# Wronskian Envelope of a Lie Algebra

Authors: Laurent Poinsot
Journal: Algebra
 (2013)

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## Abstract

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds
into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivationa↦a′, that is, a commutative differential algebra, admits a Wronskian bracket W(a,b)=ab′−a′b under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.


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## Body

## 1. Introduction

Any (associative) algebra (and more generally a Lie-admissible algebra), sayA, admits a derived structure of Lie algebra under the commutator bracket [a,b]=ab-ba, a,b∈A. This actually describes a forgetful functor, more precisely an algebraic functor, from associative to Lie algebras. This functor admits a left adjoint that enables to associate to any Lie algebra its universal associative envelope. In this way the theory of Lie algebras may be explored through (but not reduced to) that of associative algebras. A Lie algebra which embeds into its universal enveloping algebra is referred to as special. The famous Poincaré-Birkhoff-Witt theorem states that any Lie algebra which is free as a module (and therefore, any Lie algebra over a field) is special. When the Lie algebra is Abelian, then its universal enveloping algebra reduces to the symmetric algebra of its underlying module structure, and thus any commutative Lie algebra is trivially special.However, there is another way to associate an associative algebra to a Lie algebra, and reciprocally, in a functorial way. The idea does not consist anymore to consider noncommutative algebras under commutators but differential commutative algebras together with the so-called Wronskian determinant. A derivation of an algebraA is a linear map ∂:A→A that satisfies the usual Leibniz rule. An algebra with a distinguished derivation is said to be a differential algebra. For any such pair (A,∂) may be defined a bilinear map W:A2→A by W(a,b)=a∂(b)-∂(a)b. When the algebra A is commutative, then W is alternating and satisfies the Jacobi identity so that it defines a Lie bracket on A. The definition of the Lie algebra (A,W) from (A,∂) is functorial (homomorphisms of differential algebras are transformed into homomorphism of Lie algebras) as is the definition of the Lie algebra (A,[·,·]) associated to A from the classical theory. Also as in the classical case, this functor admits a left adjoint that allows us to define a Wronskian envelope for a Lie algebra, that is, a universal commutative and differential algebra.This paper contributes to the study of this functorial relation (an adjointness) between Lie and differential commutative algebras in a way parallel to the classical theory of Lie and associative algebras. In particular, we describe the construction of the Wronskian envelope, and we present a sufficient condition (Theorem16) over a field of characteristic zero for certain Lie algebras to be special, that is, to embed into their Wronskian envelopes. Contrary to the usual case, freeness of the underlying R-module is no more sufficient: there are Lie algebras, free as modules, which do not embed into their Wronksian envelopes. Indeed, special Lie algebras, in this new setting, satisfy a nontrivial relation similar to a relation that holds in Lie algebras of vector fields.


## 2. Differential Algebra

Because the notion of Wronskian envelope is based upon commutative differential algebras, we start by a basic and brief presentation of this theory that is inspired from [1]. Besides we show that any algebra (commutative or not) may be embedded into a differential algebra in a functorial way.LetR be a commutative ring with a unit (it is assumed hereafter to be nonzero), and let A be an R-algebra. An R-derivation∂ of A is an endomorphism of the underlying R-module structure of A that satisfies Leibniz’s rule
(1)∂(ab)=∂(a)b+a∂(b)
for every a,b∈R. We remark that ∂(1A)=0 (1A denoting the unit of A) for if ∂(1A)=∂(1A1A)=∂(1A)+∂(1A). A pair (A,∂) is called a differentialR-algebra. It is said to be commutative when A is so. We note that any R-algebra is actually a differential R-algebra with the zero derivation. Let (A,∂A) and (B,∂B) be two differential R-algebras, and ϕ:A→B be a (unit-preserving) homomorphism of R-algebras. It is a homomorphism of differentialR-algebras when ϕ∘∂A=∂B∘ϕ. Differential (resp., commutative) R-algebras and their homomorphisms form a category denoted by R-DiffAlg (resp., R-𝒞DiffAlg). A differential (two-sided) idealI of a differential R-algebra (A,∂A) is an (two-sided) ideal of the carrier R-algebra A such that ∂A(I)⊆I. It is clear that the quotient-algebra A/I becomes a differential R-algebra in a natural way and that the canonical epimorphism πI:A→A/I is a homomorphism of differential R-algebras. Let S⊆A be a subset. Then, the intersection of all differential ideals of (A,∂A) that contain S is again a differential ideal, called the differential ideal generated byS. We may observe that this differential ideal is the same as the (algebraic) ideal generated by {∂Ai(x):x∈S,i≥0}.The obvious forgetful functor𝒜 (resp., 𝒞𝒜) from R-DiffAlg (resp., R-𝒞DiffAlg) to the category R-Alg (resp., R-𝒞Alg) of R-algebras (resp., commutative R-algebras), that forgets the derivation, has a left adjoint (see [2] for usual category-theoretic definitions). To prove this fact, let us introduce some notations. Let X be any set and M be any monoid with unit 0. Let f∈MX. Its supportsupp(f) is the set of all i∈X such that f(i)≠0. Finitely supported maps are those maps with a finite support, and M(X) is the set of all such maps. Let V be a R-module. We denote by V(ℕ) the R-module of all finitely supported maps from ℕ to V (with point-wise addition and scalar multiplication), namely, ⨁n∈ℕV. For every x∈V and i∈ℕ, let x(i)∈V(ℕ) be defined by x(i)(j)=x if i=j and 0V otherwise; the maps qi:x↦x(i) are the canonical injections of the coproduct V(ℕ). We also denote by 𝒯(V) (resp., 𝒮(V)) the tensor (resp., symmetric) R-algebra of V (see [3]). The natural injection from V to 𝒯(V) (resp., 𝒮(V)) is denoted by nV. Now, let A be an R-algebra (resp., commutative R-algebra), and let us denote by 𝔪𝔬𝔡(A) the underlying R-module structure of A. Let ∂:𝔪𝔬𝔡(A)(ℕ)→𝔪𝔬𝔡(A)(ℕ) be the R-linear endomorphism defined by ∂(x(i))=x(i+1) for every x∈A and i∈ℕ. According to [3, Lemma 4] there exists a unique R-derivation of 𝒯(𝔪𝔬𝔡(A)(ℕ)), again denoted by ∂, that extends ∂. Let C be the two-sided ideal of 𝒯(𝔪𝔬𝔡(A)(ℕ)) generated by x⊗y-y⊗x for every x,y∈𝔪𝔬𝔡(A)(ℕ). Since ∂(C)⊆C, (𝒮(𝔪𝔬𝔡(A)(ℕ)),∂) is a commutative differential R-algebra (by abuse of notations ∂ is the derivation on the quotient algebra). Now, let us consider the (usual) ideal I of 𝒯(𝔪𝔬𝔡(A)(ℕ)) (resp., 𝒮(𝔪𝔬𝔡(A)(ℕ))) generated by (xy)(i)-∑j=0i‍(ij)x(j)⊗y(i-j) for every x,y∈A and i∈ℕ (where the product of elements in A is denoted by a juxtaposition), by (1A)(0)-1 (where 1 is the unit of 𝒯(𝔪𝔬𝔡(A)(ℕ)) and, resp., of 𝒮(𝔪𝔬𝔡(A)(ℕ))) and by (1A)(i) for every i>0. It is clear that πI∘∂:𝒯(𝔪𝔬𝔡(A)(ℕ))→𝒯(𝔪𝔬𝔡(A)(ℕ))/I (resp., πI∘∂:𝒮(𝔪𝔬𝔡(A)(ℕ))→𝒮(𝔪𝔬𝔡(A)(ℕ))/I) factors through the quotient (where πI is the canonical epimorphism from 𝒯(𝔪𝔬𝔡(A)(ℕ)) to 𝒯(𝔪𝔬𝔡(A)(ℕ))/I and, resp., from 𝒮(𝔪𝔬𝔡(A)(ℕ)) to 𝒮(𝔪𝔬𝔡(A)(ℕ))/I), and defines an R-derivation ∂¯ on 𝒯(𝔪𝔬𝔡(A)(ℕ))/I (resp., 𝒮(𝔪𝔬𝔡(A)(ℕ))/I). This differential R-algebra is denoted thereafter by (𝒟(A),∂¯) (resp., (𝒞𝒟(A),∂¯)).Theorem 1.
LetA be an R-algebra (resp., commutative R-algebra). Let (B,∂B) be a differential R-algebra (resp., commutative differential R-algebra), and let ϕ:A→B be a homomorphism of R-algebras. Then, there is a unique homomorphism ϕ¯:(𝒟(A),∂¯)→(B,∂B) of differential R-algebras (resp., ϕ¯:(𝒞𝒟(A),∂¯)→(B,∂B) commutative differential R-algebras) such that ϕ¯(πI(x(0)))=ϕ(x) for every x∈A.Proof.
Letϕ1:𝔪𝔬𝔡(A)(ℕ)→𝔪𝔬𝔡(B) be the unique R-linear map such that ϕ1(x(i))=∂Bi(ϕ(x)) for every x∈A, i≥0. Then, we may define an algebra map ϕ^1:𝒯(𝔪𝔬𝔡(A)(ℕ))→B (resp., ϕ^1:𝒮(𝔪𝔬𝔡(A)(ℕ))→B) using the universal property of the tensor algebra (resp., symmetric algebra). We have ϕ^1(x(i))=∂Bi(ϕ(x)). This map factors through πI. Indeed, let x,y∈A and i≥0. We have ϕ^1((xy)(i))=∂Bi(ϕ(xy))=∂Bi(ϕ(x)ϕ(y))=∑j=0i‍(ij)∂Bj(ϕ(x))∂Bi-j(ϕ(x))=∑j=0i‍(ij)ϕ^1(x(j))ϕ^1(y(i-j))=ϕ^1(∑j=0i‍(ij)x(j)⊗y(i-j)), also ϕ^1((1A)(0))=ϕ(1A)=1B=ϕ^1(1), and ϕ^1((1A)(i))=∂Bi(ϕ(1A))=∂Bi(1B)=0 for every i>0. Therefore, there is a unique homomorphism ϕ¯ of R-algebras (resp., commutative R-algebras) from 𝒟(A) (resp., 𝒞𝒟(A)) to B such that ϕ¯∘πI(x(0))=ϕ(x) for every x∈A. It is easily seen to be a homomorphism of differential (resp., commutative differential) R-algebras.Example 2.
LetR[X]=𝒮(RX) be the free commutative R-algebra over X. Therefore (𝒞𝒟(R[X]),∂¯) is recovered from the usual algebra of differential polynomials R{X} over R (see [4, 5] for instance), that is, the free commutative R-algebra R[X×ℕ] with the R-derivation ∂(x,i)=(x,i+1) for all x∈X, i≥0. Now, if R〈X〉=𝒯(RX) is the free R-algebra over X, then (𝒟(R〈X〉),∂¯) is the (not so wellknown, see [6] however) noncommutative counterpart of R{X}, that is, the free R-algebra R〈X×ℕ〉 with derivation ∂((x1,i1)(x2,i2)⋯(x,in))=(x1,i1+1)(x2,i2)⋯(x,in)+(x1,i1)(x2,i2+1)⋯(xn,in)+⋯+(x1,i1)(x2,i2)⋯(xn,in+1).Remark 3.
It is clear that (commutative) differential algebras form a variety (in the sense of universal algebra, see [7]), and therefore, we may define the free (commutative) differential algebra over a set X. It is not difficult to check that R{X} is the free commutative differential algebra over X and 𝒟(R〈X〉) is the free differential algebra over X. Moreover X embeds into these algebras: X⊆R{X} and X⊆𝒟(R〈X〉).Corollary 4.
The algebraA  (resp., commutative algebra A) embeds into 𝒟(A) (resp., 𝒞𝒟(A)) as a subalgebra, more precisely the map jA:A→𝒟(A) (resp., jA:A→𝒞𝒟(A)), such that jA(x)=πI(x(0)) for every x∈A is a one-to-one algebra homomorphism.Proof.
The mapjA:A→𝒟(A) (resp., jA:A→𝒞𝒟(A)) is easily seen to be an algebra map. Since (A,0) is a differential R-algebra (resp., commutative differential R-algebra) and according to Theorem 1, the identity algebra map idA extends uniquely to a homomorphism id¯A:(𝒟(A),∂¯)→(A,0) (resp., id¯A:(𝒞𝒟(A),∂¯)→(A,0)) of differential R-algebras (resp., commutative differential R-algebras) such that id¯A∘jA=idA, so that jA is one-to-one.From these results (Theorem1 and Corollary 4), we may deduce a Poincaré-Birkhoff-Witt-like theorem. Let us denote by 𝒰(𝔤) the universal enveloping algebra of a Lie algebra 𝔤, and let i𝔤:𝔤→𝒰(𝔤) denote the canonical map which is one-to-one when 𝔤 is a free module (Poincaré-Birkhoff-Witt theorem; see [8], e.g.). The underlying Lie algebra structure, denoted by 𝔩(A), of an associative algebra A is given by the usual commutation bracket.Corollary 5.
Let𝔤 be a Lie algebra (resp., commutative Lie algebra) over R which is free as a R-module. Then, 𝔤 embeds into 𝒟(𝒰(𝔤))  (resp., 𝒞𝒟(𝒰(𝔤))) as a Lie subalgebra. Moreover, if (B,∂B) is a differential R-algebra (resp., commutative differential R-algebra) and ϕ:𝔤→𝔩(B) is a homomorphism of Lie algebras, then there is a unique homomorphism ϕ¯:(𝒟(𝒰(𝔤)),∂¯)→(B,∂B)(resp.,ϕ¯:(𝒞𝒟(𝒰(𝔤)),∂¯)→(B,∂B)) such that ϕ¯∘j𝒰(𝔤)∘i𝔤.Proof.
The proof is easy (essentially a composition of left adjoints), hence omitted.Corollary5 means in particular that the forgetful functor from differential R-algebras (resp., commutative differential R-algebras) to Lie algebras (resp., commutative Lie algebras) R-LieAlg (resp., R-𝒞LieAlg), obtained by composition of the previous “forget-the-derivation” functor from R-DiffAlg to R-Alg (resp., R-𝒞DiffAlg to R-𝒞Alg) and 𝔩 from R-Alg to R-LieAlg (resp., R-𝒞Alg to R-𝒞LieAlg), has a left adjoint, also obtained by composition of the one given by Theorem 1 and the universal enveloping algebra functor. But there is another forgetful functor from R-𝒞DiffAlg to R-LieAlg given by the Wronskian, which is studied in what follows.


## 3. Wronskian Envelope

In this section the Wronskian envelope universally associated to any Lie algebra is constructed, that is, a left adjoint to the forgetful functor from commutative differential algebras to Lie algebras.Let(A,∂A) be a differential commutative R-algebra. Then, it admits a (functorial) structure of a Lie R-algebra for which the bracket is defined by the usual Wronskian determinant
(2)WA(x,y)=x∂A(y)-∂A(x)y=(idA∧∂A)(x∧y)
for every x,y∈A, where ∧ denotes the exterior product (this kind of structure has been used to define n-Lie algebras, see [9]). Let us denote by 𝔴(A,∂A) or more simply 𝔴(A) this Lie algebra structure. Let (B,∂B) be another differential commutative R-algebra, and let ϕ:(A,∂A)→(B,∂B) be a homomorphism of differential commutative R-algebras. Then, ϕ is also a homomorphism of Lie algebras from 𝔴(A) to 𝔴(B) since ϕ(WA(x,y))=ϕ(x∂A(y)-∂A(x)y)=ϕ(x)ϕ(∂A(y))-ϕ(∂A(x))ϕ(y)=ϕ(x)∂B(ϕ(y))-∂B(ϕ(x))ϕ(y)=WB(ϕ(x),ϕ(y)). Hence 𝔴:R-DiffAlg→R-LieAlg defines a (forgetful) functor.Remark 6.
We observe that the Lie algebra structure𝔴(A) associated to the differential R-algebra (A,0) is commutative. Conversely, let (A,∂A) be a commutative differential algebra such that 𝔴(A,∂A) is a commutative Lie algebra; that is, WA(x,y)=0 for every x,y∈A. Since WA(1A,x)=∂A(x), it follows that ∂A is the zero derivative.The functor𝔴 also has a left adjoint (The existence of a left adjoint is guaranteed because 𝔴 is algebraic. See, e.g., [10] for the notion of algebraic functors.) that allows us to define a notion of Wronskian universal enveloping algebra or shortly Wronskian envelope. Let 𝔤 be a Lie algebra over R. Another time let us denote by 𝔪𝔬𝔡(𝔤) its underlying R-module structure. We now consider the symmetric R-algebra 𝒮(𝔪𝔬𝔡(𝔤)(ℕ)) of 𝔪𝔬𝔡(𝔤)(ℕ). As in Section 2, let C be the two-sided ideal of 𝒯(𝔪𝔬𝔡(𝔤)(ℕ)) generated by x⊗y-y⊗x for every x,y∈𝔪𝔬𝔡(𝔤)(ℕ). We have ∂(C)⊆C where, as in Section 2, ∂ is the unique R-derivation of 𝒯(𝔪𝔬𝔡(𝔤)(ℕ)) that extends the R-linear endomorphism ∂:x(i)↦x(i+1) of 𝔪𝔬𝔡((𝔤)(ℕ)). Hence (𝒮(𝔪𝔬𝔡(𝔤)(ℕ)),∂) is a commutative differential R-algebra where, by abuse of language, ∂ is the natural derivation on the quotient algebra. Let us consider the differential ideal J𝔤 generated by [x,y](0)-x(0)⊗y(1)+x(1)⊗y(0) for every x,y∈𝔤. Let σ:𝒮(𝔪𝔬𝔡(𝔤)(ℕ))→𝒮(𝔪𝔬𝔡(𝔤)(ℕ))/J𝔤 be the canonical epimorphism, and let ∂¯ be the (unique) derivation such that ∂¯∘σ=σ∘∂. Let 𝒲(𝔤)=𝒮(𝔪𝔬𝔡(𝔤)(ℕ))/J𝔤.Remark 7.
For anyR-module V, we denote by 𝒮(V)+ the ideal of all members of 𝒮(V) with no constant term (relatively to the usual gradation of the symmetric algebra), and let 𝒮(V)0=R·1 so that 𝒮(V)=𝒮(V)0⊕𝒮(V)+ as R-module. We observe that J𝔤⊆𝒮(𝔪𝔬𝔡(𝔤)(ℕ))+ (since for every x,y∈𝔤, [x,y](0)-x(0)⊗y(1)+x(1)⊗y(0) and all their derivatives [x,y](n)-∑k=0n‍(nk)x(n-k)⊗y(k+1)+∑k=0n‍(nk)x(n-k+1)⊗y(k) belong to 𝒮(𝔪𝔬𝔡(𝔤)(ℕ))+) so that W(𝔤) is not reduced to zero (except if R itself is (0)) because there exists an algebra map from 𝒲(𝔤) onto 𝒮(𝔪𝔬𝔡(𝔤)(ℕ))/𝒮(𝔪𝔬𝔡(𝔤)(ℕ))+≅R·1. The direct sum decomposition 𝒮(𝔪𝔬𝔡(𝔤)(ℕ))=𝒮(𝔪𝔬𝔡(𝔤)(ℕ))0⊕𝒮(𝔪𝔬𝔡(𝔤)(ℕ))+ induces a decomposition 𝒲(𝔤)=𝒲(𝔤)0⊕𝒲(𝔤)+, where 𝒲(𝔤)+ and 𝒲(𝔤)0 are the canonical images of 𝒮(𝔪𝔬𝔡(𝔤)(ℕ))+ and 𝒮(𝔪𝔬𝔡(𝔤)(ℕ))0, and 𝒲(𝔤)0=R·1.In what follows we denote simply byW the Wronskian bracket W𝒮(𝔪𝔬𝔡(𝔤)(ℕ)) of the differential algebra (𝒮(𝔪𝔬𝔡(𝔤)(ℕ)),∂), and the tensor symbol “⊗” is omitted.Proposition 8.
For everyx,y∈𝔤 and every i,j,n≥0,
(3)∂n(W(x(i),y(j)))=∑k=0n‍(nk)W(x(i+n-k),y(j+k)).Proof.
We have(4)∂(W(x(i),y(j)))=∂(x(i)y(j+1)-x(i+1)y(j))=x(i+1)y(j+1)+x(i)y(j+2)-x(i+2)y(j)-x(i+1)y(j+1)=x(i)y(j+2)-x(i+1)y(j+1)+x(i+1)y(j+1)-x(i+2)y(j)=W(x(i),y(j+1))+W(x(i+1),y(j)).
Now, assume by induction on n that
(5)∂n(W(x(i),y(j)))=∑k=0n‍(nk)W(x(i+n-k),y(j+k))
(the cases n=0,1 are checked). We have
(6)∂n+1(W(x(i),y(j)))0000=∂(∂n(W(x(i),y(j))))0000=∂(∑k=0n‍(nk)W(x(i+n-k),y(j+k)))0000=∑k=0n‍(nk)(W(x(i+n+1-k),y(j+k))00000000000000+W(x(i+n-k),y(j+1+k)))0000=(n0)︸=1W(x(i+n+1),y(j))000000+(nn)︸=1W(x(i),y(j+1+n))000000+∑k=1n‍(nk)W(x(i+n+1-k),y(j+k))000000+∑k=0n-1‍(nk)W(x(i+n-k),y(j+1+k))0000=W(x(i+n+1),y(j))+W(x(i),y(j+1+n))000000+∑k=1n‍(nk)W(x(i+n+1-k),y(j+k))000000+∑k=1n‍(nk-1)W(x(i+n+1-k),y(j+k))0000=W(x(i+n+1),y(j))+W(x(i),y(j+1+n))000000+∑k+1n‍((nk-1)+(nk))︸=(n+1k)W(x(i+n+1-k),y(j+k))0000=∑k=0n+1‍(n+1k)W(x(i+n+1-k),y(j+k)).According to Proposition8, J𝔤 is actually the ideal generated by [x,y](n)-∑k=0n‍(nk)W(x(n-k),y(k)) for every x,y∈𝔤 and every n≥0.We claim that(𝒲(𝔤),∂¯) is the universal enveloping algebra of 𝔤 with respect to 𝔴. More precisely, the following holds.Theorem 9.
Let𝔤 be a Lie algebra over R. Let w𝔤:𝔤→𝒲(𝔤) be the map defined by w𝔤(x)=σ(x(0)) for every x∈𝔤. Then, w𝔤 is a homomorphism of Lie algebras from 𝔤 to 𝔴(𝒲(𝔤)). Let (A,∂A) be a commutative differential R-algebra, and let ϕ:𝔤→𝔴(A) be a homomorphism of Lie algebras. Then, there is a unique homomorphism of commutative differential R-algebras ϕ~:(𝒲(𝔤),∂¯)→(A,∂A) such that ϕ~∘w𝔤=ϕ.Proof.
It is quite clear thatw𝔤 is R-linear. Let x,y∈𝔤. Then, we have w𝔤([x,y])=σ([x,y](0))=σ(x(0)⊗y(1)-x(1)⊗y(0))=σ(x(0))σ(y(1))-σ(x(1))σ(y(0))=σ(x(0))σ(∂(y(0)))-σ(∂(x(0)))σ(y(0))=σ(x(0))∂A(σ(y(1)))-∂A(σ(x(0)))σ(y(0))=WB(σ(x(0)),σ(y(0))). This proves that w𝔤 is a homomorphism of Lie algebras. Now, let (A,∂A) be a commutative differential R-algebra, and let ϕ:𝔤→𝔴(A) be a homomorphism of Lie algebras. Let ϕ1:𝔪𝔬𝔡(𝔤)(ℕ)→𝔪𝔬𝔡(A) be the unique R-linear map such that ϕ1(x(i))=∂Ai(ϕ(x)) for every x∈𝔤, i≥0. According to the universal property of the symmetric algebra, there is a unique homomorphism of algebras ϕ^1:S(𝔪𝔬𝔡(𝔤)(ℕ))→A such that ϕ^1(x(i))=ϕ1(x(i)) for every x∈𝔤, i≥0. This map factors through the quotient by J𝔤. Indeed, ϕ^1([x,y](0))=ϕ([x,y])=WA(ϕ(x),ϕ(y))=ϕ(x)∂A(ϕ(y))-∂A(ϕ(x))ϕ(y)=ϕ^1(x(0))ϕ^1(y(1))-ϕ^1(x(1))ϕ^1(y(0))=ϕ^1(x(0)⊗y(1)-x(1)⊗y(0)). Therefore, there is a unique homomorphism of algebras ϕ~:W(𝔤)→A such that ϕ~∘σ=ϕ^1. By construction, it commutes with the derivations; hence, it is a homomorphism of differential R-algebras. Finally, ϕ~∘w𝔤=ϕ also by construction.The commutative differentialR-algebra (𝒲(𝔤),∂¯) of Theorem 9 is called the universal enveloping algebra of  𝔤with respect to𝔴 or the Wronskian envelope of  𝔤.It is easy to prove that any commutative Lie algebra embeds as a sub-Lie algebra into𝒲(𝔤). Moreover the universal enveloping algebra 𝒰(𝔤) of 𝔤 also embeds into 𝒲(𝔤) as a subalgebra but not as a differential subalgebra (when 𝒰(𝔤) is equipped with the zero derivative and 𝒲(𝔤) with its derivative ∂¯). Indeed, let j𝔤:𝔤↪𝒮(𝔪𝔬𝔡(𝔤))≅𝒰(𝔤) be the canonical injection. It is clearly a homomorphism of Lie algebras from 𝔤 to 𝔴(𝒮(𝔪𝔬𝔡(𝔤)),0)=𝔩(𝒮(𝔪𝔬𝔡(𝔤))). Hence by Theorem 9 there is a unique homomorphism of commutative differential algebras j~𝔤:(𝒲(𝔤),∂¯)→(𝒮(𝔪𝔬𝔡(𝔤)),0) such that j~𝔤∘w𝔤=j𝔤. We then deduce that w𝔤 is one-to-one (let x∈kerw𝔤, then 0=j~𝔤(w𝔤(x))=j𝔤(x) so that x=0). Moreover, since w𝔤 is a homomorphism of Lie algebras from 𝔤 to 𝔴(𝒲(𝔤),∂¯) and because 𝔤 is commutative, it is also a homomorphism of Lie algebras from 𝔤 to 𝔴(𝒲(𝔤),0)=𝔩(𝒲(𝔤)). Therefore there is a unique homomorphism of (commutative) algebras w𝔤# from 𝒮(𝔪𝔬𝔡(𝔤)) to 𝒲(𝔤) such that w𝔤#∘j𝔤=w𝔤. Thus, we obtain w𝔤#∘j~𝔤∘w𝔤=w𝔤 and also j~𝔤∘w𝔤#∘j𝔤=j𝔤. According to the universal property of the enveloping algebra, the unique homomorphism that satisfies the second equality is id𝒮(𝔪𝔬𝔡(𝔤)) so that j~𝔤∘w𝔤#=id𝒮(𝔪𝔬𝔡(𝔤)). We observe, however, that we cannot deduce from the first equality that w𝔤#∘j~𝔤=id𝒲(𝔤) because w𝔤#∘j~𝔤 is a homomorphism of commutative differential algebras from (𝒲(𝔤),∂¯) to (𝒲(𝔤),0) and not from (𝒲(𝔤),∂¯) to itself. Nevertheless from j~𝔤∘w𝔤#=id𝒮(𝔤) it follows that w𝔤# is one-to-one and j~𝔤 is onto. We observe that in general when 𝔤 is a commutative Lie algebra, then its universal enveloping algebra with respect to 𝔴 is not (𝒮(𝔪𝔬𝔡(𝔤)),0) that is, j~𝔤 is only onto and not one-to-one, while w𝔤# is only one-to-one and not onto, and the derivative ∂¯ on 𝒲(𝔤) is not the zero derivative. Indeed, for instance let 𝔤 be the free R-module R·x generated by {x} with the zero Lie bracket. The algebra R[x] of polynomials in the variable x is 𝒮(𝔪𝔬𝔡(𝔤))=𝒮(R·x). Let ϕ:R·x→R{x} be the R-module homomorphism defined by ϕ(x)=x. It is a homomorphism of Lie algebras from R·x to 𝔴(R{x}) since 0=ϕ(0)=ϕ([αx,βx]) and αβ(ϕ(x)′ϕ(x)-ϕ(x)ϕ(x)′)=0 for every α,β∈R while the algebra homomorphism extension ϕ#:R[x]→R{x} of ϕ is not a homomorphism of differential algebras from (R[x],0) to R{x} because it does not commute with the derivations (0=ϕ#(0x) and (ϕ#(x))′=x′≠0).


### 3.1. Some Remarks about the Generators ofJ𝔤

Let us denote byPn+1(x,y)=[x,y](n)-∂nux,y for every n≥0 and x,y∈𝔤, where ux,y=[x,y](0)-x(0)y(1)+x(1)y(0). We have Pn+1(x,y)=∑k=0n‍(nk)W(x(n-k),y(k)).Let us introduce the following integers:T(1,0)=-1, T(1,1)=1, and, for all n≥2 and all k=0,…,n,
(7)T(n,k)={T(n-1,0)00ifk=0,T(n-1,n-1)00ifk=n,T(n-1,k-1)+T(n-1,k)00ifk=1,…,n-1.Lemma 10.
For everyn≥1 and every k=0,…,n,
(8)T(n,k)=(nk)-2(n-1k).Proof.
Recall that(nk)=0 for every k>n≥0. We have T(1,0)=(10)-2(00)-=1-2=-1, and T(1,1)=(11)-2(01)=1. For every n≥1, T(n,0)=(n0)-2(n-10)=1-2=-1 and T(n,n)=(nn)-2(n-1n)=1. For every n≥2 and k=0,…,n, we have (n0)-2(n-10)=1-2=-1=(n-10)-2(n-20), (nn)-2(n-1n)=1=(n-1n-1)-2(n-2n-1). Now, let k=1,…,n-1. If k≤n-2, then (nk)-2(n-1k)=(n-1k-1)-2(n-2k-1)+(n-1k)-2(n-2k), and if k=n-1, then (nn-1)-2(n-1n-1)=n-2 while (n-1n-2)-2(n-2n-2)+(n-1n-1)-2(n-2n-1)=n-1-2+1-0=n-2.Lemma 11.
For everyn≥1 and every x,y∈𝔤, one has
(9)Pn(x,y)=∑k=0n‍T(n,k)x(n-k)y(k).Proof.
We haveP1(x,y)=x(0)y(1)-x(1)y(0)=T(1,0)x(1)y(0)+T(1,1)x(0)y(1). It is clear that Pn(x,y) is homogeneous of degree n (in the sense that it is a sum of word x(i)y(j) with i+j=n). It can be written as Pn(x,y)=∑k=0n‍αn,kx(n-k)y(k). We have Pn+1(x,y)=∂Pn(x,y)=∑k=0n‍αn,kx(n+1-k)y(k)+∑k=0n‍αn,kx(n-k)y(k+1)=αn,0x(n+1)+αn,ny(n+1)+∑k=1n‍αn,kx(n+1-k)y(k)+∑k=0n-1‍αn,kx(n-k)y(k+1) so that αn+1,0=αn,0, αn+1,n+1=αn,n and αn+1,k=αn,k-1+αn,k.


## 3.1. Some Remarks about the Generators ofJ𝔤

Let us denote byPn+1(x,y)=[x,y](n)-∂nux,y for every n≥0 and x,y∈𝔤, where ux,y=[x,y](0)-x(0)y(1)+x(1)y(0). We have Pn+1(x,y)=∑k=0n‍(nk)W(x(n-k),y(k)).Let us introduce the following integers:T(1,0)=-1, T(1,1)=1, and, for all n≥2 and all k=0,…,n,
(7)T(n,k)={T(n-1,0)00ifk=0,T(n-1,n-1)00ifk=n,T(n-1,k-1)+T(n-1,k)00ifk=1,…,n-1.Lemma 10.
For everyn≥1 and every k=0,…,n,
(8)T(n,k)=(nk)-2(n-1k).Proof.
Recall that(nk)=0 for every k>n≥0. We have T(1,0)=(10)-2(00)-=1-2=-1, and T(1,1)=(11)-2(01)=1. For every n≥1, T(n,0)=(n0)-2(n-10)=1-2=-1 and T(n,n)=(nn)-2(n-1n)=1. For every n≥2 and k=0,…,n, we have (n0)-2(n-10)=1-2=-1=(n-10)-2(n-20), (nn)-2(n-1n)=1=(n-1n-1)-2(n-2n-1). Now, let k=1,…,n-1. If k≤n-2, then (nk)-2(n-1k)=(n-1k-1)-2(n-2k-1)+(n-1k)-2(n-2k), and if k=n-1, then (nn-1)-2(n-1n-1)=n-2 while (n-1n-2)-2(n-2n-2)+(n-1n-1)-2(n-2n-1)=n-1-2+1-0=n-2.Lemma 11.
For everyn≥1 and every x,y∈𝔤, one has
(9)Pn(x,y)=∑k=0n‍T(n,k)x(n-k)y(k).Proof.
We haveP1(x,y)=x(0)y(1)-x(1)y(0)=T(1,0)x(1)y(0)+T(1,1)x(0)y(1). It is clear that Pn(x,y) is homogeneous of degree n (in the sense that it is a sum of word x(i)y(j) with i+j=n). It can be written as Pn(x,y)=∑k=0n‍αn,kx(n-k)y(k). We have Pn+1(x,y)=∂Pn(x,y)=∑k=0n‍αn,kx(n+1-k)y(k)+∑k=0n‍αn,kx(n-k)y(k+1)=αn,0x(n+1)+αn,ny(n+1)+∑k=1n‍αn,kx(n+1-k)y(k)+∑k=0n-1‍αn,kx(n-k)y(k+1) so that αn+1,0=αn,0, αn+1,n+1=αn,n and αn+1,k=αn,k-1+αn,k.


## 4. Embedding Conditions

In this section, we present a sufficient condition under which a Lie algebra embeds into its Wronskian envelope.Adapting the terminology from [7, 11], we call Wronskian special those Lie R-algebras that embed into their Wronskian envelope. Thus every Abelian Lie algebra is Wronskian special (see Section 3). It is quite obvious that not all Lie algebras are Wronskian special, even when they are free as R-modules and even in the case where R is a field. This can be shown as follows. For any elements x1,…,xn of a Lie algebra, we denote by [x1,…,xn] the left-normed bracket; that is, if n=2, then [x1,x2] is the Lie bracket, and for n>2, [x1,…,xn]=[[x1,…,xn-1],xn].Let(A,∂) be a commutative differential R-algebra (with a unit). Then, the Lie algebra 𝔴(A,∂) satisfies the nontrivial identity (called the “standard Lie identity of degree 5” in [12] and T4 in [13])
(10)∑σ∈𝔖4‍ϵ(σ)[z,xσ(1),xσ(2),xσ(3),xσ(4)],
where ϵ(σ) denotes the signature representation of the permutation σ, for every x1,x2,x3,x4,y∈A. (This result was noticed in [12–14], e.g.) Therefore, a necessary condition for an embedding is the following.Lemma 12.
A Wronskian special Lie algebra satisfiesT4.It is not known whether the above lemma is also a sufficient condition. Nevertheless this gives a negative answer to a question of Lawvere in his Ph.D thesis [15] where he asked whether or not any Lie algebra is Wronskian special.Following [16], let us define the Lie algebra Diff∂(A) of special derivations with respect to the signature derivation∂ as follows: let A·∂={α∂:α∈A} be the sub-A-module of DerR(A), the A-module of all R-derivations of A, generated by ∂ (it is the image of the A-module map ρ∂:A→DerR(A) given by ρ(α)=α∂). Then, Diff∂(A) is the R-module structure of A·∂ together with the usual bracket of derivations (inherited by the usual bracket of 𝔤𝔩R(A)=EndR-Mod(A) the linear endomorphisms of A). Let ann(∂)={a∈A:a∂=0} be the annihilator of ∂. It is an Abelian Lie subalgebra of 𝔴(A,∂) and ann(∂)↪𝔴(A,∂)↠ρ∂Diff∂(A) is a short exact sequence of Lie R-algebras in such a way that 𝔴(A,∂) is an extension of Diff∂(A) by ann(∂). In the case where kerρ∂=ann(∂)=0—for instance when A has no zero divisors and ∂≠0—then Diff∂(A) and 𝔴(A,∂) are isomorphic Lie R-algebras. The following result is thus obvious.Lemma 13.
LetA be a commutative integral R-domain, and let ∂ be a nonzero derivation of A. Let 𝔤 be a Lie R-algebra such that 𝔤↪Diff∂(A) (as Lie R-algebras) (we note that in particular 𝔤 satisfies T4 since it holds in Diff∂(A)). Then, 𝔤 is a Wronskian special Lie algebra.Example 14.
The Lie algebra𝔰𝔩2(𝕂) over a field of characteristic zero is simple and embeds into Diff∂(𝕂[x]) where ∂ is the usual derivation of polynomials in the variable x. Indeed, 𝔰𝔩2(𝕂) is isomorphic to the three-dimensional 𝕂-vector space generated by e=-x2∂, f=∂, h=2x∂ with the Lie algebra structure given by the usual commutator (see, e.g., [17]). Therefore, 𝔰𝔩2(𝕂) is a Wronskian special Lie algebra.We may also remark that Wronskian speciality is preserved by direct products (even infinite). More precisely, let(𝔤i)i∈I be a collection of Lie R-algebras such that for each i∈I, 𝔤i↪𝔴(𝒲(𝔤i)) (as Lie R-algebras), that is, each factor 𝔤i is a Wronskian special Lie algebra. Then, ∏i∈I‍𝔤i↪𝔴(𝒲(∏i∈I‍𝔤i)) as Lie R-algebras, where the operations (Lie brackets, derivative, and product) are considered component-wise (i.e., ∏i∈I‍𝔤i is a Wronskian special Lie algebra). Indeed, it is clear that ∏i∈I‍𝔤i↪∏i∈I‍𝔴(𝒲(𝔤i))=𝔴(∏i∈I‍𝒲(𝔤i)) as a sub-Lie algebra over R. It is also clear that any sub-Lie algebra of a Wronskian special Lie algebra is itself a Wronskian special Lie algebra. Thus, Wronskian speciality is closed under product and subalgebra.Now, we recall an important result from Razmyslov that is used hereafter to describe some basic Wronskian special Lie algebras.Theorem 15 (see [12, 16]).
Let𝕂 be a field of characteristic zero. Let 𝔤 be a simple Lie algebra that satisfies T4. Then, there exists a commutative integral 𝕂-domain A and a nonzero derivation ∂∈Der𝕂(A) such that 𝔤 embeds into Diff∂(A) as a Lie subalgebra.As a consequence of the previous theorem and Lemma13, in characteristic zero, any simple Lie algebra satisfying T4 is a Wronskian special Lie algebra. From this result we deduce the following.Theorem 16.
Let𝕂 be a field of characteristic zero. Let (𝔤i)i∈I be a family of Lie 𝕂-algebras such that for every i∈I either 𝔤i is simple and satisfies T4 or 𝔤i is Abelian. Let 𝔤 be a sub-Lie algebra of ∏i∈I‍𝔤i. Then, 𝔤 is a Wronskian special Lie algebra.Proof.
According to Razmyslov’s theorem if𝔤i is simple and satisfies T4 then 𝔤i is Wronskian special. We also know that any Abelian Lie algebra is Wronskian special. So their product ∏i∈I‍𝔤i also is. Finally any sub-Lie algebra of a Wronskian special Lie algebra also is Wronskian special.


## 5. The Free Wronskian Special LieR-Algebra

Up to now it is not known if Wronskian special Lie algebras form a variety of Lie algebras (nor even if it closed under homomorphic images). In [13] it is conjectured that (finitely generated) Wronskian special Lie algebras form the variety of all (finitely generated) Lie algebras satisfying T4 (in characteristic zero). We do not prove nor disprove this conjecture. Nevertheless we observe that Wronskian special Lie R-algebras, for some ring R, form a derived category of the variety of commutative differential algebras obtained by considering their Lie bracket as the derived operator W(x,y)=x∂(y)-∂(x)y (see [7] for definitions of derived category and derived operator). Therefore, we may speak about the free Wronskian special LieR-algebra generated by a setX (see [7, Theorem 4.4]). It is obtained as the Lie subalgebra of 𝔴(R{X}) generated by X, where we recall that R{X} is the free commutative differential algebra over X (if u∈R{X}, then u′ denotes the derivation of u in R{X} and its Wronskian commutator is denoted by W(u,v)=uv′-u′v). It is constructed by induction as follows: let
(11)X0=X,Xk+1=〈Xk∪{w∈R{X}:∃u,v∈Xk2,0000000w=W(u,v)w∈R{X}:∃u,v∈Xk2}〉00∀k≥0,
where 〈E〉 denotes the submodule of R{X} generated by a subset E. Then, the free Wronskian special Lie algebra, denoted by WspLie(X), over X is the nested union ⋃k≥0‍Xk, and its bracket is given by the Wronskian bracket W of R{X}. (It is clear that as defined previously WspLie(X) is a R-module.)Remark 17.
It is clear that ifu,v∈R{X} have no nonzero constant term, then it is also the case for W(u,v). Moreover every element of WspLie(X) has no nonzero constant term as it can be checked inductively from the previous observation and since every element of X0=X has no nonzero constant term. Therefore, WspLie(X)≠R{X} as sets so that WspLie(X)≠𝔴(R{X}) as Lie algebras. (Note that when X=∅, then WspLie(∅)=0 while R{∅}≅R.) When X is reduced to only one element x, then WspLie({x}) is the free module Rx seen as an Abelian Lie algebra and, therefore, is isomorphic to the free Lie algebra on one generator.We observe that since commutative differential algebras form a nontrivial variety (a nontrivial variety is a variety with algebras of cardinality >1), the natural set-theoretic map fromX to R{X} is one-to-one, and we may assume that X⊆WspLie(X)⊆R{X} (in particular, X is free over R in WspLie(X)). According to the property of the Wronskian envelope, there is a unique differential algebra map ϕ:𝒲(WspLie(X))→R{X} such that the following diagram commutes (where the unnamed arrows are the canonical inclusions)
(12)
Moreover the composition X↪WspLie(X)↪𝔴(𝒲(WspLie(X))) of natural embeddings gives rise to a set-theoretic map X↪𝒲(WspLie(X)). Therefore, there is a unique homomorphism of commutative differential algebras ψ:R{X}→𝒲(WspLie(X)) such that the following diagram commutes:
(13)
Both compositions ψ∘ϕ and ϕ∘ψ are the identity on X and hence, by uniqueness, the identity everywhere. Therefore, 𝒲(WspLie(X)) and R{X} are canonically isomorphic (as commutative differential algebras):
(14)W(WspLie(X))≅R{X}.Remark 18.
The previous result shares some similarity with the well-known fact that the universal enveloping algebra of the free Lie algebra generated byX is canonically isomorphic to the free associative algebra generated by X (see [3]).Remark 19.
Let us assume thatR is an integral domain. Then, R{X} is also an integral domain, and 𝔴(R{X})≅Diff∂(R{X}) (as Lie algebras over R), where ∂ is the usual derivation on R{X} (i.e, ∂(u)=u′). Since   Diff∂(R{X}) is a sub-Lie algebra of 𝔤𝔩R(R{X}) (under the commutator), it follows that 𝔴(R{X}) is also a special Lie algebra.



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