Hindawi Publishing Corporation Algebra Volume 2013, Article ID 341631, 8 pages http://dx.doi.org/10.1155/2013/341631
Research Article Wronskian Envelope of a Lie Algebra
Laurent Poinsot
Universite´ Paris 13, Sorbonne Paris Cite,´ LIPN, CNRS, UMR 7030, 93430 Villetaneuse, France
Correspondence should be addressed to Laurent Poinsot; [email protected]
Received 6 March 2013; Accepted 23 April 2013
Academic Editor: Rutwig Campoamor-Stursberg
Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The famous Poincare-Birkhoff-WitttheoremstatesthataLiealgebra,freeasamodule,embedsintoitsassociativeenvelope—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 𝑎 →𝑎, that is, a commutative differential algebra, admits a Wronskian bracket 𝑊(𝑎, 𝑏) =𝑎𝑏 −𝑎𝑏 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 maybeembeddedintoitsWronskianenvelope,andwepresenttheconstructionofthefreeLiealgebrawiththisproperty.
1. Introduction with a distinguished derivation is said to be a differential algebra. For any such pair (𝐴, 𝜕) may be defined a bilinear Any (associative) algebra (and more generally a Lie- 𝑊:𝐴2 →𝐴 𝑊(𝑎, 𝑏) = 𝑎𝜕(𝑏) −𝜕(𝑎)𝑏 𝐴 map by . When the admissible algebra), say , admits a derived structure of Lie algebra 𝐴 is commutative, then 𝑊 is alternating and satisfies algebra under the commutator bracket [𝑎, 𝑏] = 𝑎𝑏 −𝑏𝑎, 𝑎, 𝑏 ∈ 𝐴 𝐴 the Jacobi identity so that it defines a Lie bracket on .The . This actually describes a forgetful functor, more precisely definition of the Lie algebra (𝐴, 𝑊) from (𝐴, 𝜕) is functorial an algebraic functor, from associative to Lie algebras. This (homomorphisms of differential algebras are transformed functor admits a left adjoint that enables to associate to any intohomomorphismofLiealgebras)asisthedefinitionofthe Liealgebraitsuniversalassociativeenvelope.Inthiswaythe Lie algebra (𝐴, [⋅, ⋅]) associated to 𝐴 from the classical theory. theory of Lie algebras may be explored through (but not Also as in the classical case, this functor admits a left adjoint reduced to) that of associative algebras. A Lie algebra which that allows us to define a Wronskian envelope for a Lie alge- embeds into its universal enveloping algebra is referred to bra, that is, a universal commutative and differential algebra. as special. The famous Poincare-Birkhoff-Witt´ theorem states Thispapercontributestothestudyofthisfunctorial that any Lie algebra which is free as a module (and therefore, relation (an adjointness) between Lie and differential com- any Lie algebra over a field) is special. When the Lie algebra is mutative algebras in a way parallel to the classical theory of Abelian, then its universal enveloping algebra reduces to the Lieandassociativealgebras.Inparticular,wedescribethe symmetric algebra of its underlying module structure, and construction of the Wronskian envelope, and we present a thus any commutative Lie algebra is trivially special. sufficient conditionTheorem ( 16) over a field of characteristic However, there is another way to associate an associative zero for certain Lie algebras to be special, that is, to embed algebra to a Lie algebra, and reciprocally, in a functorial into their Wronskian envelopes. Contrary to the usual case, way. The idea does not consist anymore to consider non- freeness of the underlying 𝑅-module is no more sufficient: commutative algebras under commutators but differential there are Lie algebras, free as modules, which do not embed commutative algebras together with the so-called Wronskian into their Wronksian envelopes. Indeed, special Lie algebras, determinant. A derivation of an algebra 𝐴 is a linear map in this new setting, satisfy a nontrivial relation similar to a 𝜕:𝐴that →𝐴 satisfies the usual Leibniz rule. An algebra relation that holds in Lie algebras of vector fields. 2 Algebra
(N) 2. Differential Algebra that extends 𝜕.Let𝐶 be the two-sided ideal of T(mod(𝐴) ) (N) generated by 𝑥⊗𝑦−𝑦⊗𝑥for every 𝑥, 𝑦 ∈ mod(𝐴) .Since Because the notion of Wronskian envelope is based upon (N) 𝜕(𝐶) ⊆𝐶, (S(mod(𝐴) ), 𝜕) is a commutative differential commutative differential algebras, we start by a basic and 𝑅-algebra (by abuse of notations 𝜕 is the derivation on the brief presentation of this theory that is inspired from [1]. 𝐼 Besides we show that any algebra (commutative or not) quotient algebra). Now, let us consider the (usual) ideal of T( (𝐴)(N)) S( (𝐴)(N)) (𝑥𝑦) − may be embedded into a differential algebra in a functorial mod (resp., mod ) generated by (𝑖) 𝑖 𝑖 way. ∑𝑗=0 ( 𝑗 )𝑥(𝑗) ⊗𝑦(𝑖−𝑗) for every 𝑥, 𝑦 ∈𝐴 and 𝑖∈N (where 𝑅 Let be a commutative ring with a unit (it is assumed the product of elements in 𝐴 is denoted by a juxtaposition), hereafter to be nonzero), and let 𝐴 be an 𝑅-algebra. An 𝑅- (N) by (1𝐴)(0) −1(where 1 is the unit of T(mod(𝐴) ) and, derivation 𝜕 of 𝐴 is an endomorphism of the underlying 𝑅- (N) 𝐴 resp., of S(mod(𝐴) ))andby(1𝐴)(𝑖) for every 𝑖>0.It module structure of that satisfies Leibniz’s rule (N) (N) is clear that 𝜋𝐼 ∘𝜕 : T(mod(𝐴) )→T(mod(𝐴) )/𝐼 𝜕 (𝑎𝑏) =𝜕(𝑎) 𝑏+𝑎𝜕(𝑏) (1) (N) (N) (resp., 𝜋𝐼 ∘𝜕 : S(mod(𝐴) )→S(mod(𝐴) )/𝐼) factors 𝜋 for every 𝑎, 𝑏.Weremarkthat ∈𝑅 𝜕(1𝐴)=0(1𝐴 denoting through the quotient (where 𝐼 is the canonical epimor- (N) (N) the unit of 𝐴)forif𝜕(1𝐴)=𝜕(1𝐴1𝐴)=𝜕(1𝐴)+𝜕(1𝐴). phism from T(mod(𝐴) ) to T(mod(𝐴) )/𝐼 and, resp., (N) (N) Apair(𝐴, 𝜕) is called a differential 𝑅-algebra.Itissaidto from S(mod(𝐴) ) to S(mod(𝐴) )/𝐼), and defines an 𝑅- (N) (N) be commutative when 𝐴 is so. We note that any 𝑅-algebra derivation 𝜕 on T(mod(𝐴) )/𝐼 (resp., S(mod(𝐴) )/𝐼). 𝑅 is actually a differential -algebra with the zero derivation. This differential 𝑅-algebra is denoted thereafter by (D(𝐴), 𝜕) Let (𝐴,𝐴 𝜕 ) and (𝐵,𝐵 𝜕 ) be two differential 𝑅-algebras, and 𝜙: (resp., (CD(𝐴), 𝜕)). 𝐴→𝐵be a (unit-preserving) homomorphism of 𝑅-algeb- 𝑅 ras. It is a homomorphism of differential -algebras when Theorem 1. Let 𝐴 be an 𝑅-algebra (resp., commutative 𝑅- 𝜙∘𝜕𝐴 =𝜕𝐵 ∘𝜙. Differential (resp., commutative) 𝑅-algebras algebra). Let (𝐵,𝐵 𝜕 ) be a differential 𝑅-algebra (resp., com- and their homomorphisms form a category denoted by 𝑅- 𝑅 𝜙:𝐴 →𝐵 𝑅 C mutative differential -algebra), and let DiffAlg (resp., - DiffAlg). A differential (two-sided) be a homomorphism of 𝑅-algebras. Then, there is a unique ideal 𝐼 of a differential 𝑅-algebra (𝐴,𝐴 𝜕 ) is an (two-sided) homomorphism 𝜙:(D(𝐴), 𝜕) → (𝐵,𝐵 𝜕 ) of differential 𝑅- ideal of the carrier 𝑅-algebra 𝐴 such that 𝜕𝐴(𝐼) ⊆ 𝐼.Itis 𝜙:(CD(𝐴), 𝜕) → (𝐵, 𝜕 ) clear that the quotient-algebra 𝐴/𝐼 becomes a differential 𝑅- algebras (resp., 𝐵 commutative 𝑅 𝜙(𝜋 (𝑥 )) = 𝜙(𝑥) algebrainanaturalwayandthatthecanonicalepimorphism differential -algebras) such that 𝐼 (0) for every 𝑥∈𝐴 𝜋𝐼 :𝐴 → 𝐴/𝐼is a homomorphism of differential 𝑅-algebras. . Let 𝑆⊆𝐴be a subset. Then, the intersection of all differential 𝜙 : (𝐴)(N) → (𝐵) 𝑅 ideals of (𝐴,𝐴 𝜕 ) that contain 𝑆 is again a differential ideal, Proof. Let 1 mod mod be the unique - 𝜙 (𝑥 )=𝜕𝑖 (𝜙(𝑥)) 𝑥∈ called the differential ideal generated by 𝑆.Wemayobserve linear map such that 1 (𝑖) 𝐵 for every ̂ that this differential ideal is the same as the (algebraic) ideal 𝐴, 𝑖≥0. Then, we may define an algebra map 𝜙1 : {𝜕𝑖 (𝑥) : 𝑥 ∈ 𝑆, 𝑖≥0} (N) ̂ (N) generated by 𝐴 . T(mod(𝐴) )→𝐵(resp., 𝜙1 : S(mod(𝐴) )→𝐵) The obvious forgetful functor A (resp., CA)from𝑅- usingtheuniversalpropertyofthetensoralgebra(resp., 𝑅 C 𝑅 ̂ 𝑖 DiffAlg (resp., - DiffAlg)tothecategory -Alg (resp., symmetric algebra). We have 𝜙1(𝑥(𝑖))=𝜕(𝜙(𝑥)).This 𝑅 C 𝑅 𝑅 𝐵 - Alg)of -algebras (resp., commutative -algebras), that map factors through 𝜋𝐼. Indeed, let 𝑥, 𝑦 ∈𝐴 and 𝑖≥ forgets the derivation, has a left adjoint (see [2]forusual ̂ 𝑖 𝑖 0.Wehave𝜙1((𝑥𝑦)(𝑖))=𝜕𝐵(𝜙(𝑥𝑦))𝐵 =𝜕 (𝜙(𝑥)𝜙(𝑦)) = category-theoretic definitions). To prove this fact, let us intro- ∑𝑖 ( 𝑖 )𝜕𝑗 (𝜙(𝑥))𝜕𝑖−𝑗(𝜙(𝑥))𝑖 =∑ ( 𝑖 ) 𝜙̂ (𝑥 )𝜙̂ (𝑦 )= duce some notations. Let 𝑋 be any set and 𝑀 be any monoid 𝑗=0 𝑗 𝐵 𝐵 𝑗=0 𝑗 1 (𝑗) 1 (𝑖−𝑗) 𝑋 ̂ 𝑖 𝑖 ̂ with unit 0.Let𝑓∈𝑀 .Itssupport supp(𝑓) is the set of all 𝑖∈ 𝜙1(∑𝑗=0 ( 𝑗 )𝑥(𝑗) ⊗𝑦(𝑖−𝑗)),also𝜙1((1𝐴)(0))=𝜙(1𝐴)=1𝐵 = 𝑋 𝑓(𝑖) =0̸ ̂ ̂ 𝑖 𝑖 such that . Finitely supported maps are those maps 𝜙1(1),and𝜙1((1𝐴)(𝑖))=𝜕(𝜙(1𝐴)) = 𝜕 (1𝐵)=0for every (𝑋) 𝐵 𝐵 with a finite support, and 𝑀 is the set of all such maps. 𝑖>0 𝜙 𝑅 (N) . Therefore, there is a unique homomorphism of - Let 𝑉 be a 𝑅-module. We denote by 𝑉 the 𝑅-module of all algebras (resp., commutative 𝑅-algebras) from D(𝐴) (resp., N 𝑉 finitely supported maps from to (with point-wise addi- CD(𝐴))to𝐵 such that 𝜙∘𝜋𝐼(𝑥(0))=𝜙(𝑥)for every 𝑥∈𝐴. tion and scalar multiplication), namely, ⨁𝑛∈N𝑉.Forevery (N) It is easily seen to be a homomorphism of differential (resp., 𝑥∈𝑉and 𝑖∈N,let𝑥(𝑖) ∈𝑉 be defined by 𝑥(𝑖)(𝑗) = 𝑥 if 𝑖= commutative differential) 𝑅-algebras. 𝑗 and 0𝑉 otherwise; the maps 𝑞𝑖 :𝑥→𝑥(𝑖) are the canonical (N) injections of the coproduct 𝑉 .WealsodenotebyT(𝑉) Example 2. Let 𝑅[𝑋] = S(𝑅𝑋) bethefreecommutative (resp., S(𝑉)) the tensor (resp., symmetric) 𝑅-algebra of 𝑉 (see 𝑅-algebra over 𝑋. Therefore (CD(𝑅[𝑋]), 𝜕) is recovered [3]). The natural injection from 𝑉 to T(𝑉) (resp., S(𝑉))is from the usual algebra of differential polynomials 𝑅{𝑋} denoted by 𝑛𝑉.Now,let𝐴 be an 𝑅-algebra (resp., commuta- over 𝑅 (see [4, 5] for instance), that is, the free commutative tive 𝑅-algebra), and let us denote by mod(𝐴) the underlying 𝑅-algebra 𝑅[𝑋 × N] with the 𝑅-derivation 𝜕(𝑥, 𝑖) = (𝑥, 𝑖+1) (N) (N) 𝑅-module structure of 𝐴.Let𝜕:mod(𝐴) → mod(𝐴) for all 𝑥∈𝑋, 𝑖≥0.Now,if𝑅⟨𝑋⟩ = T(𝑅𝑋) is the be the 𝑅-linear endomorphism defined by 𝜕(𝑥(𝑖))=𝑥(𝑖+1) for free 𝑅-algebra over 𝑋,then(D(𝑅⟨𝑋⟩), 𝜕) is the (not so every 𝑥∈𝐴and 𝑖∈N. According to [3, Lemma 4] there exists wellknown, see [6] however) noncommutative counterpart (N) aunique𝑅-derivation of T(mod(𝐴) ), again denoted by 𝜕, of 𝑅{𝑋},thatis,thefree𝑅-algebra 𝑅⟨𝑋 × N⟩ with derivation Algebra 3
𝜕((𝑥1,𝑖1)(𝑥2,𝑖2)⋅⋅⋅(𝑥,𝑖𝑛)) = (𝑥1,𝑖1 + 1)(𝑥2,𝑖2)⋅⋅⋅(𝑥,𝑖𝑛)+ 3. Wronskian Envelope (𝑥1,𝑖1)(𝑥2,𝑖2 +1)⋅⋅⋅(𝑥𝑛,𝑖𝑛)+⋅⋅⋅+(𝑥1,𝑖1)(𝑥2,𝑖2)⋅⋅⋅(𝑥𝑛,𝑖𝑛 +1). In this section the Wronskian envelope universally associated Remark 3. It is clear that (commutative) differential algebras toanyLiealgebraisconstructed,thatis,aleftadjointtothe form a variety (in the sense of universal algebra, see [7]), and forgetful functor from commutative differential algebras to therefore, we may define the free (commutative) differential Lie algebras. algebra over a set 𝑋.Itisnotdifficulttocheckthat𝑅{𝑋} is the Let (𝐴,𝐴 𝜕 ) be a differential commutative 𝑅-algebra. Then, free commutative differential algebra over 𝑋 and D(𝑅⟨𝑋⟩) it admits a (functorial) structure of a Lie 𝑅-algebra for which isthefreedifferentialalgebraover𝑋.Moreover𝑋 embeds the bracket is defined by the usual Wronskian determinant into these algebras: 𝑋⊆𝑅{𝑋}and 𝑋⊆D(𝑅⟨𝑋⟩). 𝑊𝐴 (𝑥,) 𝑦 =𝑥𝜕𝐴 (𝑦) −𝜕𝐴 (𝑥) 𝑦=(𝑖𝑑𝐴 ∧𝜕𝐴)(𝑥∧𝑦) (2) Corollary 4. The algebra 𝐴 (resp., commutative algebra 𝐴) 𝑥, 𝑦 ∈𝐴 ∧ embeds into D(𝐴) (resp., CD(𝐴))asasubalgebra,more for every ,where denotes the exterior prod- 𝑛 precisely the map 𝑗𝐴 :𝐴 → D(𝐴) (resp., 𝑗𝐴 :𝐴 → CD(𝐴)), uct (this kind of structure has been used to define - (𝐴, 𝜕 ) such that 𝑗𝐴(𝑥) =𝐼 𝜋 (𝑥(0)) for every 𝑥∈𝐴is a one-to-one Lie algebras, see [9]). Let us denote by w 𝐴 or more (𝐴) (𝐵, 𝜕 ) algebra homomorphism. simply w this Lie algebra structure. Let 𝐵 be another differential commutative 𝑅-algebra, and let 𝜙: (𝐴, 𝜕 )→(𝐵,𝜕) Proof. The map 𝑗𝐴 :𝐴 → D(𝐴) (resp., 𝑗𝐴 :𝐴 → CD(𝐴)) 𝐴 𝐵 be a homomorphism of differential is easily seen to be an algebra map. Since (𝐴, 0) is a differential commutative 𝑅-algebras. Then, 𝜙 is also a homomorphism 𝑅-algebra (resp., commutative differential 𝑅-algebra) and of Lie algebras from w(𝐴) to w(𝐵) since 𝜙(𝑊𝐴(𝑥, 𝑦)) = 𝜙(𝑥𝜕 (𝑦) − 𝜕 (𝑥)𝑦) = 𝜙(𝑥)𝜙(𝜕 (𝑦)) − 𝜙(𝜕 (𝑥))𝜙(𝑦) = according to Theorem 1, the identity algebra map 𝑖𝑑𝐴 extends 𝐴 𝐴 𝐴 𝐴 𝜙(𝑥)𝜕 (𝜙(𝑦)) −𝜕 (𝜙(𝑥))𝜙(𝑦) =𝑊 (𝜙(𝑥), 𝜙(𝑦)) uniquely to a homomorphism 𝑖𝑑𝐴 :(D(𝐴), 𝜕) → (𝐴, 0) 𝐵 𝐵 𝐵 .Hence w :𝑅-DiffAlg →𝑅-LieAlg defines a (forgetful) functor. (resp., 𝑖𝑑𝐴 :(CD(𝐴), 𝜕) → (𝐴, 0)) of differential 𝑅-algebras (resp., commutative differential 𝑅-algebras) such that Remark 6. We observe that the Lie algebra structure w(𝐴) 𝑖𝑑 ∘𝑗 =𝑖𝑑 𝑗 𝐴 𝐴 𝐴,sothat 𝐴 is one-to-one. associated to the differential 𝑅-algebra (𝐴, 0) is commutative. Conversely, let (𝐴,𝐴 𝜕 ) be a commutative differential algebra From these results (Theorem 1 and Corollary 4), we may such that w(𝐴,𝐴 𝜕 ) is a commutative Lie algebra; that is, deduce a Poincare-Birkhoff-Witt-like´ theorem. Let us denote 𝑊 (𝑥, 𝑦) =0 𝑥, 𝑦 ∈𝐴 𝑊 (1 ,𝑥)= 𝜕 (𝑥) U( ) 𝐴 for every .Since 𝐴 𝐴 𝐴 ,it by g the universal enveloping algebra of a Lie algebra g, 𝜕 𝑖 : → U( ) follows that 𝐴 is the zero derivative. and let g g g denote the canonical map which is one-to-one when g is a free module (Poincare-Birkhoff-Witt´ The functor w also has a left adjoint (The existence of theorem; see [8], e.g.). The underlying Lie algebra structure, a left adjoint is guaranteed because w is algebraic. See, e.g., (𝐴) 𝐴 denoted by l ,ofanassociativealgebra is given by the [10] for the notion of algebraic functors.) that allows us to usual commutation bracket. define a notion of Wronskian universal enveloping algebra or shortly Wronskian envelope.Letg beaLiealgebraover Corollary 5. Let g be a Lie algebra (resp., commutative Lie 𝑅. Another time let us denote by mod(g) its underlying 𝑅- algebra) over 𝑅 whichisfreeasa𝑅-module. Then, g embeds module structure. We now consider the symmetric 𝑅-algebra into D(U(g)) (resp., CD(U(g)))asaLiesubalgebra.More- (N) (N) S(mod(g) ) of mod(g) .AsinSection 2,let𝐶 be the two- over, if (𝐵,𝐵 𝜕 ) is a differential 𝑅-algebra (resp., commutative (N) 𝑅 𝜙: → (𝐵) sided ideal of T(mod(g) ) generated by 𝑥⊗𝑦−𝑦⊗𝑥 differential -algebra) and g l is a homomorphism (N) 𝜙: for every 𝑥, 𝑦 ∈ mod(g) .Wehave𝜕(𝐶) ⊆𝐶 where, as of Lie algebras, then there is a unique homomorphism (N) 𝜕 𝑅 T(mod(g) ) (D(U(g)), 𝜕)→(𝐵,𝜕𝐵)(resp.,𝜙:(CD(U(g)), 𝜕) → in Section 2, is the unique -derivation of 𝑅 𝜕:𝑥 →𝑥 (𝐵, 𝜕 )) 𝜙∘𝑗 ∘𝑖 that extends the -linear endomorphism (𝑖) (𝑖+1) 𝐵 such that U(g) g. (N) (N) of mod((g) ).Hence(S(mod(g) ), 𝜕) is a commutative Proof. Theproofiseasy(essentiallyacompositionofleft differential 𝑅-algebra where, by abuse of language, 𝜕 is the adjoints), hence omitted. natural derivation on the quotient algebra. Let us consider 𝐽 [𝑥, 𝑦] −𝑥 ⊗𝑦 + the differential ideal g generated by (0) (0) (1) Corollary 5 means in particular that the forgetful functor (N) 𝑥(1) ⊗𝑦(0) for every 𝑥, 𝑦 ∈ g.Let𝜎:S(mod(g) )→ from differential 𝑅-algebras (resp., commutative differential (N) S(mod(g) )/𝐽 𝜕 𝑅-algebras) to Lie algebras (resp., commutative Lie algebras) g be the canonical epimorphism, and let be 𝑅-LieAlg (resp., 𝑅-CLieAlg), obtained by composition the (unique) derivation such that 𝜕∘𝜎=𝜎∘𝜕.LetW(g)= 𝑅 S( ( )(N))/𝐽 of the previous “forget-the-derivation” functor from - mod g g. DiffAlg to 𝑅-Alg (resp., 𝑅-CDiffAlg to 𝑅-CAlg)andl from 𝑅-Alg to 𝑅-LieAlg (resp., 𝑅-CAlg to 𝑅-CLieAlg), Remark 7. For any 𝑅-module 𝑉, we denote by S(𝑉)+ the ideal has a left adjoint, also obtained by composition of the of all members of S(𝑉) with no constant term (relatively one given by Theorem 1 and the universal enveloping alge- to the usual gradation of the symmetric algebra), and let bra functor. But there is another forgetful functor from S(𝑉)0 =𝑅⋅1so that S(𝑉) = S(𝑉)0 ⊕S(𝑉)+ as 𝑅-module. We 𝑅 C 𝑅 𝐽 ⊆ S( ( )(N)) 𝑥, 𝑦 ∈ - DiffAlg to -LieAlg given by the Wronskian, which observe that g mod g + (since for every g, is studied in what follows. [𝑥, 𝑦](0) −𝑥(0) ⊗𝑦(1) +𝑥(1) ⊗𝑦(0) and all their derivatives 4 Algebra
𝑛 𝑛 𝑛 𝑛 𝑛 [𝑥, 𝑦](𝑛) − ∑𝑘=0 ( 𝑘 )𝑥(𝑛−𝑘) ⊗𝑦(𝑘+1) + ∑𝑘=0 ( 𝑘 )𝑥(𝑛−𝑘+1) ⊗𝑦(𝑘) =( )𝑊(𝑥 ,𝑦 ) (N) ⏟⏟⏟⏟⏟⏟⏟0 (𝑖+𝑛+1) (𝑗) belong to S(mod(g) )+)sothat𝑊(g) is not reduced to zero (except if 𝑅 itself is (0)) because there exists an algebra map =1 (N) (N) from W(g) onto S(mod(g) )/S(mod(g) )+ ≅𝑅⋅1.The 𝑛 (N) (N) +( )𝑊(𝑥(𝑖),𝑦(𝑗+1+𝑛)) direct sum decomposition S(mod(g) )=S(mod(g) )0 ⊕ ⏟⏟⏟⏟⏟⏟⏟𝑛 (N) S(mod(g) )+ induces a decomposition W(g)=W(g)0 ⊕ =1 W(g) W(g) W(g) 𝑛 +,where + and 0 are the canonical images of 𝑛 S( ( )(N)) S( ( )(N)) W( ) =𝑅⋅1 + ∑ ( )𝑊(𝑥 ,𝑦 ) mod g + and mod g 0,and g 0 . 𝑘 (𝑖+𝑛+1−𝑘) (𝑗+𝑘) 𝑘=1 𝑊 In what follows we denote simply by the 𝑛−1 𝑊 (N) 𝑛 Wronskian bracket S(mod(g) ) of the differential algebra + ∑ ( )𝑊(𝑥 ,𝑦 ) (N) 𝑘 (𝑖+𝑛−𝑘) (𝑗+1+𝑘) (S(mod(g) ), 𝜕),andthetensorsymbol“⊗”isomitted. 𝑘=0
Proposition 8. For every 𝑥, 𝑦 ∈ g and every 𝑖, 𝑗,, 𝑛≥0 =𝑊(𝑥(𝑖+𝑛+1),𝑦(𝑗))+𝑊(𝑥(𝑖),𝑦(𝑗+1+𝑛)) 𝑛 𝑛 + ∑ ( )𝑊(𝑥 ,𝑦 ) 𝑛 𝑛 𝑘 (𝑖+𝑛+1−𝑘) (𝑗+𝑘) 𝜕𝑛 (𝑊 (𝑥 ,𝑦 )) = ∑ ( ) 𝑊 (𝑥 ,𝑦 ) . 𝑘=1 (𝑖) (𝑗) 𝑘 (𝑖+𝑛−𝑘) (𝑗+𝑘) (3) 𝑘=0 𝑛 𝑛 + ∑ ( )𝑊(𝑥 ,𝑦 ) 𝑘−1 (𝑖+𝑛+1−𝑘) (𝑗+𝑘) 𝑘=1 Proof. We have =𝑊(𝑥(𝑖+𝑛+1),𝑦(𝑗))+𝑊(𝑥(𝑖),𝑦(𝑗+1+𝑛))
𝑛 𝑛 𝑛 𝜕(𝑊(𝑥(𝑖),𝑦(𝑗))) = 𝜕 (𝑖)(𝑥 𝑦(𝑗+1) −𝑥(𝑖+1)𝑦(𝑗)) + ∑(( )+( ))𝑊(𝑥 ,𝑦 ) 𝑘−1 𝑘 (𝑖+𝑛+1−𝑘) (𝑗+𝑘) 𝑘+1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =𝑥(𝑖+1)𝑦(𝑗+1) +𝑥(𝑖)𝑦(𝑗+2) 𝑛+1 =( 𝑘 ) −𝑥(𝑖+2)𝑦(𝑗) −𝑥(𝑖+1)𝑦(𝑗+1) (4) 𝑛+1 𝑛+1 =𝑥(𝑖)𝑦(𝑗+2) −𝑥(𝑖+1)𝑦(𝑗+1) = ∑ ( )𝑊(𝑥 ,𝑦 ). 𝑘 (𝑖+𝑛+1−𝑘) (𝑗+𝑘) 𝑘=0 +𝑥 𝑦 −𝑥 𝑦 (𝑖+1) (𝑗+1) (𝑖+2) (𝑗) (6)
=𝑊(𝑥(𝑖),𝑦(𝑗+1))+𝑊(𝑥(𝑖+1),𝑦(𝑗)). 𝐽 According to Proposition 8, g is actually the ideal gener- [𝑥, 𝑦] −∑𝑛 ( 𝑛 )𝑊(𝑥 ,𝑦 ) 𝑥, 𝑦 ∈ Now, assume by induction on 𝑛 that ated by (𝑛) 𝑘=0 𝑘 (𝑛−𝑘) (𝑘) for every g and every 𝑛≥0. We claim that (W(g), 𝜕) is the universal enveloping 𝑛 𝑛 algebra of g with respect to w.Moreprecisely,thefollowing 𝜕𝑛 (𝑊 (𝑥 ,𝑦 )) = ∑ ( ) 𝑊 (𝑥 ,𝑦 ) (𝑖) (𝑗) 𝑘 (𝑖+𝑛−𝑘) (𝑗+𝑘) (5) holds. 𝑘=0 Theorem 9. 𝑅 𝑤 : → W( ) Let g be a Lie algebra over .Let g g g 𝑤 (𝑥) = 𝜎(𝑥 ) 𝑥∈ be the map defined by g (0) for every g.Then, (the cases 𝑛=0,1are checked). We have 𝑤 (W( )) g is a homomorphism of Lie algebras from g to w g . Let (𝐴,𝐴 𝜕 ) be a commutative differential 𝑅-algebra, and let 𝜙: → (𝐴) 𝜕𝑛+1 (𝑊 (𝑥 ,𝑦 )) g w be a homomorphism of Lie algebras. Then, (𝑖) (𝑗) there is a unique homomorphism of commutative differential 𝑅 𝜙:(̃ W( ), 𝜕) → (𝐴, 𝜕 ) 𝜙∘𝑤̃ =𝜙 𝑛 -algebras g 𝐴 such that g . =𝜕(𝜕 (𝑊 (𝑖)(𝑥 ,𝑦(𝑗)))) Proof. It is quite clear that 𝑤 is 𝑅-linear. Let 𝑥, 𝑦 ∈ g.Then, 𝑛 𝑛 g =𝜕(∑ ( )𝑊(𝑥 ,𝑦 )) we have 𝑤 ([𝑥, 𝑦]) = 𝜎([𝑥,(0) 𝑦] )=𝜎(𝑥(0) ⊗𝑦(1) −𝑥(1) ⊗ 𝑘 (𝑖+𝑛−𝑘) (𝑗+𝑘) g 𝑘=0 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))= 𝑛 𝑛 𝑊 (𝜎(𝑥 ), 𝜎(𝑦 )) 𝑤 = ∑ ( )(𝑊(𝑥 ,𝑦 ) 𝐵 (0) (0) .Thisprovesthat g is a homomorphism 𝑘 (𝑖+𝑛+1−𝑘) (𝑗+𝑘) (𝐴, 𝜕 ) 𝑘=0 of Lie algebras. Now, let 𝐴 be a commutative differential 𝑅-algebra, and let 𝜙:g → w(𝐴) be a homomorphism of (N) +𝑊 (𝑖+𝑛−𝑘)(𝑥 ,𝑦(𝑗+1+𝑘))) Lie algebras. Let 𝜙1 : mod(g) → mod(𝐴) be the unique Algebra 5
𝑖 𝑅-linear map such that 𝜙1(𝑥(𝑖))=𝜕𝐴(𝜙(𝑥)) for every 𝑥∈g, algebras from (𝑅[𝑥], 0) to 𝑅{𝑥} because it does not commute # # 𝑖≥0. According to the universal property of the symmetric with the derivations (0=𝜙(0𝑥) and (𝜙 (𝑥)) =𝑥 =0̸ ). ̂ algebra, there is a unique homomorphism of algebras 𝜙1 : 𝑆( ( )(N))→𝐴 𝜙̂ (𝑥 )=𝜙(𝑥 ) 𝑥∈ 𝐽 mod g such that 1 (𝑖) 1 (𝑖) for every 3.1. Some Remarks about the Generators of g. Let us denote 𝑖≥0 𝐽 𝑛 g, . This map factors through the quotient by g. Indeed, by 𝑃𝑛+1(𝑥,𝑦)=[𝑥,𝑦](𝑛) −𝜕 𝑢𝑥,𝑦 for every 𝑛≥0and 𝑥, 𝑦 ∈ g, ̂ 𝜙1([𝑥, (0)𝑦] ) = 𝜙([𝑥, 𝑦])𝐴 =𝑊 (𝜙(𝑥), 𝜙(𝑦)) 𝐴= 𝜙(𝑥)𝜕 (𝜙(𝑦)) − where 𝑢𝑥,𝑦 =[𝑥,𝑦](0)−𝑥(0)𝑦(1)+𝑥(1)𝑦(0).Wehave𝑃𝑛+1(𝑥, 𝑦) = ̂ ̂ ̂ ̂ ̂ ∑𝑛 ( 𝑛 )𝑊(𝑥 ,𝑦 ) 𝜕𝐴(𝜙(𝑥))𝜙(𝑦) = 𝜙1(𝑥(0))𝜙1(𝑦(1))−𝜙1(𝑥(1))𝜙1(𝑦(0))=𝜙1(𝑥(0) ⊗ 𝑘=0 𝑘 (𝑛−𝑘) (𝑘) . 𝑇(1, 0) = −1 𝑦(1) −𝑥(1) ⊗𝑦(0)). Therefore, there is a unique homomorphism Let us introduce the following integers: , ̃ ̃ ̂ 𝑇(1, 1),and,forall =1 𝑛≥2and all 𝑘=0,...,𝑛, of algebras 𝜙:𝑊(g)→𝐴such that 𝜙∘𝜎 = 𝜙1.Bycon- struction, it commutes with the derivations; hence, it is a 𝑇 (𝑛−1,0) 𝑅 𝜙∘𝑤̃ =𝜙 { homomorphism of differential -algebras. Finally, g { 𝑘=0, { if also by construction. { 𝑇 (𝑛−1,𝑛−1) 𝑇 (𝑛,) 𝑘 = { 𝑘=𝑛, (7) 𝑅 (W( ), 𝜕) { if The commutative differential -algebra g of { 𝑇 (𝑛−1,𝑘−1) +𝑇(𝑛−1,𝑘) Theorem 9 is called the universal enveloping algebra of g with { if 𝑘=1,...,𝑛−1. respect to w or the Wronskian envelope of g. { It is easy to prove that any commutative Lie algebra Lemma 10. For every 𝑛≥1and every 𝑘=0,...,𝑛, embedsasasub-LiealgebraintoW(g).Moreovertheuni- versal enveloping algebra U(g) of g also embeds into W(g) 𝑛 𝑛−1 𝑇 (𝑛,) 𝑘 =( )−2( ). asasubalgebrabutnotasadifferentialsubalgebra(when 𝑘 𝑘 (8) U(g) is equipped with the zero derivative and W(g) with its 𝜕 𝑗 : → S( ( )) ≅ U( ) 𝑛 derivative ). Indeed, let g g mod g g be Proof. Recall that ( 𝑘 )=0for every 𝑘>𝑛≥0.Wehave 1 0 1 the canonical injection. It is clearly a homomorphism of Lie 𝑇(1, 0) =( 0 )−2(0 )−=1−2=−1,and𝑇(1, 1) =( 1 )− (S( ( )), 0) = (S( ( ))) 0 𝑛 𝑛−1 algebras from g to w mod g l mod g .Hence 2(1 )=1.Forevery𝑛≥1, 𝑇(𝑛, 0) = ( 0 )−2 ( 0 )=1−2=−1 𝑛 𝑛−1 by Theorem 9 there is a unique homomorphism of commu- and 𝑇(𝑛, 𝑛) = ( 𝑛 ) −2( 𝑛 )=1.Forevery𝑛≥2and 𝑘= 𝑗̃ :(W( ), 𝜕) → (S( ( )), 0) 𝑛 𝑛−1 𝑛−1 𝑛−2 tative differential algebras g g mod g 0,...,𝑛,wehave( 0 )−2( 0 )=1−2=−1=( 0 )−2( 0 ), ̃ 𝑛 𝑛−1 𝑛−1 𝑛−2 such that 𝑗 ∘𝑤 =𝑗.Wethendeducethat𝑤 is one-to-one ( 𝑛 ) −2( 𝑛 )=1=(𝑛−1 )−2(𝑛−1 ).Now,let𝑘=1,...,𝑛−1. g g g g 𝑛 𝑛−1 𝑛−1 𝑛−2 𝑛−1 𝑥∈ 𝑤 0=𝑗̃ (𝑤 (𝑥)) = 𝑗 (𝑥) 𝑥=0 If 𝑘≤𝑛−2,then( 𝑘 )−2( 𝑘 )=(𝑘−1 )−2(𝑘−1 )+( 𝑘 )− (let ker g,then g g g so that ). 𝑛−2 𝑛 𝑛−1 𝑤 2( 𝑘 ),andif𝑘=𝑛−1,then( 𝑛−1 ) −2(𝑛−1 )=𝑛−2while Moreover, since g is a homomorphism of Lie algebras from ( 𝑛−1 )−2(𝑛−2 )+(𝑛−1 )−2(𝑛−2 ) = 𝑛−1−2+1−0 = 𝑛−2. g to w(W(g), 𝜕) and because g is commutative, it is also 𝑛−2 𝑛−2 𝑛−1 𝑛−1 g w(W(g), 0) = a homomorphism of Lie algebras from to Lemma 11. For every 𝑛≥1and every 𝑥, 𝑦 ∈ g,onehas l(W(g)). Therefore there is a unique homomorphism of # 𝑤 S( ( )) W( ) 𝑛 (commutative) algebras g from mod g to g such # # 𝑃 (𝑥,) 𝑦 = ∑𝑇 (𝑛,) 𝑘 𝑥 𝑦 . that 𝑤 ∘𝑗 =𝑤.Thus,weobtain𝑤 ∘ 𝑗̃ ∘𝑤 =𝑤 and also 𝑛 (𝑛−𝑘) (𝑘) (9) g g g g g g g 𝑘=0 𝑗̃ ∘𝑤# ∘𝑗 =𝑗 g g g g. According to the universal property of the enveloping algebra, the unique homomorphism that satisfies Proof. We have 𝑃1(𝑥, 𝑦)(0) =𝑥 𝑦(1)−𝑥(1)𝑦(0) = 𝑇(1, 0)𝑥(1)𝑦(0)+ ̃ # 𝑇(1, 1)𝑥 𝑦 𝑃 (𝑥, 𝑦) thesecondequalityis𝑖𝑑S( ( )) so that 𝑗 ∘𝑤 =𝑖𝑑S( ( )). (0) (1).Itisclearthat 𝑛 is homogeneous of mod g g g mod g 𝑛 𝑥 𝑦 We observe, however, that we cannot deduce from the first degree (inthesensethatitisasumofword (𝑖) (𝑗) # # 𝑤 ∘ 𝑗̃ =𝑖𝑑 𝑤 ∘ 𝑗̃ with 𝑖+𝑗). =𝑛 It can be written as 𝑃𝑛(𝑥, 𝑦) = equality that g g W(g) because g g is a homomor- 𝑛 ∑𝑘=0 𝛼𝑛,𝑘𝑥(𝑛−𝑘)𝑦(𝑘).Wehave𝑃𝑛+1(𝑥, 𝑦) =𝑛 𝜕𝑃 (𝑥, 𝑦) = phism of commutative differential algebras from (W(g), 𝜕) to 𝑛 𝑛 ∑𝑘=0 𝛼𝑛,𝑘𝑥(𝑛+1−𝑘)𝑦(𝑘) +∑𝑘=0 𝛼𝑛,𝑘𝑥(𝑛−𝑘)𝑦(𝑘+1) =𝛼𝑛,0𝑥(𝑛+1) + (W(g), 0) and not from (W(g), 𝜕) to itself. Nevertheless from 𝑛 𝑛−1 # # 𝛼 𝑦 +∑ 𝛼 𝑥 𝑦 +∑ 𝛼 𝑥 𝑦 𝑗̃ ∘𝑤 =𝑖𝑑 𝑤 𝑗̃ 𝑛,𝑛 (𝑛+1) 𝑘=1 𝑛,𝑘 (𝑛+1−𝑘) (𝑘) 𝑘=0 𝑛,𝑘 (𝑛−𝑘) (𝑘+1) so that g g S(g) it follows that g is one-to-one and g is 𝛼𝑛+1,0 =𝛼𝑛,0, 𝛼𝑛+1,𝑛+1 =𝛼𝑛,𝑛 and 𝛼𝑛+1,𝑘 =𝛼𝑛,𝑘−1 +𝛼𝑛,𝑘. onto. We observe that in general when g is a commutative Lie algebra, then its universal enveloping algebra with respect to (S( ( )), 0) 𝑗̃ w is not mod g that is, g is only onto and not one- 4. Embedding Conditions 𝑤# to-one, while g is only one-to-one and not onto, and the In this section, we present a sufficient condition under which derivative 𝜕 on W(g) is not the zero derivative. Indeed, for a Lie algebra embeds into its Wronskian envelope. instance let g be the free 𝑅-module 𝑅⋅𝑥generated by {𝑥} Adapting the terminology from [7, 11], we call Wronskian with the zero Lie bracket. The algebra 𝑅[𝑥] of polynomials in special those Lie 𝑅-algebras that embed into their Wronskian the variable 𝑥 is S(mod(g)) = S(𝑅 ⋅ 𝑥).Let𝜙 : 𝑅 ⋅ 𝑥 → 𝑅{𝑥} envelope. Thus every Abelian Lie algebra is Wronskian special be the 𝑅-module homomorphism defined by 𝜙(𝑥) =𝑥.Itis (see Section 3). It is quite obvious that not all Lie algebras a homomorphism of Lie algebras from 𝑅⋅𝑥to w(𝑅{𝑥}) since are Wronskian special, even when they are free as 𝑅-modules 0 = 𝜙(0) = 𝜙([𝛼𝑥, 𝛽𝑥]) and 𝛼𝛽(𝜙(𝑥) 𝜙(𝑥)−𝜙(𝑥)𝜙(𝑥) )=0for andeveninthecasewhere𝑅 is a field. This can be shown every 𝛼, 𝛽 ∈𝑅 while the algebra homomorphism extension as follows. For any elements 𝑥1,...,𝑥𝑛 of a Lie algebra, # 𝜙 :𝑅[𝑥]→ 𝑅{𝑥}of 𝜙 is not a homomorphism of differential we denote by [𝑥1,...,𝑥𝑛] theleft-normedbracket;thatis, 6 Algebra
if 𝑛=2,then[𝑥1,𝑥2] is the Lie bracket, and for 𝑛>2, [𝑥1, it is clear that ∏𝑖∈𝐼g𝑖 → ∏𝑖∈𝐼w(W(g𝑖)) = w(∏𝑖∈𝐼W(g𝑖)) ...,𝑥𝑛] = [[𝑥1,...,𝑥𝑛−1], 𝑥𝑛]. asasub-Liealgebraover𝑅. It is also clear that any sub- Let (𝐴, 𝜕) be a commutative differential 𝑅-algebra (with Lie algebra of a Wronskian special Lie algebra is itself a aunit).Then,theLiealgebraw(𝐴, 𝜕) satisfies the nontrivial WronskianspecialLiealgebra.Thus,Wronskianspecialityis identity (called the “standard Lie identity of degree 5” in [12] closed under product and subalgebra. and 𝑇4 in [13]) Now, we recall an important result from Razmyslov that is used hereafter to describe some basic Wronskian special Lie ∑ 𝜖 (𝜎) [𝑧, 𝑥 ,𝑥 ,𝑥 ,𝑥 ], 𝜎(1) 𝜎(2) 𝜎(3) 𝜎(4) (10) algebras. 𝜎∈S4 Theorem 15 K where 𝜖(𝜎) denotes the signature representation of the per- (see [12, 16]). Let be a field of characteristic 𝜎 𝑥 ,𝑥 ,𝑥 ,𝑥 ,𝑦 ∈ 𝐴 zero. Let g be a simple Lie algebra that satisfies 𝑇4.Then, mutation ,forevery 1 2 3 4 .(Thisresultwas K 𝐴 noticed in [12–14], e.g.) Therefore, a necessary condition for there exists a commutative integral -domain and a nonzero 𝜕∈ (𝐴) g (𝐴) an embedding is the following. derivation DerK such that embeds into Diff𝜕 as aLiesubalgebra.
Lemma 12. A Wronskian special Lie algebra satisfies 𝑇4. As a consequence of the previous theorem and Lemma 13, 𝑇 It is not known whether the above lemma is also a in characteristic zero, any simple Lie algebra satisfying 4 is sufficient condition. Nevertheless this gives a negative answer a Wronskian special Lie algebra. From this result we deduce to a question of Lawvere in his Ph.D thesis [15] where he asked the following. whether or not any Lie algebra is Wronskian special. Theorem 16. K ( ) (𝐴) Let be a field of characteristic zero. Let g𝑖 𝑖∈𝐼 Following [16], let us define the Lie algebra Diff𝜕 of K 𝑖∈𝐼 special derivations withrespecttothesignature derivation 𝜕 be a family of Lie -algebras such that for every either g 𝑇 g g as follows: let 𝐴⋅𝜕={𝛼𝜕:𝛼∈𝐴}be the sub-𝐴-module of 𝑖 is simple and satisfies 4 or 𝑖 is Abelian. Let be a sub-Lie algebra of ∏𝑖∈𝐼g𝑖.Then,g is a Wronskian special Lie algebra. Der𝑅(𝐴),the𝐴-module of all 𝑅-derivations of 𝐴, generated by 𝜕 (it is the image of the 𝐴-module map 𝜌𝜕 :𝐴 → Der𝑅(𝐴) Proof. According to Razmyslov’s theorem if g𝑖 is simple and given by 𝜌(𝛼) = 𝛼𝜕). Then, Diff𝜕(𝐴) is the 𝑅-module satisfies 𝑇4 then g𝑖 is Wronskian special. We also know that structure of 𝐴⋅𝜕together with the usual bracket of derivations (𝐴) = (𝐴) any Abelian Lie algebra is Wronskian special. So their product (inherited by the usual bracket of gl𝑅 End𝑅-Mod the ∏ g𝑖 also is. Finally any sub-Lie algebra of a Wronskian linear endomorphisms of 𝐴). Let ann(𝜕)={𝑎∈𝐴:𝑎𝜕=0} 𝑖∈𝐼 special Lie algebra also is Wronskian special. be the annihilator of 𝜕.ItisanAbelianLiesubalgebraof 𝜌𝜕 w(𝐴, 𝜕) and ann(𝜕) → w(𝐴, 𝜕) Diff𝜕(𝐴) is a short exact sequence of Lie 𝑅-algebras in such a way that w(𝐴, 𝜕) is 5. The Free Wronskian Special Lie 𝑅-Algebra an extension of Diff𝜕(𝐴) by ann(𝜕).Inthecasewhere UptonowitisnotknownifWronskianspecialLiealge- ker 𝜌𝜕 = ann(𝜕) = 0—for instance when 𝐴 has no zero divi- bras form a variety of Lie algebras (nor even if it closed sors and 𝜕 =0̸ —then Diff𝜕(𝐴) and w(𝐴, 𝜕) are isomorphic Lie 𝑅-algebras. The following result is thus obvious. under homomorphic images). In [13]itisconjecturedthat (finitely generated) Wronskian special Lie algebras form the 𝑇 Lemma 13. Let 𝐴 be a commutative integral 𝑅-domain, and variety of all (finitely generated) Lie algebras satisfying 4 let 𝜕 be a nonzero derivation of 𝐴.Letg be a Lie 𝑅-algebra (in characteristic zero). We do not prove nor disprove this such that g → Diff𝜕(𝐴) (as Lie 𝑅-algebras)(wenotethatin conjecture. Nevertheless we observe that Wronskian special 𝑅 𝑅 particular g satisfies 𝑇4 since it holds in Diff𝜕(𝐴)).Then,g is Lie -algebras, for some ring , form a derived category a Wronskian special Lie algebra. of the variety of commutative differential algebras obtained by considering their Lie bracket as the derived operator Example 14. The Lie algebra sl2(K) over a field of charac- 𝑊(𝑥, 𝑦) = 𝑥𝜕(𝑦) −𝜕(𝑥)𝑦 (see [7] for definitions of derived teristic zero is simple and embeds into Diff𝜕(K[𝑥]) where category and derived operator). Therefore, we may speak 𝜕 is the usual derivation of polynomials in the variable 𝑥. about the free Wronskian special Lie 𝑅-algebra generated by Indeed, sl2(K) is isomorphic to the three-dimensional K- aset𝑋 (see [7,Theorem4.4]).ItisobtainedastheLie 2 vector space generated by 𝑒=−𝑥𝜕, 𝑓=𝜕, ℎ=2𝑥𝜕with the subalgebra of w(𝑅{𝑋}) generated by 𝑋, where we recall that 𝑅{𝑋} is the free commutative differential algebra over 𝑋 (if Lie algebra structure given by the usual commutator (see, e.g., sl2(K) 𝑢∈𝑅{𝑋},then𝑢 denotes the derivation of 𝑢 in 𝑅{𝑋} and its [17]). Therefore, is a Wronskian special Lie algebra. Wronskian commutator is denoted by 𝑊(𝑢, V)=𝑢V −𝑢V). We may also remark that Wronskian speciality is pre- It is constructed by induction as follows: let served by direct products (even infinite). More precisely, let (g𝑖)𝑖∈𝐼 be a collection of Lie 𝑅-algebras such that for each 𝑋0 =𝑋, 𝑖∈𝐼, g𝑖 → w(W(g𝑖)) (as Lie 𝑅-algebras), that is, each factor g𝑖 is a Wronskian special Lie algebra. Then, ∏𝑖∈𝐼g𝑖 → 2 𝑋𝑘+1 =⟨𝑋𝑘 ∪{𝑤∈𝑅{𝑋} :∃𝑢,V ∈𝑋𝑘, w(W(∏𝑖∈𝐼g𝑖)) as Lie 𝑅-algebras, where the operations (Lie (11) brackets, derivative, and product) are considered component- 𝑤=𝑊(𝑢, V) }⟩ ∀𝑘 ≥ 0, wise (i.e., ∏𝑖∈𝐼g𝑖 is a Wronskian special Lie algebra). Indeed, Algebra 7
where ⟨𝐸⟩ denotes the submodule of 𝑅{𝑋} generated by w(𝑅{𝑋}) ≅ Diff𝜕(𝑅{𝑋}) (as Lie algebras over 𝑅), where asubset𝐸.Then,thefreeWronskianspecialLiealgebra, 𝜕 istheusualderivationon𝑅{𝑋} (i.e, 𝜕(𝑢) =𝑢 ). denoted by WspLie(𝑋),over𝑋 is the nested union ⋃𝑘≥0 𝑋𝑘, Since Diff𝜕(𝑅{𝑋}) isasub-Liealgebraofgl𝑅(𝑅{𝑋}) and its bracket is given by the Wronskian bracket 𝑊 of 𝑅{𝑋}. (under the commutator), it follows that w(𝑅{𝑋}) is also a (It is clear that as defined previously WspLie(𝑋) is a 𝑅- special Lie algebra. module.) Acknowledgments Remark 17. It is clear that if 𝑢, V ∈ 𝑅{𝑋} have no nonzero 𝑊(𝑢, V) constantterm,thenitisalsothecasefor .Moreover The author wishes to thank Professor Huishi Li from the (𝑋) every element of WspLie has no nonzero constant term as Department of Applied Mathematics of the College of it can be checked inductively from the previous observation Information Science and Technology of Hainan University 𝑋 =𝑋 and since every element of 0 has no nonzero (China) for discussions related the Wronskian envelope and (𝑋) =𝑅{𝑋}̸ constant term. Therefore, WspLie as sets so that noncommutative Groebner bases, Professor Hans-E. Porst (𝑋) ≠ (𝑅{𝑋}) 𝑋= WspLie w as Lie algebras. (Note that when from the University of Bremen (Germany) for his advices 0 (0) = 0 𝑅{0} ≅ 𝑅 𝑋 ,thenWspLie while .) When is reduced to concerning left adjoints and algebraic functors, and Professor 𝑥 ({𝑥}) 𝑅𝑥 only one element ,thenWspLie isthefreemodule G.M. Bergman from the Department of Mathematics of the seen as an Abelian Lie algebra and, therefore, is isomorphic University of California (USA) for a fruitful correspondence tothefreeLiealgebraononegenerator. andforpointingtotheknowledgeoftheauthortheworksof the Russian mathematicians, Kirillov and Razmyslov, about We observe that since commutative differential alge- Lie algebras of vector fields. bras form a nontrivial variety (a nontrivial variety is a variety with algebras of cardinality >1), the natural set- theoretic map from 𝑋 to 𝑅{𝑋} is one-to-one, and we may References assume that 𝑋⊆WspLie(𝑋) ⊆ 𝑅{𝑋} (in particular, 𝑋 is free 𝑅 (𝑋) [1] I. Kaplansky, An Introduction to Differential Algebra,Publica- over in WspLie ). According to the property of the tions de l’Institut Mathematiques´ de l’UniversitedeNancago,´ Wronskian envelope, there is a unique differential algebra Hermann, 1957. 𝜙:W( (𝑋)) → 𝑅{𝑋} map WspLie such that the following [2] S. MacLane, Categories for the Working Mathematician,vol.5of diagram commutes (where the unnamed arrows are the Graduate Texts in Mathematics, Springer, 1971. canonical inclusions) [3] N. Bourbaki, Elements of Mathematics, Algebra,chapter1–3, Springer, 1998. (𝑋) 𝔴(𝑅{𝑋}) WspLie [4] M. van der Put and M. F. Singer, Galois Theory of Linear Dif- 𝔴(𝜙) (12) ferential Equations,vol.328ofGrundlehren der Mathematis- 𝔴(𝒲(WspLie(𝑋))) chen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, 2003. [5] J. F. Ritt, Differential Equations from the Algebraic Standpoint, Moreover the composition 𝑋→WspLie(𝑋) → vol. 14 of American Mathematical Society, Colloquium Publica- w(W(WspLie(𝑋))) of natural embeddings gives rise to a tions, 1932. set-theoretic map 𝑋→W(WspLie(𝑋)). Therefore, there is a [6] Y. Chen, Y. Chen, and Y. Li, “Composition-diamond lemma for unique homomorphism of commutative differential algebras differential algebras,” TheArabianJournalforScienceandEngi- 𝜓 : 𝑅{𝑋} → W(WspLie(𝑋)) such that the following dia- neering A,vol.34,no.2,pp.135–145,2009. gram commutes: [7] P. M. Cohn, Universal Algebra,vol.6ofMathematics and its Applications,Kluwer,1981. 𝑋 𝒲(WspLie(𝑋)) [8] N. Bourbaki, Elements of Mathematics, Lie Groups and Lie 𝜓 (13) Algebras, chapter 1, Springer, 1998. 𝑅{𝑋} [9] A. S. Dzhumadil’cprimedaev, “𝑛-Lie structures generated by Wronskians,” Sibirski˘ı Matematicheski˘ıZhurnal,vol.46,no.4, Both compositions 𝜓∘𝜙and 𝜙∘𝜓are the identity on 𝑋 pp.601–612,2005. and hence, by uniqueness, the identity everywhere. Therefore, [10] E. G. Manes, Algebraic Theories,vol.26ofGraduate Texts in W(WspLie(𝑋)) and 𝑅{𝑋} are canonically isomorphic (as Mathematics,, Springer, 1976. commutative differential algebras): [11] Yu. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, 1987. 𝑊 (WspLie (𝑋)) ≅𝑅{𝑋} . (14) [12] Yu. P. Razmyslov, “Simple Lie algebras satisfying the standard Lie identity of degree 5,” Mathematics of the USSR-Izvestiya,vol. Remark 18. The previous result shares some similarity with 26, no. 1, 1986. the well-known fact that the universal enveloping algebra of [13] A. A. Kirillov, V. Yu. Ovsienko, and O. D. Udalova, “Identities thefreeLiealgebrageneratedby𝑋 is canonically isomorphic in the Lie algebra of vector fields on the real line,” Selecta Math- to the free associative algebra generated by 𝑋 (see [3]). ematica Formerly Sovietica,vol.10,no.1,pp.7–17,1991. 𝑛 [14] G. M. Bergman, The Lie Algebra of Vector Fields in R Satisfies Remark 19. Let us assume that 𝑅 is an integral Polynomial Identities, University of California, Berkeley, Calif, domain. Then, 𝑅{𝑋} is also an integral domain, and USA, 1979. 8 Algebra
[15] F. W. Lawvere, Functorial semantics of algebraic theories [Ph.D. thesis], Columbia University, 1963. [16] Yu. P. Razmyslov, Identities of Algebras and Their Represen- tations,vol.138ofTranslations of Mathematical Monographs, American Mathematical Society, 1994. [17] P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, 1993. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014
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