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 derivation , that is, a commutative differential algebra, admits a Wronskian bracket 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.

1. Introduction

Any (associative) algebra (and more generally a Lie-admissible algebra), say , admits a derived structure of Lie algebra under the commutator bracket , . 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 algebra is a linear map that satisfies the usual Leibniz rule. An algebra with a distinguished derivation is said to be a differential algebra. For any such pair may be defined a bilinear map by . When the algebra is commutative, then is alternating and satisfies the Jacobi identity so that it defines a Lie bracket on . The definition of the Lie algebra from is functorial (homomorphisms of differential algebras are transformed into homomorphism of Lie algebras) as is the definition of the Lie algebra associated to 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 (Theorem 16) 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 -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.

Let be a commutative ring with a unit (it is assumed hereafter to be nonzero), and let be an -algebra. An -derivation of is an endomorphism of the underlying -module structure of that satisfies Leibniz’s rule for every . We remark that ( denoting the unit of ) for if . A pair is called a differential -algebra. It is said to be commutative when is so. We note that any -algebra is actually a differential -algebra with the zero derivation. Let and be two differential -algebras, and be a (unit-preserving) homomorphism of -algebras. It is a homomorphism of differential -algebras when . Differential (resp., commutative) -algebras and their homomorphisms form a category denoted by - (resp., ). A differential (two-sided) ideal of a differential -algebra is an (two-sided) ideal of the carrier -algebra such that . It is clear that the quotient-algebra becomes a differential -algebra in a natural way and that the canonical epimorphism is a homomorphism of differential -algebras. Let be a subset. Then, the intersection of all differential ideals of that contain is again a differential ideal, called the differential ideal generated by . We may observe that this differential ideal is the same as the (algebraic) ideal generated by .

The obvious forgetful functor (resp., ) from - (resp., ) to the category - (resp., ) of -algebras (resp., commutative -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 be any set and be any monoid with unit . Let . Its support is the set of all such that . Finitely supported maps are those maps with a finite support, and is the set of all such maps. Let be a -module. We denote by the -module of all finitely supported maps from to (with point-wise addition and scalar multiplication), namely, . For every and , let be defined by if and otherwise; the maps are the canonical injections of the coproduct . We also denote by (resp., ) the tensor (resp., symmetric) -algebra of (see [3]). The natural injection from to (resp., ) is denoted by . Now, let be an -algebra (resp., commutative -algebra), and let us denote by the underlying -module structure of . Let be the -linear endomorphism defined by for every and . According to [3, Lemma 4] there exists a unique -derivation of , again denoted by , that extends . Let be the two-sided ideal of generated by for every . Since , is a commutative differential -algebra (by abuse of notations is the derivation on the quotient algebra). Now, let us consider the (usual) ideal of (resp., ) generated by for every and (where the product of elements in is denoted by a juxtaposition), by (where is the unit of and, resp., of ) and by for every . It is clear that (resp., ) factors through the quotient (where is the canonical epimorphism from to and, resp., from to ), and defines an -derivation on (resp., ). This differential -algebra is denoted thereafter by (resp., ).

Theorem 1. Let be an -algebra (resp., commutative -algebra). Let be a differential -algebra (resp., commutative differential -algebra), and let be a homomorphism of -algebras. Then, there is a unique homomorphism of differential -algebras (resp., commutative differential -algebras) such that for every .

Proof. Let be the unique -linear map such that for every , . Then, we may define an algebra map (resp., ) using the universal property of the tensor algebra (resp., symmetric algebra). We have . This map factors through . Indeed, let and . We have , also , and for every . Therefore, there is a unique homomorphism of -algebras (resp., commutative -algebras) from (resp., ) to such that for every . It is easily seen to be a homomorphism of differential (resp., commutative differential) -algebras.

Example 2. Let be the free commutative -algebra over . Therefore is recovered from the usual algebra of differential polynomials over (see [4, 5] for instance), that is, the free commutative -algebra with the -derivation for all , . Now, if is the free -algebra over , then is the (not so wellknown, see [6] however) noncommutative counterpart of , that is, the free -algebra with derivation .

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 . It is not difficult to check that is the free commutative differential algebra over and is the free differential algebra over . Moreover embeds into these algebras: and .

Corollary 4. The algebra   (resp., commutative algebra ) embeds into (resp., ) as a subalgebra, more precisely the map (resp., ), such that for every is a one-to-one algebra homomorphism.

Proof. The map (resp., ) is easily seen to be an algebra map. Since is a differential -algebra (resp., commutative differential -algebra) and according to Theorem 1, the identity algebra map extends uniquely to a homomorphism (resp., of differential -algebras (resp., commutative differential -algebras) such that , so that is one-to-one.

From these results (Theorem 1 and Corollary 4), we may deduce a Poincaré-Birkhoff-Witt-like theorem. Let us denote by the universal enveloping algebra of a Lie algebra , and let denote the canonical map which is one-to-one when is a free module (Poincaré-Birkhoff-Witt theorem; see [8], e.g.). The underlying Lie algebra structure, denoted by , of an associative algebra is given by the usual commutation bracket.

Corollary 5. Let be a Lie algebra (resp., commutative Lie algebra) over which is free as a -module. Then, embeds into   (resp., ) as a Lie subalgebra. Moreover, if is a differential -algebra (resp., commutative differential -algebra) and is a homomorphism of Lie algebras, then there is a unique homomorphism ., such that .

Proof. The proof is easy (essentially a composition of left adjoints), hence omitted.

Corollary 5 means in particular that the forgetful functor from differential -algebras (resp., commutative differential -algebras) to Lie algebras (resp., commutative Lie algebras) - (resp., - ), obtained by composition of the previous “forget-the-derivation” functor from - to - (resp., to ) and from - to - (resp., to - ), 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 to - 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 be a differential commutative -algebra. Then, it admits a (functorial) structure of a Lie -algebra for which the bracket is defined by the usual Wronskian determinant for every , where denotes the exterior product (this kind of structure has been used to define -Lie algebras, see [9]). Let us denote by or more simply this Lie algebra structure. Let be another differential commutative -algebra, and let be a homomorphism of differential commutative -algebras. Then, is also a homomorphism of Lie algebras from to since . Hence defines a (forgetful) functor.

Remark 6. We observe that the Lie algebra structure associated to the differential -algebra is commutative. Conversely, let be a commutative differential algebra such that is a commutative Lie algebra; that is, for every . Since , it follows that is the zero derivative.

The functor also has a left adjoint (The existence of a left adjoint is guaranteed because 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 be a Lie algebra over . Another time let us denote by its underlying -module structure. We now consider the symmetric -algebra of . As in Section 2, let be the two-sided ideal of generated by for every . We have where, as in Section 2, is the unique -derivation of that extends the -linear endomorphism of . Hence is a commutative differential -algebra where, by abuse of language, is the natural derivation on the quotient algebra. Let us consider the differential ideal generated by for every . Let be the canonical epimorphism, and let be the (unique) derivation such that . Let .

Remark 7. For any -module , we denote by the ideal of all members of with no constant term (relatively to the usual gradation of the symmetric algebra), and let so that as -module. We observe that (since for every , and all their derivatives belong to ) so that is not reduced to zero (except if itself is ) because there exists an algebra map from onto . The direct sum decomposition induces a decomposition , where and are the canonical images of and , and .

In what follows we denote simply by the Wronskian bracket of the differential algebra , and the tensor symbol “ ” is omitted.

Proposition 8. For every and every ,

Proof. We have Now, assume by induction on that (the cases are checked). We have

According to Proposition 8, is actually the ideal generated by for every and every .

We claim that is the universal enveloping algebra of with respect to . More precisely, the following holds.

Theorem 9. Let be a Lie algebra over . Let be the map defined by for every . Then, is a homomorphism of Lie algebras from to . Let be a commutative differential -algebra, and let be a homomorphism of Lie algebras. Then, there is a unique homomorphism of commutative differential -algebras such that .

Proof. It is quite clear that is -linear. Let . Then, we have . This proves that is a homomorphism of Lie algebras. Now, let be a commutative differential -algebra, and let be a homomorphism of Lie algebras. Let be the unique -linear map such that for every , . According to the universal property of the symmetric algebra, there is a unique homomorphism of algebras such that for every , . This map factors through the quotient by . Indeed, . Therefore, there is a unique homomorphism of algebras such that . By construction, it commutes with the derivations; hence, it is a homomorphism of differential -algebras. Finally, also by construction.

The commutative differential -algebra of Theorem 9 is called the universal enveloping algebra of   with respect to or the Wronskian envelope of   .

It is easy to prove that any commutative Lie algebra embeds as a sub-Lie algebra into . Moreover the universal enveloping algebra of also embeds into as a subalgebra but not as a differential subalgebra (when is equipped with the zero derivative and with its derivative ). Indeed, let be the canonical injection. It is clearly a homomorphism of Lie algebras from to . Hence by Theorem 9 there is a unique homomorphism of commutative differential algebras such that . We then deduce that is one-to-one (let , then so that ). Moreover, since is a homomorphism of Lie algebras from to and because is commutative, it is also a homomorphism of Lie algebras from to . Therefore there is a unique homomorphism of (commutative) algebras from to such that . Thus, we obtain and also . According to the universal property of the enveloping algebra, the unique homomorphism that satisfies the second equality is so that . We observe, however, that we cannot deduce from the first equality that because is a homomorphism of commutative differential algebras from to and not from to itself. Nevertheless from it follows that is one-to-one and is onto. We observe that in general when is a commutative Lie algebra, then its universal enveloping algebra with respect to is not that is, is only onto and not one-to-one, while is only one-to-one and not onto, and the derivative on is not the zero derivative. Indeed, for instance let be the free -module generated by with the zero Lie bracket. The algebra of polynomials in the variable is . Let be the -module homomorphism defined by . It is a homomorphism of Lie algebras from to since and for every while the algebra homomorphism extension of is not a homomorphism of differential algebras from to because it does not commute with the derivations ( and ).

3.1. Some Remarks about the Generators of

Let us denote by for every and , where . We have .

Let us introduce the following integers: , , and, for all and all ,

Lemma 10. For every and every ,

Proof. Recall that for every . We have , and . For every , and . For every and , we have , . Now, let . If , then , and if , then while .

Lemma 11. For every and every , one has

Proof. We have . It is clear that is homogeneous of degree (in the sense that it is a sum of word with ). It can be written as . We have so that , and .

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 -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 -modules and even in the case where is a field. This can be shown as follows. For any elements of a Lie algebra, we denote by the left-normed bracket; that is, if , then is the Lie bracket, and for , .

Let be a commutative differential -algebra (with a unit). Then, the Lie algebra satisfies the nontrivial identity (called the “standard Lie identity of degree 5” in [12] and in [13]) where denotes the signature representation of the permutation , for every . (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 satisfies .

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 of special derivations with respect to the signature derivation as follows: let be the sub- -module of , the -module of all -derivations of , generated by (it is the image of the -module map given by ). Then, is the -module structure of together with the usual bracket of derivations (inherited by the usual bracket of the linear endomorphisms of ). Let be the annihilator of . It is an Abelian Lie subalgebra of and is a short exact sequence of Lie -algebras in such a way that is an extension of by . In the case where —for instance when has no zero divisors and —then and are isomorphic Lie -algebras. The following result is thus obvious.

Lemma 13. Let be a commutative integral -domain, and let be a nonzero derivation of . Let be a Lie -algebra such that (as Lie -algebras) (we note that in particular satisfies since it holds in . Then, is a Wronskian special Lie algebra.

Example 14. The Lie algebra over a field of characteristic zero is simple and embeds into where is the usual derivation of polynomials in the variable . Indeed, is isomorphic to the three-dimensional -vector space generated by , , with the Lie algebra structure given by the usual commutator (see, e.g., [17]). Therefore, is a Wronskian special Lie algebra.

We may also remark that Wronskian speciality is preserved by direct products (even infinite). More precisely, let be a collection of Lie -algebras such that for each , (as Lie -algebras), that is, each factor is a Wronskian special Lie algebra. Then, as Lie -algebras, where the operations (Lie brackets, derivative, and product) are considered component-wise (i.e., is a Wronskian special Lie algebra). Indeed, it is clear that as a sub-Lie algebra over . 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]). Let be a field of characteristic zero. Let be a simple Lie algebra that satisfies . Then, there exists a commutative integral -domain and a nonzero derivation such that embeds into as a Lie subalgebra.

As a consequence of the previous theorem and Lemma 13, in characteristic zero, any simple Lie algebra satisfying is a Wronskian special Lie algebra. From this result we deduce the following.

Theorem 16. Let be a field of characteristic zero. Let be a family of Lie -algebras such that for every either is simple and satisfies or is Abelian. Let be a sub-Lie algebra of . Then, is a Wronskian special Lie algebra.

Proof. According to Razmyslov’s theorem if is simple and satisfies then is Wronskian special. We also know that any Abelian Lie algebra is Wronskian special. So their product also is. Finally any sub-Lie algebra of a Wronskian special Lie algebra also is Wronskian special.

5. The Free Wronskian Special Lie -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 (in characteristic zero). We do not prove nor disprove this conjecture. Nevertheless we observe that Wronskian special Lie -algebras, for some ring , form a derived category of the variety of commutative differential algebras obtained by considering their Lie bracket as the derived operator (see [7] for definitions of derived category and derived operator). Therefore, we may speak about the free Wronskian special Lie -algebra generated by a set (see [7, Theorem 4.4]). It is obtained as the Lie subalgebra of generated by , where we recall that is the free commutative differential algebra over (if , then denotes the derivation of in and its Wronskian commutator is denoted by ). It is constructed by induction as follows: let where denotes the submodule of generated by a subset . Then, the free Wronskian special Lie algebra, denoted by , over is the nested union , and its bracket is given by the Wronskian bracket of . (It is clear that as defined previously is a -module.)

Remark 17. It is clear that if have no nonzero constant term, then it is also the case for . Moreover every element of has no nonzero constant term as it can be checked inductively from the previous observation and since every element of has no nonzero constant term. Therefore, as sets so that as Lie algebras. (Note that when , then while .) When is reduced to only one element , then is the free module 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 from to is one-to-one, and we may assume that (in particular, is free over in ). According to the property of the Wronskian envelope, there is a unique differential algebra map such that the following diagram commutes (where the unnamed arrows are the canonical inclusions) xy(12) Moreover the composition of natural embeddings gives rise to a set-theoretic map . Therefore, there is a unique homomorphism of commutative differential algebras such that the following diagram commutes: xy(13) Both compositions and are the identity on and hence, by uniqueness, the identity everywhere. Therefore, and are canonically isomorphic (as commutative differential algebras):

Remark 18. The previous result shares some similarity with the well-known fact that the universal enveloping algebra of the free Lie algebra generated by is canonically isomorphic to the free associative algebra generated by (see [3]).

Remark 19. Let us assume that is an integral domain. Then, is also an integral domain, and (as Lie algebras over ), where is the usual derivation on (i.e, ). Since    is a sub-Lie algebra of (under the commutator), it follows that is also a special Lie algebra.

Acknowledgments

The author wishes to thank Professor Huishi Li from the Department of Applied Mathematics of the College of Information Science and Technology of Hainan University (China) for discussions related the Wronskian envelope and noncommutative Groebner bases, Professor Hans-E. Porst from the University of Bremen (Germany) for his advices concerning left adjoints and algebraic functors, and Professor G.M. Bergman from the Department of Mathematics of the University of California (USA) for a fruitful correspondence and for pointing to the knowledge of the author the works of the Russian mathematicians, Kirillov and Razmyslov, about Lie algebras of vector fields.