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

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.


Introduction
Throughout, we let k be an algebraically closed field of characteristic 0. All vector spaces are k-vector spaces.
Since its inception in the late 1960s [GF68,GF69], the study of cohomology of infinitedimensional 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) ∼ = Der(L)/ Inn(L), where Der(L) is the space of derivations of L and 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 ≥−1 = Der(k[t]) be the one-sided Witt algebra and let W = Der(k[t, t −1 ]) be the Witt algebra (or centerless Virasoro algebra).These are infinite-dimensional Lie algebras which are naturally modules over k[t] and k[t, t −1 ], respectively.We are interested in infinite-dimensional subalgebras of W ≥−1 and W .Of particular interest are those subalgebras of W ≥−1 and W which are also k[t]-or k[t, t −1 ]-submodules, which we call submodule-subalgebras.They are denoted by W ≥−1 (f ) := f W ≥−1 and W (g) := gW for f ∈ k[t] and g ∈ k[t, t −1 ].All subalgebras of W ≥−1 of finite codimension are "essentially" submodule-subalgebras: if L is such a subalgebra, then there exists f ∈ k[t] such that W ≥−1 (f ) ⊆ L ⊆ W ≥−1 (rad(f )), where rad(f ) = ξ∈V (f ) (t − ξ) [PS23, Proposition 3.2.7].Furthermore, by [BB], any infinite-dimensional subalgebra of W ≥−1 is isomorphic to a subalgebra of finite codimension.We can therefore extract a lot of information about infinite-dimensional subalgebras of W ≥−1 by first studying submodule-subalgebras.
Of the submodule-subalgebras of W ≥−1 , those of the form W ≥n := W ≥−1 (t n+1 ) for n ∈ N are of particular significance and have been extensively studied.For example, W ≥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 ≥1 ; W ≥1 ) was computed.Similar results for the Lie algebras W ≥2 and W ≥3 were established in [FP01] and [Koc06].
Submodule-subalgebras of W ≥−1 and W form a large and interesting class of infinitedimensional Lie algebras of linear growth.More generally, one could study submodulesubalgebras of Der(A), where A is an associative k-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 Der(k[X 1 ]) and Der(k[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 ≥−1 .
In particular, , where rad(f ) = ξ∈V (f ) (t − ξ).As a result, For a Lie algebra L, we have where k is the one-dimensional trivial representation of L. We can generalise Theorem 0.1 by computing Ext 1 U (W ≥−1 (f )) (M, W ≥−1 (f )), where f ∈ k[t] and M is an arbitrary one-dimensional representation of W ≥−1 (f ).An element of Ext 1 U (L) (M, L) can be viewed as a non-split short exact sequence of L-representations 0 → L → L → M → 0.
We study such short exact sequences in Section 3 for L a subalgebra of W ≥−1 of finite codimension.In particular, we prove that L can be canonically embedded in W ≥−1 (see Proposition 3.7).
The above results suggest that subalgebras of W ≥−1 of finite codimension are "rigid", in the sense that they inherit their properties from W ≥−1 .We explain this by showing that W ≥−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 ≥−1 of finite codimension.Then W ≥−1 is the universal completely non-split extension of L, in the following sense: W ≥−1 is a completely non-split extension of L, and if L is another completely non-split extension of L, then L can be uniquely embedded in W ≥−1 such that the diagram Given Siebert's result, the rigidity noted above, and the close relationship between subalgebras of W ≥−1 of finite codimension and submodule-subalgebras, it is natural to ask whether any isomorphism between subalgebras of W ≥−1 of finite codimension extends to an automorphism of W ≥−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 ≥−1 of finite codimension and suppose there is an isomorphism of Lie algebras ϕ : L 1 → L 2 .Then ϕ extends to an automorphism of W ≥−1 .
The idea of the proof of Theorem 0.3 is as follows: let ϕ : L 1 → L 2 be an isomorphism between proper subalgebras of W ≥−1 of finite codimension.We show that there exist subalgebras L i of W ≥−1 with L i ⊆ L i ⊆ W ≥−1 and dim(L i /L i ) = 1 for i = 1, 2 such that ϕ extends to an isomorphism ϕ : L 1 → L 2 .The proof then follows by induction on codim W ≥−1 (L 1 ).
In order to prove the existence of the isomorphism ϕ : L 1 → L 2 , we apply our study of non-split short exact sequences of the form 0 → L → L → L/L → 0 where L is a subalgebra of W ≥−1 of finite codimension and dim(L/L) = 1.In other words, Theorem 0.3 follows from the computation of Ext 1 U (L) (M, L), where M is a onedimensional 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 Ext 1 U (W ≥−1 (f )) (M, W ≥−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 ≥−1 (f ) for f ∈ k[t], which corresponds to the case where M = k is the trivial representation.This is because It is easy to show that all derivations of W ≥−1 are inner.Given the rigidity of the situation, it is natural to ask if derivations of submodule-subalgebras of W ≥−1 also arise from the adjoint action of elements of As the Lie algebra W ≥−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 ≥n for n ∈ 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 ≥−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 Ext 1 Consequently, there are many one-dimensional representations of W ≥−1 (f ), corresponding to the representations of the abelian Lie algebra W ≥−1 (f )/W ≥−1 (f 2 ).In other words, the one-dimensional representations of W ≥−1 (f ) are parametrised by A deg(f ) .However, the only one-dimensional modules M which give rise to non-split extensions In this case, the unique non-split extension of 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 ), 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 ≥−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 ≥−1 has a one-dimensional extension contained in W ≥−1 .We then prove the universal property using results from Section 3 about one-dimensional extensions of subalgebras of W ≥−1 .
We then briefly consider subalgebras of W ≥−1 of infinite codimension.Any infinitedimensional subalgebra of W ≥−1 is isomorphic to a subalgebra of W ≥−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 ≥−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 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 Der(W ) = 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.

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.
for a, b ∈ A. We write Der(A) for the set of derivations of A.
We write ]∂ for the onesided Witt algebra and the Witt algebra, respectively, where ∂ = d dt .The Witt algebra is sometimes called the centerless Virasoro algebra.
We let e n = t n+1 ∂ ∈ W for n ∈ Z.The Lie bracket is given by We will also consider the Lie algebra Der(k(t)) = k(t)∂, and view W ≥−1 and W as subalgebras of k(t)∂.

Our main goal in the first part of the paper is to compute Ext
We devote the first two sections to studying the easiest case, where 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 ∈ N, is defined as We are therefore interested in computing H 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 Der(L, M)/ Inn(L, M), defined below.
If L is a Lie algebra and M is a representation of L, we can construct elements of Ext 1 U (L) (k, M) from derivations of L with values in M as follows: given a derivation d ∈ Der(L, M), consider the vector space X = M ⊕ kd with L-action given by the usual action of L on M and w • d = −d(w) for w ∈ 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 Furthermore, we easily see that (1.3.1)splits if and only if d ∈ Inn(L, M).
We therefore see that computing ) is equivalent to computing derivations of W ≥−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.The Lie algebras W and W ≥−1 are Z-graded, with e n having degree n.However, submodule-subalgebras of W ≥−1 are not graded in general: the graded ones are W ≥n for n ≥ −1.Still, ungraded submodule-subalgebras are filtered, with associated graded algebras isomorphic to subalgebras of the form W ≥n .It is therefore natural to begin our study of derivations of submodule-subalgebras of W ≥−1 by considering the graded ones, and later attempt to extract as much information as we can for general submodulesubalgebras via an associated graded argument.
The goal for this section is to compute Der(W ≥n , W ) for n ∈ N, where we view W as a representation of W ≥−1 under the adjoint action.We start with the cases n = −1, 0, 1.The computation of derivations of W ≥−1 was already a well-known folklore result, but we still give a proof here.However, derivations of W ≥n for arbitrary n were not known before.
Proposition 1.5.We have We will use a theorem of Farnsteiner to prove Proposition 1.5.
Theorem 1.6 ([Far88, Proposition 1.2]).Let L = n∈Z L n be a Z-graded finitely generated Lie algebra and let M = n∈Z M n be a graded representation of L such that (1) H 1 (L 0 ; M n ) = 0 for all n ∈ Z \ {0}.
To end the proof, we can easily see that Hom ke 0 (ke n , ke m ) = 0 for n = m, where n ∈ Z ≥−1 and m ∈ Z, since e n and e m have different eigenvalues with respect to the action of e 0 .
The computation of derivations of W ≥0 is almost identical to derivations of W ≥−1 .We therefore omit the proof.
We now move on to computing Der(W ≥n , W ) for n ≥ 2.
Proposition 1.10.For n ∈ N, we have Der(W ≥n , W ) = Inn(W ≥n , W ). Consequently, In order to prove Proposition 1.10, it suffices to compute graded derivations of W ≥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 ≥1 .In other words, we exploited the relation For the proof of Proposition 1.10, we first consider derivations of W ≥n of degree 0 and then find a relation in W ≥n .It is not difficult to show that any derivation of W ≥n of degree 0 must be a multiple of ad e 0 .
Proof.The result has already been shown for n = 0, 1, so we may assume that n ≥ 2.
Let d ∈ Der(W ≥n , W ) 0 .Then for all k ≥ n there exists On the other hand, Combining the above, we conclude that λ k + λ ℓ = λ k+ℓ for all k = ℓ.
and thus Then, for m ≥ n, But we also know that λ n+m+1 = λ m+1 + λ n , so Since λ n = nλ and λ m+1 = λ m + λ for all m ≥ n, it follows by induction that λ m = mλ for all m ≥ n, proving the claim.
We can now prove Proposition 1.10.(1.12.5)Substituting d(e r ) = λ r e r+k and d(e s ) = λ s e s+k into (1.12.5) and simplifying, Now, k(s − r) 3 rs = 0, since r, s, k = 0 and r = s.Hence, for all r, s ≥ n with r = s.For brevity, let We want to show that we still have

Derivations of general submodule-subalgebras
We now consider derivations of general submodule-subalgebras of W ≥−1 .It is standard that Ext groups of a filtered object are related to 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 = n i=m λ i e i ∈ W , where n, m ∈ Z with n > m, and λ i ∈ k with λ n = 0, we call n the degree of w and write deg(w) = n.Furthermore, we call λ n e n the leading term of w and write LT(w where k, ℓ ∈ N and r, s ∈ k[t] do not vanish at ξ.The order of vanishing of f at ξ is ord ξ (f ) = k − ℓ.If ord ξ (f ) < 0, we say f has a pole of order − ord ξ (f ).By convention, ord ξ (0) = ∞ for all ξ ∈ k.Furthermore, for the derivation f ∂ ∈ Der(k(t)), we simply write ord ξ (f ∂) = ord ξ (f ).

The goal of this section is to compute Der
Remark 2.3.A complete Lie algebra is a Lie algebra with trivial center whose derivations are all inner.Theorem 2.2 implies that In order to understand derivations of W ≥−1 (f ), we will construct associated graded derivations, which are graded derivations of W ≥n , where n := deg(f ∂).This will essentially be done by only considering the leading terms of elements of W ≥−1 (f ).The following result is the first step in the construction of associated graded derivations.
By convention, the degree of the zero derivation is −∞.Given Lemma 2.6, we can now make the following definition.

We now consider leading terms of elements
for some (and therefore all) We now complete the construction of associated graded derivations.
The rest of the proof now follows by a straightforward case-by-case analysis, depending on whether k and ℓ are d-compatible.We explicitly show the case where k and ℓ are both d-compatible and leave the rest of the cases to the reader.
We will therefore need to divide the proof into two cases.Case 2: k + N = ℓ and ℓ + N = k.
We have This completes the proof.
We now study Der(W ≥−1 (f ), W ≥−1 ).This is not strictly necessary to prove Theorem 2.2, which only considers Der(W ≥−1 (f )), but it is not much more difficult than considering Der(W ≥−1 (f )), and will be useful for the next section.
Proof.Consider gr(d) ∈ Der(W ≥n , W ≥−1 ).By Proposition 2.9, gr(d) = λ 0 ad e k 0 for some for all m ∈ N, where we set , and thus Combining the above, we conclude that h , then Proposition 2.10 tells us that d = ad g∂ for some g ∈ k(t).For the proof of Theorem 2.2, we need to show that g ∈ rad(f )k[t], in other words, that g ∈ k[t] and that g vanishes at all the roots of f .In order to achieve this, we will consider orders of vanishing of [f ∂, g∂] at roots of f .The following easy lemma computes ord Proof.Write f = (t − ξ) k p and g = (t − ξ) ℓ q, where p ∈ k[t], q ∈ k(t) and ord ξ (p) = ord ξ (q) = 0, so that k = ord ξ (f ) and ℓ = ord ξ (g).Differentiating f and g, we get which concludes the proof.
We are now ready to prove Theorem 2.2.

Extensions of submodule-subalgebras
for M = k, we move on to the computation of the Ext group when M is an arbitrary one-dimensional representation of W ≥−1 (f ).We may view an element of Ext 1 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 ).Therefore, regarding (3.0.1) as a sequence of U(W ≥−1 (f 2 ))-modules, we get an element of ) from the beginning of Section 1).We can use Proposition 2.10 to compute Der(W ≥−1 (f 2 ), W ≥−1 (f )), which then allows us to determine the structure of X as a representation of W ≥−1 (f 2 ).This will give a lot of information into the structure of X as a representation of W ≥−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 → L → L π − → M → 0 be a short exact sequence of U(L)-modules, where L is a Lie algebra and dim(M) = 1.Writing L = L ⊕ kx as a vector space (where π(kx) = M), there is a unique Lie bracket on L extending the action of L on L. This is given by so (3.1.1)holds.
Lemma 3.1 says that studying the group Ext 1 where M is a one-dimensional U(W ≥−1 (f ))-module, is essentially the same as studying Lie algebras L containing W ≥−1 (f ) such that dim(L/W ≥−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 L containing L such that dim(L/L) = 1 and the short exact sequence of U(L)-modules We therefore consider one-dimensional extensions of W ≥−1 (f ), where f ∈ k[t] \ {0}.There are two types of "obvious" one-dimensional extensions of W ≥−1 (f ).Firstly, suppose f is non-reduced, so that there exists w ).The other "obvious" one-dimensional extensions come from removing non-repeated roots of f .In other words, if The main goal of this section is to prove that all one-dimensional extensions of W ≥−1 (f ) are one of these two types.
In this case, the unique non-split extension of The first step in proving Theorem 3.3 is computing the derived subalgebra of W ≥−1 (f ).This will be used to deduce that any one-dimensional representation of W ≥−1 (f ) is trivial when viewed as a representation of

Proof. The result follows by noting that
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 ∈ N. We say that a Lie algebra L is residually nilpotent ) for all n ≥ 1.On the other hand, when f ∈ k[t] is not reduced, one can easily show that W ≥−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 ≥−1 with a nearly identical proof.Proposition 3.6 ([PS23, Proposition 3.2.7]).Let L be a subalgebra of W ≥−1 of finite codimension.Then there exists a monic polynomial f ∈ k[t] \ {0} such that 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 ≥−1 (f ) are contained in W ≥−1 .Note that the proposition considers arbitrary subalgebras of W ≥−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 ≥−1 of finite codimension, and let L be a one-dimensional extension of L. Then L can be uniquely embedded in W ≥−1 such that the diagram Hence, ad x W ≥−1 (f 2 ) ∈ Der(W ≥−1 (f 2 ), W ≥−1 (rad(f ))).By Proposition 2.10, we have where we used that [u, v] ∈ W ≥−1 (f 2 ) in the second equality.Let M = L/L as a U(L)-module.We can regard the short exact sequence , where we view L as a subalgebra of k(t)∂.Under the map ϕ, the short exact sequence (3.7.2) gets mapped to where X is the U(L)-module k(t)∂ ⊕ kx with L acting by its adjoint action on k(t)∂ and the action of L on x agreeing with the bracket on L. We claim that (3.7.3) splits, in other words, that the image of (3.7.2) under ϕ is zero.This will then allow us to conclude that L can be embedded into k(t)∂.
Let y = x − g∂ ∈ X.By (3.7.1), we have [v, [u, y]] = 0 for all u ∈ L \ {0}, v ∈ W ≥−1 (f 2 ) \ {0}.There exist k-linear maps a : L → k(t)∂ and λ : L → k such that [u, y] = a(u) + λ(u)x for all u ∈ L. We can also write where in the final equality we used that [v, x] = [v, g∂] for all v ∈ W ≥−1 (f 2 ).It follows that a(u) + λ(u)g∂ ∈ k(t)∂ is a scalar multiple of v, since the centraliser of any nonzero element of k(t)∂ is one-dimensional.But this must hold for all v ∈ W ≥−1 (f 2 ) \ {0}, so a(u) + λ(u)g∂ = 0. Now, we have for all u ∈ L. Letting x be the image of x in M, the map is a section of (3.7.3), so (3.7.3) splits, as claimed.Suppose g∂ ∈ L. Then y ∈ L, so we could take the codomain of s to be L.In this case, s : M → 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∂ ∈ L, and therefore L ∼ = L ⊕ kg∂.Without loss of generality, we make the identification L = L ⊕ g∂, so we view L as a subalgebra of k(t)∂.
We claim that ord ξ (g) ≥ 0 for all ξ ∈ V (f ).Since g = h f 2 can only have poles at roots of f , this will imply that g ∈ k[t].In other words, this will show that L ⊆ W ≥−1 , which is enough to finish the proof.
Applying the above inductively, we deduce that m − kn ∈ S for all m ∈ S, k ∈ N such that m > kn.Since 2 ∈ S, it follows that S does not contain any elements which are 2 modulo n.However, this contradicts Z ≥ord ξ (f ) ⊆ S. It follows that ord ξ (g) ≥ 0 for all ξ ∈ V (f ), as claimed.
A special case of Proposition 3.7 is a description of derivations of arbitrary subalgebras of W ≥−1 of finite codimension.
Corollary 3.8.Let L be a subalgebra of W ≥−1 of finite codimension.Then In other words, derivations of L are restrictions of derivations of W ≥−1 .Consequently, Having shown that all one-dimensional extensions of W ≥−1 (f ) are contained in W ≥−1 , we can now explicitly compute them to prove Theorem 3.3.
Proof of Theorem 3.3.If M ∼ = k then the result follows by Theorem 2.2.So, assume M is nontrivial as a representation of W ≥−1 (f ).
To end the proof, we claim that g∂ We have already shown that ord µ (g The following corollary, which follows immediately from Theorem 3.3, summarises the two types of one-dimensional extensions which W ≥−1 (f ) has.

Isomorphisms between subalgebras of finite codimension
As an application of our results on extensions of subalgebras of W ≥−1 , we study isomorphisms between subalgebras of W ≥−1 of finite codimension.We prove that any isomorphism between subalgebras of finite codimension extends to an automorphism of W ≥−1 , which allows us to classify submodule-subalgebras of W ≥−1 up to isomorphism and compute automorphism groups of submodule-subalgebras.This will be done by an inductive argument: if we have an isomorphism ϕ : L 1 → L 2 between subalgebras of W ≥−1 , we will show that ϕ 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 ≥−1 of finite codimension and suppose there is an isomorphism of Lie algebras ϕ : L 1 → L 2 .Then ϕ extends to an automorphism of W ≥−1 .
In particular, if f ∈ k[t]\{0}, then Theorem 4.1 implies that automorphisms of W ≥−1 (f ) are restrictions of automorphisms of W ≥−1 .This is similar to Theorem 2.2, which says that derivations of W ≥−1 (f ) are restrictions of derivations of W ≥−1 .
Theorem 4.1 also implies that if L 1 and L 2 are isomorphic subalgebras of Remark 4.2.Theorem 4.1 is not true for arbitrary infinite-dimensional subalgebras of W ≥−1 .Taking f ∈ k[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 ≥−1 , and there exists h ∈ k[t] such that L(f ) ∼ = W ≥−1 (h), by [Buz22, Lemma 4.12].However, this isomorphism clearly does not extend to an automorphism of W ≥−1 .In fact, it does not even extend to an endomorphism of W ≥−1 , since any nonzero endomorphism of W ≥−1 is an automorphism [Du04].
If we have a submodule-subalgebra W ≥−1 (f ), where f ∈ k[t] is non-constant, then we know that W ≥−1 (f ) has one-dimensional extensions (for example, we can take W ≥−1 ( f t−ξ ), where f (ξ) = 0).However, it is not immediately clear whether an arbitrary subalgebra of W ≥−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.
for the derived series of L, where n ∈ N.
Remark 4.4.Analogously to residual nilpotence, we say a Lie algebra L 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 ≥−1 is not residually solvable, since D n (W ≥−1 ) = W ≥−1 for all n ∈ N.
In fact, we will see that W ≥−1 (f ) satisfies a stronger property defined below, provided f ∈ k[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 ⊆ W ≥−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 finitedimensional irreducible representations of L are one-dimensional.
Lemma 4.6.If L is a proper subalgebra of W ≥−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 Since W ≥−1 (rad(f )) is residually solvable, it follows that L is also residually solvable.It remains to show that L/I is solvable.
Note that contradicting the minimality of ord ξ (w).Therefore, rad(g) = rad(f ).In particular, this means that there exists n ∈ N such that g divides rad(f ) n , in other words, we have Corollary 4.7.Let L be a subalgebra of W ≥−1 of finite codimension.Then all finitedimensional irreducible representations of L are one-dimensional.
Proof.Let M be a finite-dimensional irreducible representation of L. Then Ann where ρ w (m) = w • m for m ∈ M. Therefore, L/ Ann(M) is solvable by Lemma 4.6.Note that M is a finite-dimensional irreducible representation of L/ 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 ≥−1 of finite codimension has one-dimensional extensions.
Proposition 4.9.Let L be a proper subalgebra of W ≥−1 of finite codimension.Then L has a one-dimensional extension L such that the short exact sequence of U(L)-modules

any such extension can be uniquely embedded in W
Proof.We start by considering the case where dim(W ≥−1 /L) = 1, so that W ≥−1 is a one-dimensional extension of L. In this case, the short exact sequence is non-split for the same reason as above: W ≥−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 1 is also a one-dimensional extension of L 2 .Therefore, Proposition 3.7 implies that there is an injective map ϕ : L 1 ֒→ W ≥−1 such that the diagram , we conclude that ϕ extends to an automorphism of W ≥−1 .
Theorem 4.1 allows us to determine when two submodule-subalgebras of W ≥−1 are isomorphic.We now describe the automorphism group of W ≥−1 .The most "obvious" automorphisms of W ≥−1 are those which are induced by automorphisms of k[t].We introduce notation for these automorphisms of W ≥−1 .Notation 4.10.For n ≥ −1 and x ∈ k, we let e n (x) = (t − x) n+1 ∂.Letting α ∈ k * and x ∈ k, we define a linear map In fact, all automorphisms of W ≥−1 are of the form ρ x;α for some x ∈ k, α ∈ k * .This result appeared without proof in [Rud86].For a proof, see [Bav17].
As an immediate consequence, we determine exactly when two submodule-subalgebras of W ≥−1 are isomorphic, and compute the automorphism group of W ≥−1 (f ).
Furthermore, the group of automorphisms of Proof.Suppose there is an isomorphism ϕ : W ≥−1 (f ) → W ≥−1 (g).Then Theorem 4.1 and Proposition 4.11 imply that there exist The computation of Aut(W ≥−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 ≥−1 of finite codimension, then L 1 is a submodule-subalgebra of W ≥−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 ≥−1 which are submodules and those which are not.One possible way of distinguishing between these is given in the next result 1 .Proposition 4.13.Let L be a subalgebra of 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 gr(L) = W ≥r for any r, then it is not hard to show that dim(L ab ) < codim W ≥−1 (L) by simply looking at degrees of elements of [L, L].Therefore, the difficult case is when gr(L) = W ≥r for some r ∈ N.
First, we introduce some notation.
We now prove the first case of Proposition 4.13, when gr(L) = W ≥r for any r, in other words, when deg(L) = Z ≥r for any r.
Proof.We know that deg(L) ⊇ Z ≥r for some r ∈ N. Choose r minimal with this property.As a vector space, we can write L = L ≥r ⊕ span{g 1 ∂, . . ., g k ∂}, where is some subspace of L spanned by elements of degrees r and above, deg(f i ∂) = i for all i, and . Taking brackets of g 1 ∂ with elements in L ≥r , we can get elements of any degree greater than or equal to n 1 + r, in other words, Z ≥n 1 +r ⊆ deg([L, L]).In [L, L], we also have the elements [g 1 ∂, g i ∂] for i = 2, . . ., k.These elements have degrees n 1 + n i , which are distinct integers less than n 1 + r.In other words, 1 Proposition 4.13 is a result of a collaboration with Jason Bell.
We therefore see that codim The remaining case is when gr(L) = W ≥r for some r ∈ N.For this case, we will need the following easy lemma.Lemma 4.16.Let L be a subalgebra of W ≥−1 of finite codimension such that deg(L) = Z ≥r for some r ∈ N and codim W ≥−1 (L) = dim(L ab ).Let g∂ be an element of L of degree r.
so [g∂, L] has elements of degrees 2r + 1 and above.Hence, We are now ready to prove Proposition 4.13.
Conversely, suppose L is not a submodule-subalgebra.By Lemma 4.15, we may assume that deg(L) = Z ≥r for some r ∈ N. Let g∂ ∈ L be an element of degree r, and write where θ i ∈ C(x).Since gθ i ∈ C[x], the poles of θ i are limited to roots of g.Furthermore, since L is not a submodule-subalgebra, it must be the case that at least one θ i has a pole at some root λ of g.Let p i be the order of vanishing of θ i at λ (which is negative if θ i has a pole at λ), and let V = {p 1 , p 2 , p 3 , . ..}. Relabeling if necessary, assume that p 1 < p 2 < p 3 < . ... In particular, p 1 < 0, in other words, θ 1 has a pole at λ.
Assume, for a contradiction, that dim for all i, j.Note that, for i = j, the order of vanishing of [θ i ∂, θ j ∂] at λ 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 ∈ 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 .
. .}, leading to a contradiction.This concludes the proof.

Universal property
Up until now, all our results suggest that submodule-subalgebras of W ≥−1 are very rigid, in the sense that their Lie algebraic properties are controlled by W ≥−1 .Indeed, we have shown that their derivations and automorphisms are restrictions of derivations and automorphisms of W ≥−1 , and that their one-dimensional extensions are all contained in W ≥−1 .Therefore, even when studying submodule-subalgebras of W ≥−1 as abstract Lie algebras, they still seem to remember W ≥−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 * ≥−1 which vanish on a nontrivial Poisson ideal of S(W ≥−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 ≥−1 (f )), where f ∈ k[t]: the only functions in W ≥−1 (f ) * which vanish on a nontrivial Poisson ideal of S(W ≥−1 (f )) are restrictions of local functions.
In this section, we show that W ≥−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 ≥−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 ≥−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 L is a completely non-split extension of L if L is a Lie algebra containing L such that there exists a chain of Lie algebras 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: Applying Proposition 4.9 inductively, we can easily see that W ≥−1 is a completely nonsplit extension of any of its subalgebras of finite codimension.
Corollary 5.2.Let L be a subalgebra of W ≥−1 of finite codimension.Then W ≥−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 ≥−1 , since W ≥−1 is simple.By Proposition 1.5, Ext 1 U (W ≥−1 ) (k, W ≥−1 ) = 0, so W ≥−1 does not have any completely non-split extensions.Given a subalgebra L of W ≥−1 of finite codimension, it is therefore reasonable to expect W ≥−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 ≥−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 ≥−1 of finite codimension.Then W ≥−1 is the universal completely non-split extension of L, in the following sense: if L is another completely non-split extension of L, then L can be uniquely embedded in W ≥−1 such that the diagram Proof.By Corollary 5.2, we know that W ≥−1 is a completely non-split extension of L.
Suppose L is another completely non-split extension of L. Therefore, we have a chain of Lie algebras 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 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 ≥−1 in a way that makes the following diagram commute: , by Proposition 3.7.Continuing inductively, we conclude that there exists an injective homomorphism ϕ : L → W ≥−1 such that (5.3.1) To prove uniqueness of ϕ, suppose ψ : L → W ≥−1 is another injective homomorphism making (5.3.1)commute.Let L 1 = ϕ(L) and L 2 = ψ(L).These are two isomorphic subalgebras of W ≥−1 which contain L. Let ρ = ψ • ϕ −1 be the isomorphism between L 1 and L 2 .Since ρ is an isomorphism between two subalgebras of W ≥−1 of finite codimension, Theorem 4.1 and Proposition 4.11 imply that ρ = ρ x;α L 1 for some x ∈ k, α ∈ k * .Note that ρ x;α L = ρ L = id L , since ϕ L = ψ L .This forces α = 1 and x = 0, so that ρ x;α = ρ 0;1 = id W ≥−1 .Hence, ψ = ϕ.

Subalgebras of infinite codimension
Having studied derivations and extensions of subalgebras of W ≥−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].
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 ≥−1 .
In some sense, these are infinite codimension analogues of submodule-subalgebras: all infinite-dimensional subalgebras of W ≥−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).
In particular, L has finite codimension in L(f ), so L is isomorphic to a subalgebra of W ≥−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 ≥−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: As a consequence of our study of derivations of submodule-subalgebras of W ≥−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 Proof.Certainly, in order to have f ′ g f = h(f ), it must be the case that f ′ divides h(f ).In particular, this implies that h must vanish on f (V (f ′ )) = {f (λ) | f ′ (λ) = 0}.We claim that these are all the roots of h, in other words, h is a scalar multiple of the polynomial 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 As mentioned in the first paragraph, any element of I(f ) must vanish on f (V (f ′ )), so h divides h.Therefore, it suffices to show that h for all λ ∈ V (f ′ ).This implies that f ′ divides 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 ∈ k[t] \ k.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 ≥−1 , generalising Corollary 3.8.This further highlights the rigidity exhibited by W ≥−1 : derivations of any infinite-dimensional subalgebra of W ≥−1 are controlled by W ≥−1 .Theorem 6.7.Let L be an infinite-dimensional subalgebra of W ≥−1 .Then

In other words, derivations of L are restrictions of derivations of W
By Proposition 3.6, there exists g so h divides g.By Proposition 6.5, h is reduced, so h divides rad(g).Therefore, Hence, in order to prove the claim, it suffices to prove that w ∈ W ≥−1 (rad(g)), in other words, that ord ξ ( w) ≥ 1 for all ξ ∈ V (g).
Remark 6.8.In the proof of Theorem 6.7, it was shown that if L is a subalgebra of W ≥−1 , and Another consequence of our previous work is that any infinite-dimensional subalgebra L ⊆ W ≥−1 has a universal completely non-split extension isomorphic to W ≥−1 , which follows by combining Theorems 5.3 and 6.3.Corollary 6.9.Let L be an infinite-dimensional subalgebra of W ≥−1 .Then L has a universal completely non-split extension L: if X is another completely non-split extension of L then X embeds in L such that the following diagram commutes: Furthermore, L ∼ = W ≥−1 as Lie algebras.
We now highlight the difference between the situations in finite and infinite codimension.Consider an infinite-dimensional subalgebra L ⊆ W ≥−1 of infinite codimension.Theorem 6.3 shows that L is isomorphic to a subalgebra of W ≥−1 of finite codimension.Considering the universal property of Theorem 5.3, it is natural to ask if the universal completely nonsplit extension L of L embeds in W ≥−1 in a way that makes the diagram (6.9.1) We show that this is not the case.
Proposition 6.10.Let L be an infinite-dimensional subalgebra of W ≥−1 of infinite codimension.Then the universal completely non-split extension L of L cannot be embedded in W ≥−1 in a way that makes (6.9.1) commute.
Proof.Suppose we can embed L in W ≥−1 such that (6.9.1) commutes.Certainly, the image of L in W ≥−1 must have infinite codimension in W ≥−1 .Therefore, the map is a nonzero endomorphism of W ≥−1 which is not an automorphism.By [Du04, Theorem 2.3], this is impossible.

Finite codimension subalgebras of the full Witt algebra
We define submodule-subalgebras of W analogously to those of W ≥−1 : they are Lie subalgebras of W which are also submodules over k[t, t −1 ].They are denoted by W (f ) := f W , where f ∈ k[t, t −1 ].Since submodule-subalgebras of W correspond to ideals of k[t, t −1 ], we will usually assume that f ∈ k[t] and that f (0) = 0.
In this section, we prove results about derivations and extensions of submodule-subalgebras of the Witt algebra similar to those of W ≥−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.
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 ∈ Der(W (f ), W ), we define deg 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 ≥−1 is that the associated graded algebra of W (f ) is W .Therefore, in order to compute the associated graded derivation of d ∈ Der(W (f ), W ), we only need to know about Der(W ), rather than Der(W ≥n , W ) as was the case in Section 2, resulting in a much simpler picture.
The computation of Der(W ) is similar to that of Der(W ≥−1 ) in Proposition 1.5: we simply apply Theorem 1.6.Proposition 7.2.We have Der(W ) = 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.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 ord ξ for ξ ∈ V (f ).Since we assumed that f (0) = 0, we have 0 ∈ 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 ≥−1 , one-dimensional extensions of W (f ) are either contained in W (rad(f )) or they come from non-split extensions of W/W (t − ξ) by W (f ) for ξ ∈ V (f ) with ord ξ (f ) = 1.We explain the proof of this result, with emphasis on the difference between this situation and the proof of Theorem 3.3.Lemma 7.6.For f ∈ k[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 L be a one-dimensional extension of L. Then L can be uniquely embedded in W such that the diagram L W 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 ∈ k[t] such that W (f ) ⊆ L ⊆ W (rad(f )), we need to assume that f (0) = 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 ≥−1 , we summarise the computation of one-dimensional extensions of submodule-subalgebras of W .
Corollary 7.8.Suppose L is a one-dimensional extension of W (f ), where f ∈ k[t] and f (0) = 0. Then either L ⊆ W (rad(f )) or L = W ( f t−ξ ), where ξ ∈ V (f ) with ord ξ (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 ϕ : L 1 → L 2 .Then ϕ 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 ≥−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 finitedimensional 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 L such that the short exact sequence of U(L)-modules 0 → L → L → L/L → 0 is non-split.Furthermore, by Proposition 7.7, all such extensions can be uniquely embedded in W such that the diagram L W L commutes.
Automorphisms of W are different to those of W ≥−1 : in W , we no longer have access to the map e n → e n (x) for x ∈ k \ {0}, since (t − x) n ∂ ∈ W for n < 0. Furthermore, in k[t, t −1 ] we have an automorphism t → t −1 , which induces an automorphism of W .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 ≥−1 , we have W (f ) = W (t n f ) for f ∈ k[t, t −1 ] and n ∈ Z.
7.4.Universal property.We proceed similarly to Section 5 to prove that the Witt algebra satisfies a similar universal property to W ≥−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 ≥−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 ≥−1 of finite codimension.Then W ≥−1 is the universal completely non-split extension of L, in the following sense: if L is another completely non-split extension of L, then L can be uniquely embedded in W ≥−1 such that the diagram 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 L ֒→ 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.
Definition 1.3.A derivation of a Lie algebra L with values in an L-module M is a linear map d : L → M such that d([x, y]) = x • d(y) − y • d(x) for x, y ∈ L. We write Der(L, M) for the set of derivations L with values in M, and simply write Der(L) instead of Der(L, L).A derivation d ∈ Der(L, M) is inner if there exists m ∈ M such that d(x) = x • m.We write Inn(L, M) for the set of inner derivations of L with values in M. As before, we write Inn(L) instead of Inn(L, L).
Proposition 1.4 ([Far88, Proposition 1.1]).If L = n∈Z L n is a Z-graded finitely generated Lie algebra and M = n∈Z M n is a Z-graded representation of L, then Der(L, M) = n∈Z Der(L, M) n .

Proposition 7. 3 .
Let d ∈ Der(W (f ), W ) and define a k-linear map gr(d) : W → W by gr(d)(e k ) = LT(d(x)), where x ∈ W (f ) and LT(x) = e k , if k is d-compatible, 0, if k is not d-compatible, for k ∈ Z. Then gr(d) = λ ad e k ∈ Der(W ) for some λ ∈ k and k ∈ Z.The next result is analogous to Proposition 2.10.Proposition 7.4.Let d ∈ Der(W (f ), W ). Then d = adh f ∂ for some h ∈ k[t, t −1 ].We can now prove Theorem 7.1.
Then gr(d) = λ ad e N ∈ Der(W ≥n , W ) for some λ ∈ k and N ∈ Z. Proof.First of all, by Lemma 2.6, gr(d) is well-defined.If we prove that gr(d) ∈ Der(W ≥n , W ), then the result will follow from Proposition 1.10.Therefore, it suffices to show that gr(d) is a derivation.In other words, it is enough to show that gr(d)([e k , e ℓ ]) = [e k , gr(d)(e ℓ )] + [gr(d)(e k ), e ℓ ], for all k, ℓ ∈ Z ≥n .Let k, ℓ ∈ Z ≥n be distinct, and let u