W_{\geq-1}(f))$ are restrictions of local functions.

In this section, we show that $W_{\geq-1}$ can be intrinsically reconstructed
from any of its subalgebras of finite codimension, purely by considering their
Lie algebraic structure. In particular, we prove that $W_{\geq-1}$ is, in the
appropriate sense, a universal finite-dimensional extension of any of its
subalgebras of finite codimension, which provides an explanation for the
rigidity observed above.

Our goal is to state and prove the universal property satisfied by
$W_{\geq-1}$ as an extension of any of its subalgebras of finite codimension.
We first define the types of extensions we want to consider.

######  Definition 5.1.

Let $L$ be a Lie algebra. We say that $\overline{L}$ is a _completely non-
split extension_ of $L$ if $\overline{L}$ is a Lie algebra containing $L$ such
that there exists a chain of Lie algebras

$L_{0}=L\subseteq L_{1}\subseteq L_{2}\subseteq\ldots\subseteq
L_{n}=\overline{L},$

where for all $i$, the representation $L_{i+1}/L_{i}$ of $L_{i}$ is finite-
dimensional and irreducible, and the following short exact sequence of
$L_{i}$-representations is non-split:

$0\to L_{i}\to L_{i+1}\to L_{i+1}/L_{i}\to 0.$

Applying Proposition 4.9 inductively, we can easily see that $W_{\geq-1}$ is a
completely non-split extension of any of its subalgebras of finite
codimension.

######  Corollary 5.2.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then
$W_{\geq-1}$ is a completely non-split extension of $L$. ∎

Note that the trivial one-dimensional representation is the only irreducible
finite-dimensional representation of $W_{\geq-1}$, since $W_{\geq-1}$ is
simple. By Proposition 1.5,

$\operatorname{Ext}_{U(W_{\geq-1})}^{1}(\Bbbk,W_{\geq-1})=0,$

so $W_{\geq-1}$ does not have any completely non-split extensions. Given a
subalgebra $L$ of $W_{\geq-1}$ of finite codimension, it is therefore
reasonable to expect $W_{\geq-1}$ to be the unique maximal completely non-
split extension of $L$. We will see that this is indeed the case.

Thanks to Corollary 4.7, in order to study completely non-split extensions of
subalgebras of $W_{\geq-1}$, we only need to consider one-dimensional
extensions. In particular, we will be able to use our results from Section 3.

######  Theorem 5.3.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then
$W_{\geq-1}$ is the universal completely non-split extension of $L$, in the
following sense: if $\overline{L}$ is another completely non-split extension
of $L$, then $\overline{L}$ can be uniquely embedded in $W_{\geq-1}$ such that
the diagram

${L}$${W_{\geq-1}}$${\overline{L}}$

commutes.

###### Proof.

By Corollary 5.2, we know that $W_{\geq-1}$ is a completely non-split
extension of $L$.

Suppose $\overline{L}$ is another completely non-split extension of $L$.
Therefore, we have a chain of Lie algebras

$L_{0}=L\subseteq L_{1}\subseteq L_{2}\subseteq\ldots\subseteq
L_{n}=\overline{L},$

where for all $i$, the representation $L_{i+1}/L_{i}$ of $L_{i}$ is finite-
dimensional and irreducible, and the short exact sequence of
$L_{i}$-representations

$0\to L_{i}\to L_{i+1}\to L_{i+1}/L_{i}\to 0$

is non-split. By Corollary 4.7, $L_{1}/L$ is one-dimensional, so $L_{1}$ is a
one-dimensional extension of $L$, and thus $L_{1}$ can be uniquely embedded in
$W_{\geq-1}$ in a way that makes the following diagram commute:

${L}$${W_{\geq-1}}$${L_{1}}$

,

by Proposition 3.7. Continuing inductively, we conclude that there exists an
injective homomorphism $\varphi\colon\overline{L}\to W_{\geq-1}$ such that

(5.3.1) ${L}$${W_{\geq-1}}$${\overline{L}}$$\scriptstyle{\varphi}$

commutes.

To prove uniqueness of $\varphi$, suppose $\psi\colon\overline{L}\to
W_{\geq-1}$ is another injective homomorphism making (5.3.1) commute. Let
$L_{1}=\varphi(\overline{L})$ and $L_{2}=\psi(\overline{L})$. These are two
isomorphic subalgebras of $W_{\geq-1}$ which contain $L$. Let
$\rho=\psi\circ\varphi^{-1}$ be the isomorphism between $L_{1}$ and $L_{2}$.
Since $\rho$ is an isomorphism between two subalgebras of $W_{\geq-1}$ of
finite codimension, Theorem 4.1 and Proposition 4.11 imply that
$\rho={\left.\kern-1.2pt\rho_{x;\alpha}\vphantom{\big{|}}\right|_{L_{1}}}$ for
some $x\in\Bbbk,\alpha\in\Bbbk^{*}$. Note that
${\left.\kern-1.2pt\rho_{x;\alpha}\vphantom{\big{|}}\right|_{L}}={\left.\kern-1.2pt\rho\vphantom{\big{|}}\right|_{L}}=\operatorname{id}_{L}$,
since
${\left.\kern-1.2pt\varphi\vphantom{\big{|}}\right|_{L}}={\left.\kern-1.2pt\psi\vphantom{\big{|}}\right|_{L}}$.
This forces $\alpha=1$ and $x=0$, so that
$\rho_{x;\alpha}=\rho_{0;1}=\operatorname{id}_{W_{\geq-1}}$. Hence,
$\psi=\varphi$. ∎

##  6\. Subalgebras of infinite codimension

Having studied derivations and extensions of subalgebras of $W_{\geq-1}$ of
finite codimension, we now consider the situation in infinite codimension.
Although there are some similarities with the situation in finite codimension,
we will see that there are many differences.

We will need the following subalgebras, first defined in [Buz22].

######  Notation 6.1.

For $f,g\in\Bbbk[t]\setminus\\{0\\}$ such that $f^{\prime}g\in\Bbbk[f]$ (in
other words, $f^{\prime}g=h(f)$ for some $h\in\Bbbk[t]$), we write
$L(f,g)=\Bbbk[f]g\partial$.

Letting $g_{f}\in\Bbbk[t]$ be the unique monic polynomial of minimal degree
such that $f^{\prime}g_{f}\in\Bbbk[f]$, we write $L(f)$ instead of
$L(f,g_{f})$. By [Buz22, Proposition 4.13], $L(f,g)\subseteq L(f)$ for all
$f,g\in\Bbbk[t]$ such that $f^{\prime}g\in\Bbbk[f]$.

Let $f,g\in\Bbbk[t]\setminus\\{0\\}$ such that $f^{\prime}g\in\Bbbk[f]$.
Certainly, if $\deg(f)\geq 2$, then $L(f,g)$ has infinite codimension in
$W_{\geq-1}$. In this case, $L(f,g)$ is clearly not a submodule-subalgebra of
$W_{\geq-1}$, but it is still a $\Bbbk[f]$-submodule. Note that if
$\deg(f)=1$, then $L(f,g)=W_{\geq-1}(g)$.

In [Buz22], it was shown that $L(f,g)$ is a Lie algebra and that $L(f,g)$ has
a familiar Lie algebra structure: it is isomorphic to a submodule-subalgebra
of $W_{\geq-1}$.

######  Lemma 6.2 ([Buz22, Lemma 4.12]).

Let $f,g\in\Bbbk[t]\setminus\\{0\\}$ such that $f^{\prime}g=h(f)\in\Bbbk[f]$
for some $h\in\Bbbk[t]$. Then $L(f,g)$ is a Lie subalgebra of $W_{\geq-1}$ and
$L(f,g)\cong W_{\geq-1}(h)$.

In some sense, these are infinite codimension analogues of submodule-
subalgebras: all infinite-dimensional subalgebras of $W_{\geq-1}$ are very
similar to subalgebras of the form $L(f,g)$, just like subalgebras of finite
codimension are very similar to submodule-subalgebras (cf. Proposition 3.6).

######  Theorem 6.3 ([BB]).

Let $L$ be an infinite-dimensional subalgebra of $W_{\geq-1}$. Then there
exist $f,g\in\Bbbk[t]$ such that $f^{\prime}g\in\Bbbk[f]$ and

$L(f,g)\subseteq L\subseteq L(f).$

In particular, $L$ has finite codimension in $L(f)$, so $L$ is isomorphic to a
subalgebra of $W_{\geq-1}$ of finite codimension.

Theorem 6.3 implies that our results on derivations and extensions can be
translated to arbitrary infinite-dimensional subalgebras of $W_{\geq-1}$.
Therefore, there are many similarities between infinite codimension and finite
codimension subalgebras. However, there are also some key differences, which
we highlight at the end of the section.

For example, Lemma 4.6 and Theorem 6.3 immediately imply:

######  Corollary 6.4.

If $L$ is a proper infinite-dimensional subalgebra of $W_{\geq-1}$, then $L$
is strongly residually solvable. ∎

As a consequence of our study of derivations of submodule-subalgebras of
$W_{\geq-1}$, we can deduce that $L(f)$ has no outer derivations. In order to
see this, we require the following result, which shows that for any
$f\in\Bbbk[t]\setminus\Bbbk$, the subalgebra $L(f)$ is isomorphic to a
submodule-subalgebra $W_{\geq-1}(h)$ with $h\in\Bbbk[t]$ a reduced polynomial.

######  Proposition 6.5.

Let $f\in\Bbbk[t]\setminus\Bbbk$ and let $h\in\Bbbk[t]$ such that
$f^{\prime}g_{f}=h(f)\in\Bbbk[f]$. Then $h$ is reduced (i.e. $h$ is a scalar
multiple of $\operatorname{rad}(h)$).

###### Proof.

Certainly, in order to have $f^{\prime}g_{f}=h(f)$, it must be the case that
$f^{\prime}$ divides $h(f)$. In particular, this implies that $h$ must vanish
on $f(V(f^{\prime}))=\\{f(\lambda)\mid f^{\prime}(\lambda)=0\\}$. We claim
that these are all the roots of $h$, in other words, $h$ is a scalar multiple
of the polynomial

$\widetilde{h}=\prod_{\lambda\in f(V(f^{\prime}))}(t-\lambda).$

This will certainly imply that $h$ is reduced, as required.

In the proof of [Buz22, Proposition 4.13], it was shown that $h$ is the
generator of the ideal

$I(f)\coloneqq\\{p\in\Bbbk[t]\mid f^{\prime}\text{ divides }p(f)\\}$

of $\Bbbk[t]$. As mentioned in the first paragraph, any element of $I(f)$ must
vanish on $f(V(f^{\prime}))$, so $\widetilde{h}$ divides $h$. Therefore, it
suffices to show that $\widetilde{h}\in I(f)=(h)$, that is, $f^{\prime}$
divides $\widetilde{h}(f)=\prod_{\lambda\in f(V(f^{\prime}))}(f-\lambda)$.

Let $\lambda\in V(f^{\prime})$. Then $(t-f(\lambda))$ divides $\widetilde{h}$
by definition, so $f_{\lambda}\coloneqq f-f(\lambda)$ divides
$\widetilde{h}(f)$. Note that
$\operatorname{ord}_{\lambda}(f_{\lambda}^{\prime})=\operatorname{ord}_{\lambda}(f_{\lambda})-1$,
since $f_{\lambda}$ vanishes at $\lambda$. But
$f_{\lambda}^{\prime}=f^{\prime}$, so this means that
$\operatorname{ord}_{\lambda}(f^{\prime})=\operatorname{ord}_{\lambda}(f_{\lambda})-1$.
Therefore,

$\operatorname{ord}_{\lambda}(\widetilde{h}(f))\geq\operatorname{ord}_{\lambda}(f_{\lambda})>\operatorname{ord}_{\lambda}(f^{\prime})$

for all $\lambda\in V(f^{\prime})$. This implies that $f^{\prime}$ divides
$\widetilde{h}(f)$, as claimed. ∎

Theorem 2.2, Lemma 6.2, and Proposition 6.5 immediately imply that $L(f)$ has
no outer derivations.

######  Corollary 6.6.

Let $f\in\Bbbk[t]\setminus\Bbbk$. Then all derivations of $L(f)$ are inner. In
other words, $H^{1}(L(f);L(f))=0$. ∎

Furthermore, we can compute derivations of arbitrary infinite-dimensional
subalgebras of $W_{\geq-1}$, generalising Corollary 3.8. This further
highlights the rigidity exhibited by $W_{\geq-1}$: derivations of any
infinite-dimensional subalgebra of $W_{\geq-1}$ are controlled by
$W_{\geq-1}$.

######  Theorem 6.7.

Let $L$ be an infinite-dimensional subalgebra of $W_{\geq-1}$. Then

$\operatorname{Der}(L)=\\{\operatorname{ad}_{w}\mid w\in W_{\geq-1}\text{ such
that }\operatorname{ad}_{w}(L)\subseteq L\\}\cong N_{W_{\geq-1}}(L).$

In other words, derivations of $L$ are restrictions of derivations of
$W_{\geq-1}$. Consequently, $H^{1}(L;L)\cong N_{W_{\geq-1}}(L)/L$.

###### Proof.

Let $f\in\Bbbk[t]$ such that $L$ has finite codimension in $L(f)$, which
exists by Theorem 6.3. Let $h\in\Bbbk[t]$ such that $f^{\prime}g_{f}=h(f)$, so
that $L(f)\cong W_{\geq-1}(h)$. Let $\varphi\colon L(f)\to W_{\geq-1}(h)$ be
an isomorphism, and let $\widetilde{L}=\varphi(L)\subseteq W_{\geq-1}(h)$,
which is a subalgebra of $W_{\geq-1}$ of finite codimension.

Let $d\in\operatorname{Der}(L)$, and let $\widetilde{d}=\varphi\circ
d\circ\varphi^{-1}\in\operatorname{Der}(\widetilde{L})$. By Corollary 3.8,
$\widetilde{d}=\operatorname{ad}_{\widetilde{w}}$ for some $\widetilde{w}\in
W_{\geq-1}$ such that
$\operatorname{ad}_{\widetilde{w}}(\widetilde{L})\subseteq\widetilde{L}$. We
claim that $\widetilde{w}\in W_{\geq-1}(h)$.

By Proposition 3.6, there exists $g\in\Bbbk[t]$ such that
$W_{\geq-1}(g)\subseteq\widetilde{L}\subseteq
W_{\geq-1}(\operatorname{rad}(g))$. Furthermore,
$W_{\geq-1}(g)\subseteq\widetilde{L}\subseteq W_{\geq-1}(h)$, so $h$ divides
$g$. By Proposition 6.5, $h$ is reduced, so $h$ divides
$\operatorname{rad}(g)$. Therefore,

$W_{\geq-1}(g)\subseteq\widetilde{L}\subseteq
W_{\geq-1}(\operatorname{rad}(g))\subseteq W_{\geq-1}(h).$

Hence, in order to prove the claim, it suffices to prove that
$\widetilde{w}\in W_{\geq-1}(\operatorname{rad}(g))$, in other words, that
$\operatorname{ord}_{\xi}(\widetilde{w})\geq 1$ for all $\xi\in V(g)$.

Assume, for a contradiction, that $\operatorname{ord}_{\xi}(\widetilde{w})=0$
for some $\xi\in V(g)$. Let $\widetilde{x}\in\widetilde{L}$ such that
$\operatorname{ord}_{\xi}(\widetilde{x})$ is minimal. Note that
$\operatorname{ord}_{\xi}(\widetilde{x})\geq 1$, since $\widetilde{x}\in
W_{\geq-1}(\operatorname{rad}(g))$. By Lemma 2.11,
$\operatorname{ord}_{\xi}([\widetilde{w},\widetilde{x}])=\operatorname{ord}_{\xi}(\widetilde{x})-1$.
But $[\widetilde{w},\widetilde{x}]\in\widetilde{L}$, since $\widetilde{w}\in
N_{W_{\geq-1}}(\widetilde{L})$, contradicting the minimality of
$\operatorname{ord}_{\xi}(\widetilde{x})$. This proves the claim.

Now, we have

$d=\varphi^{-1}\circ\widetilde{d}\circ\varphi=\varphi^{-1}\circ\operatorname{ad}_{\widetilde{w}}\circ\varphi.$

Let $w=\varphi^{-1}(\widetilde{w})\in L(f)$. Then,

$d(x)=\varphi^{-1}([\widetilde{w},\varphi(x)])=[w,x]=\operatorname{ad}_{w}(x)$

for all $x\in L$. Hence, $d=\operatorname{ad}_{w}$, which concludes the proof.
∎

######  Remark 6.8.

In the proof of Theorem 6.7, it was shown that if $L$ is a subalgebra of
$W_{\geq-1}$, and $f\in\Bbbk[t]$ is a reduced polynomial such that $L\subseteq
W_{\geq-1}(f)$, then $N_{W_{\geq-1}}(L)\subseteq W_{\geq-1}(f)$.

Another consequence of our previous work is that any infinite-dimensional
subalgebra $L\subseteq W_{\geq-1}$ has a universal completely non-split
extension isomorphic to $W_{\geq-1}$, which follows by combining Theorems 5.3
and 6.3.

######  Corollary 6.9.

Let $L$ be an infinite-dimensional subalgebra of $W_{\geq-1}$. Then $L$ has a
universal completely non-split extension $\overline{L}$: if $X$ is another
completely non-split extension of $L$ then $X$ embeds in $\overline{L}$ such
that the following diagram commutes:

${{L}}$${\overline{L}}$${X}$

.

Furthermore, $\overline{L}\cong W_{\geq-1}$ as Lie algebras. ∎

We now highlight the difference between the situations in finite and infinite
codimension. Consider an infinite-dimensional subalgebra $L\subseteq
W_{\geq-1}$ of infinite codimension. Theorem 6.3 shows that $L$ is isomorphic
to a subalgebra of $W_{\geq-1}$ of finite codimension. Considering the
universal property of Theorem 5.3, it is natural to ask if the universal
completely non-split extension $\overline{L}$ of $L$ embeds in $W_{\geq-1}$ in
a way that makes the diagram

(6.9.1) ${L}$${W_{\geq-1}}$${\overline{L}}$

commute. We show that this is not the case.

######  Proposition 6.10.

Let $L$ be an infinite-dimensional subalgebra of $W_{\geq-1}$ of infinite
codimension. Then the universal completely non-split extension $\overline{L}$
of $L$ cannot be embedded in $W_{\geq-1}$ in a way that makes (6.9.1) commute.

###### Proof.

Suppose we can embed $\overline{L}$ in $W_{\geq-1}$ such that (6.9.1)
commutes. Certainly, the image of $\overline{L}$ in $W_{\geq-1}$ must have
infinite codimension in $W_{\geq-1}$. Therefore, the map

$W_{\geq-1}\cong\overline{L}\hookrightarrow W_{\geq-1}$

is a nonzero endomorphism of $W_{\geq-1}$ which is not an automorphism. By
[Du04, Theorem 2.3], this is impossible. ∎

##  7\. Finite codimension subalgebras of the full Witt algebra

We define submodule-subalgebras of $W$ analogously to those of $W_{\geq-1}$:
they are Lie subalgebras of $W$ which are also submodules over
$\Bbbk[t,t^{-1}]$. They are denoted by $W(f)\coloneqq fW$, where
$f\in\Bbbk[t,t^{-1}]$. Since submodule-subalgebras of $W$ correspond to ideals
of $\Bbbk[t,t^{-1}]$, we will usually assume that $f\in\Bbbk[t]$ and that
$f(0)\neq 0$.

In this section, we prove results about derivations and extensions of
submodule-subalgebras of the Witt algebra similar to those of $W_{\geq-1}$.
Most proofs are omitted, since they are identical to those in Sections 2–5.

###  7.1. Derivations

We start by considering derivations of submodule-subalgebras of the Witt
algebra, similarly to Section 2.

######  Theorem 7.1.

Let $f\in\Bbbk[t]$ such that $f(0)\neq 0$. We have
$\operatorname{Der}(W(f))=\\{\operatorname{ad}_{w}\mid w\in
W(\operatorname{rad}(f))\\}$. Consequently, $\operatorname{Der}(W(f))\cong
W(\operatorname{rad}(f))$, and

$\dim H^{1}(W(f);W(f))=\deg(f)-\deg(\operatorname{rad}(f)).$

We firstly note that the notions of degree and $d$-compatibility are still
well-defined for derivations of submodule-subalgebras of $W$. In other words,
for $d\in\operatorname{Der}(W(f),W)$, we define

$\deg(d)=\max\\{\deg(d(x))-\deg(x)\mid x\in W(f)\\}.$

Since $W(f)$ is a finitely generated Lie algebra, $\deg(d)$ is well-defined by
a similar proof to Lemma 2.4. Similarly, for $d$-compatibility we simply note
that an analogous result to Lemma 2.6 holds in this situation.

One key difference between submodule-subalgebras of $W$ and of $W_{\geq-1}$ is
that the associated graded algebra of $W(f)$ is $W$. Therefore, in order to
compute the associated graded derivation of $d\in\operatorname{Der}(W(f),W)$,
we only need to know about $\operatorname{Der}(W)$, rather than
$\operatorname{Der}(W_{\geq n},W)$ as was the case in Section 2, resulting in
a much simpler picture.

The computation of $\operatorname{Der}(W)$ is similar to that of
$\operatorname{Der}(W_{\geq-1})$ in Proposition 1.5: we simply apply Theorem
1.6.

######  Proposition 7.2.

We have $\operatorname{Der}(W)=\operatorname{Inn}(W)$. ∎

This has been known for a long time. For example, a proof can be found in
[IK90], where the authors compute derivations of a class of Lie algebras known
as _generalised Witt algebras_.

Now we can define associated graded derivations similarly to Section 2. The
proof is identical to Proposition 2.9.

######  Proposition 7.3.

Let $d\in\operatorname{Der}(W(f),W)$ and define a $\Bbbk$-linear map
$\operatorname{gr}(d)\colon W\to W$ by

$\operatorname{gr}(d)(e_{k})=\begin{cases}\operatorname{LT}(d(x)),&\text{where
}x\in W(f)\text{ and }\operatorname{LT}(x)=e_{k},\text{ if }k\text{ is
}d\text{-compatible,}\\\ 0,&\text{if }k\text{ is not
}d\text{-compatible,}\end{cases}$

for $k\in\mathbb{Z}$. Then
$\operatorname{gr}(d)=\lambda\operatorname{ad}_{e_{k}}\in\operatorname{Der}(W)$
for some $\lambda\in\Bbbk$ and $k\in\mathbb{Z}$. ∎

The next result is analogous to Proposition 2.10.

######  Proposition 7.4.

Let $d\in\operatorname{Der}(W(f),W)$. Then
$d=\operatorname{ad}_{\frac{h}{f}\partial}$ for some $h\in\Bbbk[t,t^{-1}]$. ∎

We can now prove Theorem 7.1.

###### Proof of Theorem 7.1.

The proof follows in a similar way to Theorem 2.2. The only difference is that
in $W$ we allow poles at 0, so we need to be a bit more careful. Note that the
proof of Theorem 2.2 only considers $\operatorname{ord}_{\xi}$ for $\xi\in
V(f)$. Since we assumed that $f(0)\neq 0$, we have $0\not\in V(f)$, so we do
not run into any problems by allowing poles at 0. ∎

###  7.2. Extensions

We proceed similarly to Section 3 to compute one-dimensional extensions of
submodule-subalgebras of the Witt algebra. Just like submodule-subalgebras of
$W_{\geq-1}$, one-dimensional extensions of $W(f)$ are either contained in
$W(\operatorname{rad}(f))$ or they come from non-split extensions of
$W/W(t-\xi)$ by $W(f)$ for $\xi\in V(f)$ with $\operatorname{ord}_{\xi}(f)=1$.
We explain the proof of this result, with emphasis on the difference between
this situation and the proof of Theorem 3.3.

######  Theorem 7.5.

Let $f\in\Bbbk[t]$ with $f(0)\neq 0$. If $M\cong W/W(t-\xi)$ for some $\xi\in
V(f)$ with $\operatorname{ord}_{\xi}(f)=1$, then

$\dim(\operatorname{Ext}_{U(W(f))}^{1}(W/W(t-\xi),W(f)))=1.$

In this case, the unique non-split extension of $W/W(t-\xi)$ by $W(f)$ is
$W(\frac{f}{t-\xi})$. If $M\cong\Bbbk$ is trivial, we have

$\dim(\operatorname{Ext}_{U(W(f))}^{1}(\Bbbk,W(f)))=\deg(f)-\deg(\operatorname{rad}(f)).$

Otherwise, $\operatorname{Ext}_{U(W(f))}^{1}(M,W(f))=0$.

As in Section 3, we need to compute the derived subalgebra of $W(f)$ for
$f\in\Bbbk[t]$. Just like in Lemma 3.4, we can easily see that
$[W(f),W(f)]=W(f^{2})$.

######  Lemma 7.6.

For $f\in\Bbbk[t,t^{-1}]$, the derived subalgebra of $W(f)$ is $W(f^{2})$. ∎

We now need to show that one-dimensional extensions of submodule-subalgebras
of $W$ are contained in $W$. The proof of this is nearly identical to
Proposition 3.7.

######  Proposition 7.7.

Let $L$ be a Lie subalgebra of $W$ of finite codimension, and let
$\overline{L}$ be a one-dimensional extension of $L$. Then $\overline{L}$ can
be uniquely embedded in $W$ such that the diagram

${L}$${W}$${\overline{L}}$

commutes. ∎

The only subtlety in the proof of Proposition 7.7 is that, when applying
[PS23, Proposition 3.2.7] to deduce that there exists $f\in\Bbbk[t]$ such that

$W(f)\subseteq L\subseteq W(\operatorname{rad}(f)),$

we need to assume that $f(0)\neq 0$. After this, the proof is exactly the same
as Proposition 3.7, since we never consider any orders of vanishing at 0. The
proof of Theorem 7.5 now follows similarly to Theorem 3.3.

As we did with $W_{\geq-1}$, we summarise the computation of one-dimensional
extensions of submodule-subalgebras of $W$.

######  Corollary 7.8.

Suppose $\overline{L}$ is a one-dimensional extension of $W(f)$, where
$f\in\Bbbk[t]$ and $f(0)\neq 0$. Then either $\overline{L}\subseteq
W(\operatorname{rad}(f))$ or $\overline{L}=W(\frac{f}{t-\xi})$, where $\xi\in
V(f)$ with $\operatorname{ord}_{\xi}(f)=1$. ∎

###  7.3. Isomorphisms between subalgebras of the Witt algebra of finite
codimension

As we did in Section 4, we explain how any isomorphism between subalgebras of
$W$ of finite codimension extends to an automorphism of $W$, analogously to
Theorem 4.1.

######  Theorem 7.9.

Let $L_{1}$ and $L_{2}$ be subalgebras of $W$ of finite codimension and
suppose there is an isomorphism of Lie algebras $\varphi\colon L_{1}\to
L_{2}$. Then $\varphi$ extends to an automorphism of $W$.

The proof of Theorem 7.9 follows by showing that any subalgebra of $W$ of
finite codimension has a one-dimensional extension contained in $W$. As we did
with $W_{\geq-1}$, we can achieve this by showing that any such subalgebra of
$W$ is strongly residually solvable. The proof is exactly the same as Lemma
4.6.

######  Lemma 7.10.

If $L$ is a proper subalgebra of $W$ of finite codimension then $L$ is
strongly residually solvable. ∎

Applying Lie’s theorem, we deduce that all finite-dimensional irreducible
representations of $L$ are one-dimensional, just like Corollary 4.7.

######  Corollary 7.11.

Let $L$ be a subalgebra of $W$ of finite codimension. Then all finite-
dimensional irreducible representations of $L$ are one-dimensional. ∎

We can now show that any subalgebra of $W$ of finite codimension has a one-
dimensional extension. The proof is identical to that of Proposition 4.9.

######  Proposition 7.12.

Let $L$ be a proper subalgebra of $W$ of finite codimension. Then $L$ has a
one-dimensional extension $\overline{L}$ such that the short exact sequence of
$U(L)$-modules

$0\to L\to\overline{L}\to\overline{L}/L\to 0$

is non-split. Furthermore, by Proposition 7.7, all such extensions can be
uniquely embedded in $W$ such that the diagram

${L}$${W}$${\overline{L}}$

commutes. ∎

Automorphisms of $W$ are different to those of $W_{\geq-1}$: in $W$, we no
longer have access to the map $e_{n}\mapsto e_{n}(x)$ for
$x\in\Bbbk\setminus\\{0\\}$, since $(t-x)^{n}\partial\not\in W$ for $n<0$.
Furthermore, in $\Bbbk[t,t^{-1}]$ we have an automorphism $t\mapsto t^{-1}$,
which induces an automorphism of $W$.

######  Notation 7.13.

For $\alpha\in\Bbbk^{*}$. Define linear maps $\sigma_{\alpha},\tau\colon W\to
W$ by

$\sigma_{\alpha}(e_{n})=\alpha^{n}e_{n},\quad\tau(e_{n})=-e_{-n},$

for $n\in\mathbb{Z}$.

Similarly to $W_{\geq-1}$, these give rise to all automorphisms of $W$.

######  Proposition 7.14 ([Bav16, Theorem 1.1]).

We have
$\operatorname{Aut}(W)=\\{\sigma_{\alpha}\mid\alpha\in\Bbbk^{*}\\}\rtimes\\{\tau^{\pm
1}\\}$.

This allows us to determine when two submodule-subalgebras of the Witt algebra
are isomorphic. It is worth noting that, unlike submodule-subalgebras of
$W_{\geq-1}$, we have $W(f)=W(t^{n}f)$ for $f\in\Bbbk[t,t^{-1}]$ and
$n\in\mathbb{Z}$.

######  Corollary 7.15.

Let $f,g\in\Bbbk[t]$ such that $f(0)\neq 0$ and $g(0)\neq 0$ and suppose
$W(f)\cong W(g)$ as Lie algebras. Then one of the following is true:

  1. (1)

There exist $\alpha,\beta\in\Bbbk^{*}$ such that $f(\alpha t)=\beta g(t)$;

  2. (2)

There exist $\alpha,\beta\in\Bbbk^{*}$ and $n\in\mathbb{N}$ such that
$t^{n}f(\alpha t^{-1})=\beta g(t)$. ∎

###  7.4. Universal property

We proceed similarly to Section 5 to prove that the Witt algebra satisfies a
similar universal property to $W_{\geq-1}$. We easily see that $W$ is a
completely non-split extension of any of its subalgebras of finite codimension
by applying Proposition 7.12 inductively.

######  Corollary 7.16.

Let $L$ be a subalgebra of $W$ of finite codimension. Then $W$ is a completely
non-split extension of $L$. ∎

Just like $W_{\geq-1}$, the Witt algebra can be reconstructed from any of its
subalgebras of finite codimension as the universal completely non-split
extension.

######  Theorem 7.17.

Let $L$ be a subalgebra of $W_{\geq-1}$ of finite codimension. Then
$W_{\geq-1}$ is the universal completely non-split extension of $L$, in the
following sense: if $\overline{L}$ is another completely non-split extension
of $L$, then $\overline{L}$ can be uniquely embedded in $W_{\geq-1}$ such that
the diagram

${L}$${W_{\geq-1}}$${\overline{L}}$

commutes.

###### Proof.

The proof follows by an inductive application of Proposition 7.7, similarly to
the proof of Theorem 5.3. As before, uniqueness of the embedding
$\overline{L}\hookrightarrow W$ follows from Theorem 7.9. ∎

We remark that we do not have a classification of subalgebras of $W$ of
infinite codimension, so we do not have any analogous results to Section 6 for
the Witt algebra. This is the subject of ongoing research.

## References

  * [Bav16] Vladimir Bavula, _The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras_, Czechoslovak Math. J. 66 (141) (2016), no. 4, 1129–1141.
  * [Bav17] by same author, _The group of automorphisms of the Lie algebra of derivations of a polynomial algebra_ , J. Algebra Appl. 16 (2017), no. 5, 1750088, 8.
  * [BB] Jason Bell and Lucas Buzaglo, _Maximal dimensional subalgebras of general Cartan type Lie algebras_ , in preparation.
  * [Bjö79] Jan-Erik Björk, _Rings of differential operators_ , North-Holland Mathematical Library, vol. 21, North-Holland Publishing Co., Amsterdam-New York, 1979.
  * [Buz22] Lucas Buzaglo, _Enveloping algebras of Krichever-Novikov algebras are not noetherian_ , Algebr. Represent. Theory (2022), electronic.
  * [DL57] Jacques Dixmier and William Lister, _Derivations of nilpotent Lie algebras_ , Proc. Amer. Math. Soc. 8 (1957), no. 1, 155–158.
  * [Du04] Hong Du, _Endomorphisms of Lie algebra $F[t]\frac{d}{dt}$_, J. Syst. Sci. Complex. 17 (2004), no. 1, 143–146.
  * [DZ98] Dragomir Djoković and Kaiming Zhao, _Derivations, isomorphisms, and second cohomology of generalized Witt algebras_ , Trans. Amer. Math. Soc. 350 (1998), no. 2, 643–664.
  * [Far88] Rolf Farnsteiner, _Derivations and central extensions of finitely generated graded Lie algebras_ , J. Algebra 118 (1988), no. 1, 33–45.
  * [FF80] Boris Feigin and Dmitry Fuks, _Homology of the Lie algebra of vector fields on the line_ , Funktsional. Anal. i Prilozhen. 14 (1980), no. 3, 45–60, 96.
  * [Fia83] Alice Fialowski, _Deformations of a Lie algebra of vector fields on the line_ , Uspekhi Mat. Nauk 38 (1983), no. 1(229), 201–202.
  * [FL96] Boris Feigin and Sergey Loktev, _Cohomology of nilpotent subalgebras of the algebra $\widehat{\rm sl}_{2}$_, Funktsional. Anal. i Prilozhen. 30 (1996), no. 4, 80–83.
  * [FM06] Alice Fialowski and Dmitry Millionschikov, _Cohomology of graded Lie algebras of maximal class_ , J. Algebra 296 (2006), no. 1, 157–176.
  * [FP01] Alice Fialowski and Gerhard Post, _Versal deformation of the Lie algebra $L_{2}$_, J. Algebra 236 (2001), no. 1, 93–109.
  * [FSW13] Jörg Feldvoss, Salvatore Siciliano, and Thomas Weigel, _Outer restricted derivations of nilpotent restricted Lie algebras_ , Proc. Amer. Math. Soc. 141 (2013), no. 1, 171–179.
  * [Fuk86] Dmitry Fuks, _Cohomology of infinite-dimensional Lie algebras_ , Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986, Translated from the Russian by A. B. Sosinskiĭ.
  * [GF68] Israel Gel’fand and Dmitry Fuks, _Cohomologies of the Lie algebra of vector fields on the circle_ , Funktsional. Anal. i Prilozhen. 2 (1968), no. 4, 92–93.
  * [GF69] by same author, _Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold_ , Funktsional. Anal. i Prilozhen. 3 (1969), no. 3, 32–52.
  * [GJP11] Shoulan Gao, Cuipo Jiang, and Yufeng Pei, _Low-dimensional cohomology groups of the Lie algebras $W(a,b)$_, Comm. Algebra 39 (2011), no. 2, 397–423.
  * [Gon73] Lyudmila Goncharova, _Cohomology of Lie algebras of formal vector fields on the line_ , Funktsional. Anal. i Prilozhen. 7 (1973), no. 2, 6–14.
  * [Gra78] Janusz Grabowski, _Isomorphisms and ideals of the Lie algebras of vector fields_ , Invent. Math. 50 (1978), no. 1, 13–33.
  * [Gra93] by same author, _Lie algebras of vector fields and generalized foliations_ , Publ. Mat. 37 (1993), no. 2, 359–367.
  * [Hoc42] Gerhard Hochschild, _Semi-simple algebras and generalized derivations_ , Amer. J. Math. 64 (1942), no. 1, 677–694.
  * [IK90] Toshiharu Ikeda and Naoki Kawamoto, _On the derivations of generalized Witt algebras over a field of characteristic zero_ , Hiroshima Math. J. 20 (1990), no. 1, 47–55.
  * [Jac37] Nathan Jacobson, _Abstract derivation and Lie algebras_ , Trans. Amer. Math. Soc. 42 (1937), no. 2, 206–224.
  * [Koc06] Yuri Kochetkov, _Deformations of the infinite-dimensional Lie algebra $L_{3}$_, Funktsional. Anal. i Prilozhen. 40 (2006), no. 3, 75–81.
  * [Leg53] George Leger, _A note on the derivations of Lie algebras_ , Proc. Amer. Math. Soc. 4 (1953), no. 4, 511–514.
  * [Mil97] Dmitry Millionshchikov, _Krichever-Novikov algebras and the cohomology of the algebra of meromorphic vector fields_ , Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, pp. 101–108.
  * [MSS13] Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki, _Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra_ , Duke Math. J. 162 (2013), no. 5, 965–1002.
  * [PS23] Alexey Petukhov and Susan Sierra, _The Poisson spectrum of the symmetric algebra of the Virasoro algebra_ , Compos. Math. 159 (2023), no. 5, 933–984.
  * [Rud86] Aleksei Rudakov, _Subalgebras and automorphisms of Lie algebras of Cartan type_ , Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 83–84.
  * [Sch14] Martin Schlichenmaier, _Krichever-Novikov type algebras: Theory and applications_ , De Gruyter Studies in Mathematics, vol. 53, De Gruyter, Berlin, 2014.
  * [Sie96] Thomas Siebert, _Lie algebras of derivations and affine algebraic geometry over fields of characteristic $0$_, Math. Ann. 305 (1996), no. 2, 271–286.
  * [SSM17] Farshid Saeedi and Sedigheh Sheikh-Mohseni, _A characterization of stem algebras in terms of central derivations_ , Algebr. Represent. Theory 20 (2017), no. 5, 1143–1150.
  * [Wag99] Friedrich Wagemann, _Some remarks on the cohomology of Krichever-Novikov algebras_ , Lett. Math. Phys. 47 (1999), no. 2, 173–177.
  * [Wag12] by same author, _More remarks on the cohomology of Krichever-Novikov algebras_ , Afr. Mat. 23 (2012), no. 1, 41–52.