Universal enveloping algebras of Lie-Rinehart algebras as a left adjoint functor
aa r X i v : . [ m a t h . R A ] F e b UNIVERSAL ENVELOPING ALGEBRAS OF LIE-RINEHARTALGEBRAS AS A LEFT ADJOINT FUNCTOR
PAOLO SARACCO
Abstract.
We prove how the universal enveloping algebra constructions for Lie-Rinehartalgebras and anchored Lie algebras are naturally left adjoint functors. This provides aconceptual motivation for the universal properties these constructions satisfy. As a sup-plement, the categorical approach offers new insights into the definitions of Lie-Rinehartalgebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gaugealgebra of a module.
Contents
Introduction 11. Preliminaries 32. The universal enveloping A e -ring as a left adjoint functor 73. The universal enveloping A -ring as a left adjoint functor 124. On morphisms, modules and the infinitesimal gauge algebra 13References 16 Introduction
It is well-known that any associative and unital algebra A over a field k admits a naturalLie algebra structure with bracket given by the commutator bracket. Usually, one writes L ( A ) for the Lie algebra associated with A and the assignment A
7→ L ( A ) turns out to befunctorial from the category of k -algebras Alg k to the category of Lie algebras Lie k . In theopposite direction, one defines the universal enveloping k -algebra of a given a Lie algebra L over k to be an associative and unital algebra U ( L ) together with a natural morphism ofLie algebras L → L ( U ( L )) which is universal from L to the functor L (see, for instance, [2,§2.1]). Equivalently, the universal enveloping algebra construction provides a left adjoint U : Lie k → Alg k to the associated Lie algebra functor L . Mathematics Subject Classification.
Primary: 16B50; 16S10; 16S30; 16W25; 18A40. Secondary:17A30; 17B66.
Key words and phrases.
Lie-Rinehart algebras; Anchored Lie algebras; universal enveloping algebras;universal properties; adjoint functors; Connes-Moscovici’s bialgebroid; Atiyah algebra.Paolo Saracco is a Charg´e de Recherches of the Fonds de la Recherche Scientifique - FNRS and a mem-ber of the “National Group for Algebraic and Geometric Structures and their Applications” (GNSAGA-INdAM).
Among Lie algebras, a family that particularly attracted the attention of the communityis that of Lie-Rinehart algebras. Informally, a Lie-Rinehart algebra over a commutativealgebra R (also called ( k , R )-Lie algebra) is a Lie algebra L together with an additionalstructure that reflects the interaction between the algebra of smooth functions on a smoothmanifold M and the Lie algebra of smooth vector fields on M . Rinehart himself gave anexplicit construction of what he called the universal enveloping algebra U ( R, L ) of a Lie-Rinehart algebra in [14] and proved a Poincar´e-Birkhoff-Witt theorem for the latter. Otherequivalent constructions are provided in [5, §3.2], [8, page 64], [16, §18]. The universalproperty of U ( R, L ) as an algebra is spelled out in [8, page 64] and [12, page 174] (whereit is attributed to Feld’man). Its universal property as an R -bialgebroid is codified inthe Cartier-Milnor-Moore Theorem for U ( R, L ) proved in [13, §3], where it is shown thatthe construction of the universal enveloping algebra provides a left adjoint to the functorsending any cocommutative bialgebroid to its Lie-Rinehart algebra of primitive elements.Further algebraic and categorical properties and applications are investigated in [1, 4, 6, 7,8].However, as far as the author is aware, nowhere it is shown that the construction ofthe so-called “universal enveloping algebra” provides a left adjoint functor. In fact, theuniversal property of U ( R, L ) mentioned above involves at the same time a morphism of k -algebras R → U ( R, L ) and a morphism of Lie algebras L → L (cid:16) U ( R, L ) (cid:17) which need tobe compatible in a suitable way and there seems to be no natural functor from the categoryof k -algebras (or that of R -rings) to the category of Lie-Rinehart algebras over R whichmay play the role of the right adjoint functor.Our aim is to make up for this lack by explicitly providing such a right adjoint. As aconsequence, our results provide a conceptual motivation for addressing U ( R, L ) as theuniversal enveloping R -ring of L . Notice also that being able to identify the universalenveloping algebra functor as a left adjoint offers a number of additional informationsfrom the categorical point of view. For instance, that it preserves all colimits that existsin the category of Lie-Rinehart algebras over R . Furthermore, we will comment on howthe categorical approach and the constructions we introduce allow us to re-interpret andexplain the notions of modules over a Lie-Rinehart algebra L , of morphisms of Lie-Rinehartalgebras over different bases and of the infinitesimal gauge algebra of an A -module M withrespect to a Lie-Rinehart algebra L , as introduced in [8].Even if our main objective concerns Lie-Rinehart algebras and their universal envelopingalgebras, we will initially work in the more general framework of anchored Lie algebrasas introduced in [15, §2]. Our motivation for this choice is twofold. Firstly, the non-commutative setting of anchored Lie algebras is, at the same time, general enough tobecome easier to be handle and close enough to our target to provide most of the resultswe need in order to deal with the commutative setting. Secondly, anchored Lie algebrassubsume several important examples of Lie algebras acting by derivations on associativealgebras which are not necessarily commutative (any Lie algebra acts by derivations onits universal enveloping algebra and any associative algebra acts by inner derivations onitself) and they proved to be of key importance in the study of the structure of primitively EAS OF LRAS AS LEFT ADJOINT 3 generated bialgebroids over a non-commutative base (see [15, §4]). Therefore, they deserveto be studied more closely. In addition, it is noteworthy that our approach via anchored Liealgebras allows also to re-interpret the algebra of differential operators of a representationof a Lie algebra, as introduced by Jacobson in [9, page 175], and to provide a conceptualinterpretation for its universal property (see discussion after Theorem 2.6).Concretely, after an introductory Section 1, where we collect the basics on anchored Liealgebras and Lie-Rinehart algebras that we need to keep the presentation self-contained, inSection 2 we explicitly construct a functor L A : Ring A e → AnchLie A which we prove to be thenatural right adjoint to the universal enveloping A e -ring functor U A : AnchLie A → Ring A e provided by the Connes-Moscovici’s bialgebroid construction in [15, §2.2] (see Theorem2.8). Section 3 is devoted to adapt the results of Section 2 to the commutative frameworkof Lie-Rinehart algebras and so to construct a right adjoint L A : Ring A → LieRin A tothe universal enveloping algebra functor U A : LieRin A → Ring A (see Theorem 3.4). Weconclude with a brief reflection in Section 4 concerning the definitions of morphisms ofLie-Rinehart algebras over different bases, of modules over a Lie-Rinehart algebra and ofthe Lie-Rinehart algebra of infinitesimal gauge transformations of an A -module. Conventions.
All over the paper, we assume a certain familiarity of the reader with thelanguage of monoidal categories and of (co)monoids therein (see, for example, [11, VII]).We work over a ground field k of characteristic 0. All vector spaces are assumed to beover k . The unadorned tensor product ⊗ stands for ⊗ k . All (co)algebras and bialgebras areintended to be k -(co)algebras and k -bialgebras, that is to say, (co)algebras and bialgebrasin the symmetric monoidal category of vector spaces ( Vect k , ⊗ , k ). Every (co)module hasan underlying vector space structure. Identity morphisms Id V are often denoted simply by V .If f : U → V is a k -linear map, we denote by f ∗ the morphism Hom k ( V, W ) → Hom k ( U, W ) , g g ◦ f, and by f ∗ the morphism Hom k ( W, U ) → Hom k ( W, V ) , g f ◦ g .Unless stated otherwise, A denotes a not necessarily commutative algebra over k and A o denotes its opposite algebra. When looking at a ∈ A as an element in A o , we will denoteit by a o . The Lie algebra associated with A , which by abuse of notation we denote by A again, is the Lie algebra which has as underlying k -vector space A itself and bracket givenby the commutator [ a, b ] = ab − ba for all a, b ∈ A .If C is a k -coalgebra, we take advantage of the Heyneman-Sweedler notation ∆( c ) = P c ⊗ c to deal explicitly with the comultiplication.Finally, we will make use of the fact that algebraic forgetful functors create limits inthe sense of [11, V.1, Definition]. In particular, the pullback of a diagram of vector spaces(or modules) can be computed by performing the corresponding pullback in the categoryof sets and by endowing it with the unique structure that makes of it a vector space (ormodule). 1. Preliminaries
We collect some facts about bimodules, rings, Lie-Rinehart algebras and anchored Liealgebras that will be needed in the sequel. The aim is to keep the exposition self-contained.
PAOLO SARACCO A -bimodules and A -rings. Given a k -algebra A , the category of A -bimodules formsa non-strict monoidal category ( A Mod A , ⊗ A , A, a , l , r ). Nevertheless, all over the paper wewill behave as if the structural natural isomorphisms a M,N,P : ( M ⊗ A N ) ⊗ A P → M ⊗ A ( N ⊗ A P ) , ( m ⊗ A n ) ⊗ A p → m ⊗ A ( n ⊗ A p ) , l M : A ⊗ A M → M, a ⊗ A m a · m, and r M : M ⊗ A A → M, m ⊗ A a m · a, were “the identities”, that is, as if A Mod A was a strict monoidal category.An A -ring is a monoid in ( A Mod A , ⊗ A , A ). Equivalently, it is a k -algebra R togetherwith a morphism of k -algebras φ : A → R . In what follows we will often write that ( R, φ )is an A -ring when we need to stress the role played by φ . A morphism of A -rings from( R, φ ) to (
S, ψ ) is a k -algebra morphism ϕ : R → S such that ϕ ◦ φ = ψ . The category of A -rings and their morphisms will be denoted by Ring A .1.2. Lie-Rinehart algebras. A Lie-Rinehart algebra over a commutative k -algebra A (called in this way in honour of Rinehart, who studied them in [14] under the name of( K, R )-Lie algebras) is a Lie algebra L endowed with a (left) A -module structure A ⊗ L → L, a ⊗ X a · X, and with a Lie algebra morphism ω : L → Der k ( A ) such that ω ( a · X ) = a · ω ( X ) and [ X, a · Y ] = a · [ X, Y ] + ω ( X ) ( a ) · Y (1)for all a ∈ A and X, Y ∈ L . A morphism of Lie-Rinehart algebras over A from ( L, ω ) to( L ′ , ω ′ ) is a Lie algebra morphism f : L → L ′ which is also left A -linear and such that ω ′ ◦ f = ω . With these morphisms, Lie-Rinehart algebras over A form a category thatwe denote by LieRin A . For the sake of brevity, we may simply write that ( A, L, ω ) is aLie-Rinehart algebra to mean that (
L, ω ) is a Lie-Rinehart algebra over A . When A and ω are clear from the context, we may simply write L instead of ( A, L, ω ). For instance, wemay often write that f : L → L ′ is a morphism of Lie-Rinehart algebras over A to meanthat f is a morphism in LieRin A as above. Example 1.1.
The smooth global sections of a Lie algebroid over a real smooth manifold M naturally form a Lie-Rinehart algebra over C ( M ). In particular, the smooth vectorfields on M give rise to the Lie-Rinehart algebra (cid:16) Der k ( C ( M )) , Id (cid:17) .Given a Lie-Rinehart algebra ( A, L, ω ), a universal enveloping algebra of L is an A -ring (cid:16) U A ( L ) , ι A (cid:17) together with a morphism of Lie algebras ι L : L → U A ( L ) such that ι L ( a · X ) = ι A ( a ) ι L ( X ) and h ι L ( X ) , ι A ( a ) i = ι A (cid:16) ω ( X )( a ) (cid:17) (2)for all a ∈ A , X ∈ L , and which is universal with respect to this property. That is to say,for any A -ring ( R, φ A ) together with a Lie algebra morphism φ L : L → R such that φ L ( a · X ) = φ A ( a ) φ L ( X ) and h φ L ( X ) , φ A ( a ) i = φ A (cid:16) ω ( X )( a ) (cid:17) (3)for all a ∈ A , X ∈ L , there exists a unique morphism of A -rings Φ : U A ( L ) → R such thatΦ ◦ ι L = φ L . It follows that the universal enveloping algebra construction induces a functor U A : LieRin A → Ring A . EAS OF LRAS AS LEFT ADJOINT 5
Anchored Lie algebras.
Recall from [15, §2.1] that an anchored Lie algebra over anon-commutative k -algebra A (also called A -anchored Lie algebra) is a Lie algebra L over k together with a Lie algebra morphism ω : L → Der k ( A ), called the anchor . A morphismof A -anchored Lie algebras from ( L, ω ) to ( L ′ , ω ′ ) is a Lie algebra morphism f : L → L ′ such that ω ′ ◦ f = ω . The category of A -anchored Lie algebras and their morphisms willbe denoted by AnchLie A . As a matter of notation, we may write ( A, L, ω ) to mean the A -anchored Lie algebra ( L, ω ). Again, if A and ω are clear from the context, we maysimply write L instead of ( A, L, ω ). Example 1.2.
The associated Lie algebra of a k -algebra A becomes an A -anchored Liealgebra with anchor ̟ A : A → Der k ( A ) , a [ a, − ].Given an A -anchored Lie algebra ( L, ω ), we have a construction for a universal envelopingalgebra of L . This is the A e -ring A ⊙ U ( L ) ⊙ A obtained by the Connes-Moscovici’sbialgebroid construction in [15, §2.2]. Explicitly, it is the k -algebra with underlying vectorspace A ⊗ U ( L ) ⊗ A , unit 1 A ⊗ U ⊗ A , multiplication uniquely determined by( a ⊗ u ⊗ b )( a ′ ⊗ v ⊗ b ′ ) = X a ( u · a ′ ) ⊗ u v ⊗ ( u · b ′ ) b and A e -ring structure given by J A : A e → A ⊙ U ( L ) ⊙ A, a ⊗ b o a ⊗ U ⊗ b . It comeswith a morphism of Lie algebras J L : L → A ⊙ U ( L ) ⊙ A, X A ⊗ X ⊗ A , such that h J L ( X ) , J A ( a ⊗ b o ) i = J A (cid:16) ω ( X )( a ) ⊗ b o + a ⊗ ω ( X )( b ) o (cid:17) for all a, b ∈ A , X ∈ L , and which is universal with respect to this property. That is tosay, for any A e -ring ( R, φ A ) together with a Lie algebra morphism φ L : L → R such that h φ L ( X ) , φ A ( a ⊗ b o ) i = φ A (cid:16) ω ( X )( a ) ⊗ b o + a ⊗ ω ( X )( b ) o (cid:17) for all a, b ∈ A , X ∈ L , there exists a unique morphism of A e -rings Φ : A ⊙ U ( L ) ⊙ A → R such that Φ ◦ J L = φ L . It follows that this construction induces a functor AnchLie A → Ring A e , ( L, ω ) (cid:16) A ⊙ U ( L ) ⊙ A, J A (cid:17) . (4)One of our main aims is to show that the universal property characterizing A ⊙ U ( L ) ⊙ A is expressing the fact that the functor (4) is a left adjoint functor. As a consequence, wewill refer to A ⊙ U ( L ) ⊙ A as the universal enveloping A e -ring of ( A, L, ω ).The following results clarify the relationship between A -anchored Lie algebras and A e -anchored Lie algebras that will be needed in the sequel. Lemma 1.3.
Any δ ∈ Der k ( A ) induces a derivation δ ⊗ ∈ Der k ( A e ) uniquely determinedby δ ⊗ (cid:16) a ⊗ b o (cid:17) = δ ( a ) ⊗ b o + a ⊗ δ ( b ) o for all a, b ∈ A . This induces a Lie algebra morphism e : Der k ( A ) → Der k ( A e ) , δ δ ⊗ .Proof. Straightforward. (cid:3)
Proposition 1.4.
Any A -anchored Lie algebra ( L, ω ) is an A e -anchored Lie algebra via ω ⊗ : L → Der k ( A e ) , X ω ( X ) ⊗ . The assignment ( L, ω ) ( L, ω ⊗ ) provides a functor E A : AnchLie A → AnchLie A e . PAOLO SARACCO
Proof.
In view of Lemma 1.3, the composition L ω −→ Der k ( A ) e −→ Der k ( A e ) mapping X to ω ( X ) ⊗ is a Lie algebra map. Thus, the assignment ( L, ω ) ( L, ω ⊗ ) is well-defined and itis clearly functorial (it acts as the identity on morphisms). (cid:3) The functor E A of Proposition 1.4 naturally admits a left adjoint functor. In order tointroduce it, we take advantage of the construction of the product in the category AnchLie A . Proposition 1.5.
Let ( L, ω ) and ( L ′ , ω ′ ) be A -anchored Lie algebras. The product of ( L, ω ) and ( L ′ , ω ′ ) in AnchLie A exists and can be computed as the pullback of vector spaces L × Der k ( A ) L ′ q / / q (cid:15) (cid:15) y L ω (cid:15) (cid:15) L ′ ω ′ / / Der k ( A ) , (5) with component-wise bracket and anchor ω × := ω ◦ q = ω ′ ◦ q .Proof. Since the pullback in vector spaces of a diagram of Lie algebras is naturally a Liealgebra with component-wise bracket, L × Der k ( A ) L ′ is a Lie algebra and q , q are Lie algebramaps. The fact that the A -anchored Lie algebra (cid:16) L × Der k ( A ) L ′ , ω × (cid:17) satisfies the universalproperty of the product is an easy check that we leave to the interested reader. (cid:3) Let (
M, ̟ ) be an A e -anchored Lie algebra. By Lemma 1.3, (cid:16) Der k ( A ) , e (cid:17) is an A e -anchored Lie algebra as well and we can consider their product in AnchLie A e : F A ( M, ̟ ) q / / q (cid:15) (cid:15) y M ̟ (cid:15) (cid:15) Der k ( A ) e / / Der k ( A e )with component-wise bracket and anchor e ◦ q = ̟ ◦ q . Concretely, F A ( M, ̟ ) = (cid:26) ( X, δ ) ∈ M × Der k ( A ) (cid:12)(cid:12)(cid:12) ̟ ( X )( a ⊗ b o ) = δ ( a ) ⊗ b o + a ⊗ δ ( b ) o , ∀ a, b ∈ A (cid:27) . Theorem 1.6.
For every A e -anchored Lie algebra ( M, ̟ ) , the k -vector space F A ( M, ̟ ) is an A -anchored Lie algebra with component-wise bracket and anchor q . The assignment ( M, ̟ ) (cid:16) F A ( M, ̟ ) , q (cid:17) induces a functor F A : AnchLie A e → AnchLie A which is rightadjoint to the functor E A : AnchLie A → AnchLie A e .Proof. We already know that F A ( M, ̟ ) is a Lie algebra and that q is a morphism of Liealgebras. Moreover, the assignment ( M, ̟ ) (cid:16) F A ( M, ̟ ) , q (cid:17) is functorial.To show that F A is right adjoint to E A : AnchLie A → AnchLie A e , notice that L itself with q = Id and q = ω is a pullback of the pair ( e, ω ⊗ ). Therefore, the identity morphism of L plays the role of the component of the unit at L and it is a universal arrow from L to F A .In view of [11, III.1, Theorem 2(i)], the pair E A , F A forms an adjoint pair of functors. (cid:3) EAS OF LRAS AS LEFT ADJOINT 7
Summing up, for every k -algebra A we have an adjunction AnchLie A e F A (cid:6) (cid:6) AnchLie A . E A E E (6)2. The universal enveloping A e -ring as a left adjoint functor Let
A, R be k -algebras and let ( M, ̟ ) be an R -anchored Lie algebra. Assume that amorphism of k -algebras φ : A → R has been given. In this way, R becomes an A -ring andwe can consider the k -vector space Der k ( A, R ) = (cid:26) f ∈ Hom k ( A, R ) (cid:12)(cid:12)(cid:12) f ( ab ) = f ( a ) φ ( b ) + φ ( a ) f ( b ) for all a, b ∈ A (cid:27) . Define A RA ( M, ̟ ) to be the pullback A RA ( M, ̟ ) y p / / p (cid:15) (cid:15) M ̟ (cid:15) (cid:15) Der k ( R ) φ ∗ (cid:15) (cid:15) Der k ( A ) φ ∗ / / Der k ( A, R ) (7)computed in the category of k -vector spaces. Concretely, A RA ( M, ̟ ) = (cid:26) ( X, δ ) ∈ M × Der k ( A ) (cid:12)(cid:12)(cid:12) ̟ ( X ) (cid:16) φ ( a ) (cid:17) = φ (cid:16) δ ( a ) (cid:17) for all a ∈ A (cid:27) . (8) Lemma 2.1.
Let ( R, φ ) be an A -ring and ( M, ̟ ) be an R -anchored Lie algebra. Thenthe k -vector space A RA ( M, ̟ ) is an A -anchored Lie algebra with anchor p : A RA ( M, ̟ ) → Der k ( A ) and with component-wise bracket. Furthermore, p is a Lie algebra morphism.Proof. Let (
X, δ ) , ( X ′ , δ ′ ) ∈ A RA ( M, ̟ ). By using (8) one checks that ̟ (cid:16) [ X, X ′ ] (cid:17)(cid:16) φ ( a ) (cid:17) = φ (cid:18) [ δ, δ ′ ]( a ) (cid:19) for all a ∈ A . Therefore (cid:16) [ X, X ′ ] , [ δ, δ ′ ] (cid:17) ∈ A RA ( M, ̟ ), which shows that A RA ( M, ̟ ) is a Liealgebra. Clearly, p is a morphism of Lie algebras with respect to this structure and hence A RA ( M, ̟ ) is an A -anchored Lie algebra, as claimed. (cid:3) Proposition 2.2.
For any A -ring ( R, φ ) , the assignment ( M, ̟ )
7→ A RA ( M, ̟ ) induces afunctor A RA : AnchLie R → AnchLie A .Proof. We already know how A RA acts on objects. To see how it acts on morphism, noticethat it f : ( M, ̟ ) → ( M ′ , ̟ ′ ) is a morphism of R -anchored Lie algebras then the external PAOLO SARACCO hexagon in the following diagram A RA (cid:16) M, ̟ (cid:17) A RA ( f ) ' ' p / / p $ $ M f ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ A RA (cid:16) M ′ , ̟ ′ (cid:17) y p ′ / / p ′ (cid:15) (cid:15) M ′ ̟ ′ (cid:15) (cid:15) Der k ( R ) φ ∗ (cid:15) (cid:15) Der k ( A ) φ ∗ / / Der k ( A, R ) , commutes by definition of A RA ( M, ̟ ). Thus, by the universal property of the pullback,there exists a unique morphism of k -vector spaces A RA ( f ) : A RA (cid:16) M, ̟ (cid:17) → A RA (cid:16) M ′ , ̟ ′ (cid:17) suchthat p ′ ◦ A RA ( f ) = p and p ′ ◦ A RA ( f ) = f ◦ p , which is explicitly given by A RA ( f ) : (cid:16) X, δ (cid:17) (cid:16) f ( X ) , δ (cid:17) . (9)Since f is a morphism of Lie algebras, it follows that A RA ( f ) is of Lie algebras as well. Thecompatibility with the anchors is clear, whence A RA ( f ) is of A -anchored Lie algebras. Sinceon arrows A RA is defined in terms of a universal property, it is functorial. (cid:3) Corollary 2.3.
Let ( R, φ ) be an A -ring and denote by ̟ R : R → Der k ( R ) the structure of R -anchored Lie algebra induced on R by the commutator bracket as in Example 1.2. The k -vector space A RA ( R, ̟ R ) is an A -anchored Lie algebra with anchor p : A RA ( R, ̟ R ) → Der k ( A ) and with component-wise bracket. It follows from Corollary 2.3 that to any A -ring R we may assign an A -anchored Liealgebra (cid:16) A RA ( R, ̟ R ) , p (cid:17) . Theorem 2.4.
The assignment R
7→ A RA ( R, ̟ R ) induces a functor L A : Ring A → AnchLie A .Proof. We already know how L A acts on objects. To see how it acts on morphisms, noticethat it ϕ : ( R, φ ) → ( S, ψ ) is a morphism of A -rings then the right-most square and thelowest triangle in the following diagram commute: A RA (cid:16) R, ̟ R (cid:17) y p / / p (cid:15) (cid:15) R ̟ R (cid:15) (cid:15) ϕ ' ' PPPPPPPPPPPPPP
Der k ( R ) φ ∗ (cid:15) (cid:15) S ̟ S (cid:15) (cid:15) Der k ( A ) φ ∗ / / ψ ∗ Der k ( A, R ) ϕ ∗ ( ( ◗◗◗◗◗◗◗◗ Der k ( S ) ψ ∗ (cid:15) (cid:15) Der k ( A, S ) . EAS OF LRAS AS LEFT ADJOINT 9
Therefore, by the universal property of the pullback, there exists a unique morphism of k -vector spaces L A ( ϕ ) : A RA (cid:16) R, ̟ R (cid:17) → A SA (cid:16) S, ̟ S (cid:17) , which is explicitly given by L A ( ϕ ) : ( r, δ ) (cid:16) ϕ ( r ) , δ (cid:17) . (10)Since ϕ is a k -algebra morphism satisfying ϕ ◦ φ = ψ , it follows that L A ( ϕ ) is a Lie algebramorphism. The compatibility with the anchors is clear, whence L A ( ϕ ) is of A -anchoredLie algebras. Since on arrows L A is defined in terms of a universal property, it is functorialand so we have a well-defined functor L A : Ring A → AnchLie A . (cid:3) Remark . Let ϕ : ( R, φ ) → ( S, ψ ) be a morphism of A -rings, as in the proof of Theorem2.4. Set A SR : AnchLie S → AnchLie R , A RA : AnchLie R → AnchLie A and A SA : AnchLie S → AnchLie A for the functors induced by ϕ , φ and ψ respectively. The universal property of thepullback gives a morphism of R -anchored Lie algebras ˜ ϕ : ( R, ̟ R ) → A SR ( S, ̟ S ) inducedby ϕ and a morphism of A -anchored Lie algebras χ : A RA (cid:16) A SR ( S, ̟ S ) (cid:17) → A SA ( S, ̟ S ) inducedby the composition A RA (cid:16) A SR ( S, ̟ S ) (cid:17) → A SR ( S, ̟ S ) → S . The interested reader may checkthat the morphism L A ( ϕ ) coincides with the composition χ ◦ A RA (cid:16) ˜ ϕ (cid:17) . However, we believethat the elementary proof we gave of Theorem 2.4 is more straightforward.Now assume that an A -anchored Lie algebra ( L, ω ) has been given. By the universalproperty of U ( L ) there exists a unique k -algebra extension Ω : U ( L ) → End k ( A ) of ω whichmakes of A a U ( L )-module algebra with X · a = ω ( X )( a ) , X · A = 0 and u · a = Ω( u )( a ) (11)for all X ∈ L , u ∈ U ( L ) and a ∈ A (see [3, Example 6.1.13(3)], for instance). As aconsequence, we may consider the A -ring A U ( L ) with underlying vector space A ⊗ U ( L ),unit 1 A ⊗ U , multiplication uniquely determined by( a ⊗ u )( b ⊗ v ) = X a ( u · b ) ⊗ u v (12)for all a, b ∈ A , u, v ∈ U ( L ), and A -ring structure j A : A → A U ( L ) , a a ⊗ U . Theorem 2.6.
The assignment ( L, ω ) A U ( L ) induces a functor U A : AnchLie A → Ring A which is left adjoint to L A : Ring A → AnchLie A .Proof. Let (
L, ω ) be an A -anchored Lie algebra and set U A ( L ) := A U ( L ). Consider theassignment j L : L → U A ( L ) , X A ⊗ X. In view of (11) and of (12), j L is a morphism ofLie algebras and h j L ( X ) , j A ( a ) i = h A ⊗ X, a ⊗ U i = ω ( X )( a ) ⊗ U = j A (cid:16) ω ( X )( a ) (cid:17) in U A ( L ), for all X ∈ L and a ∈ A . Therefore, j L and ω induce a k -linear morphism η L : L → L A (cid:16) U A ( L ) (cid:17) , X (cid:16) ι L ( X ) , ω ( X ) (cid:17) , (13)via the universal property of the pullback. It is easy to check that η L is of Lie algebras andthat it is compatible with the anchors, thus it is a morphism of A -anchored Lie algebras.We claim that η L is a universal map from L to L A in the sense of [11, III.1, Definition]. Assume then that (
R, φ ) is a A -ring and that f : L → L A ( R ) is a morphism of A -anchoredLie algebras. By definition of L A ( R ), f ( X ) = (cid:16) ˜ f ( X ) , ω ( X ) (cid:17) and h ˜ f ( X ) , φ ( a ) i = ̟ R (cid:16) ˜ f ( X ) (cid:17)(cid:16) φ ( a ) (cid:17) (8) = φ (cid:16) ω ( X )( a ) (cid:17) (14)for all X ∈ L and a ∈ A , where ˜ f := p ◦ f . As in the proof of [15, Theorem 2.9], astraightforward check using (14) and induction on a PBW basis of U ( L ) shows that F : A U ( L ) → R, a ⊗ u φ ( a ) U (cid:16) ˜ f (cid:17) ( u ) , is a morphism of A -rings which satisfies F (cid:16) ι L ( X ) (cid:17) = ˜ f ( X ) for all X ∈ L . Moreover, L A ( F ) (cid:16) η L ( X ) (cid:17) (13) = L A ( F ) (cid:16) ι L ( X ) , ω ( X ) (cid:17) (9) = (cid:18) F (cid:16) ι L ( X ) (cid:17) , ω ( X ) (cid:19) = (cid:16) ˜ f ( X ) , ω ( X ) (cid:17) = f ( X )for all X ∈ L and F is the unique A -ring map satisfying the latter relation. Therefore, U A , L A form an adjoint pair by [11, IV.1, Theorem 2(ii)]. (cid:3) Notice that Theorem 2.6 is expressing the fact that (cid:16) A U ( L ) , j A (cid:17) is the universalenveloping A -ring of ( A, L, ω ). In [9, page 175], the algebra A U ( L ) is called the algebraof differential operators of the representation ω of L (our construction differs slightly fromthe one in [9], because of the different choice of sides for the modules). Theorem 2.6provides then a conceptual explanation for the universal property of A U ( L ) describedin [9, V.6, Proposition 2] and a new proof of the latter. Remark . For the sake of future reference, if f : ( L, ω ) → ( L ′ , ω ′ ) is a morphism of A -anchored Lie algebras, then the induced morphism of A -rings U A ( f ) : U A ( L ) → U A ( L ′ )is explicitly given by U A ( f )( a ⊗ u ) = a ⊗ U ( f )( u ) for all a ∈ A and u ∈ U ( L ).As a particular case of Theorem 2.6, we have an adjunction Ring A e L Ae (cid:6) (cid:6) AnchLie A e . U Ae F F If we compose it with the adjunction (6), we obtain a new adjunction
Ring A e L A (cid:6) (cid:6) AnchLie A . U A F F (15)where U A := U A e ◦ E A and L A := F A ◦ L A e . Notice that U A ( L ) = A e U ( L ), where the U ( L )-module structure on A e is that of a tensor product of U ( L )-modules. Theorem 2.8.
There is an isomorphism of A e -rings A e U ( L ) → A ⊙ U ( L ) ⊙ A, ( a ⊗ b o ) ⊗ u a ⊗ u ⊗ b. (16) In particular, (15) exhibits the Connes-Moscovici’s bialgebroid construction of [15, §2] asa left adjoint functor.
EAS OF LRAS AS LEFT ADJOINT 11
Proof.
A straightforward computation by means of the cocommutativity of U ( L ) showsthat (16) is, in fact, a morphism of A e -rings, where the U ( L )-module structure on A e isgiven by the diagonal action u · ( a ⊗ b o ) = P ( u · a ) ⊗ ( u · b ) o for all u ∈ U ( L ), a, b ∈ A . (cid:3) The universal property of [15, Theorem 2.9] (see §1.3) expresses exactly the fact that forany morphism of A -anchored Lie algebras L → L A ( R ), there exists a unique morphism of A e -rings A ⊙ U ( L ) ⊙ A → R extending it, as the following proposition states. Proposition 2.9.
For any A e -ring ( R, φ A ) , the A -anchored Lie algebra L A ( R ) can berealized as the following pullback of k -vector spaces L A ( R ) y ρ / / ρ (cid:15) (cid:15) R ̟ R (cid:15) (cid:15) Der k ( R ) φ A ∗ (cid:15) (cid:15) Der k ( A ) e / / Der k ( A e ) φ A ∗ / / Der k ( A e , R ) with component-wise bracket and anchor ρ . Concretely, L A ( R ) = (cid:26) ( r, δ ) ∈ R × Der k ( A ) (cid:12)(cid:12)(cid:12) h r, φ A ( a ⊗ b o ) i = φ A (cid:16) δ ( a ) ⊗ b o + a ⊗ δ ( b ) o (cid:17) , ∀ a, b ∈ A (cid:27) . The datum of a morphism of A -anchored Lie algebras L → L A ( R ) is therefore equivalentto the datum of a morphism of Lie algebras φ L : L → R such that for all X ∈ L , a, b ∈ A , h φ L ( X ) , φ A ( a ⊗ b o ) i = φ A (cid:16) X · ( a ⊗ b o ) (cid:17) . Proof.
The first claim follows from the pasting law for pullbacks. The second claim is astraightforward check. (cid:3)
If we define an (
A, L, ω )-module to be an A -bimodule together with a Lie algebra mor-phism ρ : L → End k ( M ) such that ρ ( X )( a · m · b ) = ω ( X )( a ) · m · b + a · ρ ( X )( m ) · b + a · m · ω ( X )( b ) (17)as in [15, Corollary 2.10], then we have the following expected result. Proposition 2.10.
For an A -bimodule M , the datum of a left ( A, L, ω ) -module structureis equivalent to the datum of a morphism L → L A (cid:16) End k ( M ) (cid:17) of A -anchored Lie algebras.Proof. Recall that if M is an A -bimodule, then End k ( M ) has a natural A e -ring structureinduced by left and right multiplication by A : φ : A e → End k ( M ) , a ⊗ b o l a ◦ r b , where l a ( m ) := a · m and r a ( m ) := m · a for all a ∈ A , m ∈ M . By the universalproperty of the pullback, giving a morphism ̺ : L → L A (cid:16) End k ( M ) (cid:17) of A -anchored Liealgebras is equivalent to giving a morphism of Lie algebras ρ : L → End k ( M ) such that φ ∗ ◦ ̟ End k ( M ) ◦ ρ = φ ∗ ◦ e ◦ ω , which is exactly (17). (cid:3) As a consequence, Theorem 2.8 and Proposition 2.10 provide a conceptual proof of theequivalence between the category of (
A, L, ω )-modules and the category of U A ( L )-modulesalready observed in [15, Corollary 2.10].3. The universal enveloping A -ring as a left adjoint functor Henceforth, A is a commutative k -algebra. Notice that if a morphism of k -algebras φ : A → R has been given, then Der k ( A, R ) becomes a left A -module with A -action( a · f )( b ) := φ ( a ) f ( b ) for all a, b ∈ A , f ∈ Der k ( A, R ). Proposition 3.1.
Let ( R, φ ) be an A -ring. The k -vector space A RA ( R, ̟ R ) of (8) is a Lie-Rinehart algebra over A with anchor p : A RA ( R, ̟ R ) → Der k ( A ) and with component-wisebracket and left A -action. Furthermore, p is a left A -linear and Lie algebra morphism.Proof. We already know from Lemma 2.1 and Corollary 2.3 that A RA ( R, ̟ R ) is an A -anchored Lie algebra with component-wise bracket and anchor p and we know that p is a Lie algebra morphism. We only need to check the A -module properties. Since, in thiscase, (7) is also a diagram of left A -modules and left A -linear morphisms, A RA ( R, ̟ R ) is aleft A -module itself with component-wise A -action and p , p are left A -linear. Moreover (cid:20) ( r, δ ) , a · ( r ′ , δ ′ ) (cid:21) = (cid:20) ( r, δ ) , ( φ ( a ) r ′ , a · δ ′ ) (cid:21) = (cid:18) rφ ( a ) r ′ − φ ( a ) r ′ r, h δ, a · δ ′ i(cid:19) (8) = (cid:18) φ ( a )[ r, r ′ ] + φ (cid:16) δ ( a ) (cid:17) r ′ , a · h δ, δ ′ i + δ ( a ) · δ ′ (cid:19) = a · (cid:16) [ r, r ′ ] , [ δ, δ ′ ] (cid:17) + δ ( a ) · (cid:16) r ′ , δ ′ (cid:17) = a · h ( r, δ ) , ( r ′ , δ ′ ) i + p (cid:16) ( r, δ ) (cid:17) ( a ) · ( r ′ , δ ′ )for all a ∈ A , r, r ′ ∈ R , δ, δ ′ ∈ Der k ( A ), which entails that the Leibniz rule (1) is satisfiedand hence that A RA ( R, ̟ R ) is a Lie-Rinehart algebra over A . (cid:3) Theorem 3.2.
The assignment R
7→ A RA ( R, ̟ R ) induces a functor L A : Ring A → LieRin A .Proof. It follows from Theorem 2.4 and Proposition 3.1. (cid:3)
Concretely, L A ( R ) = (cid:26) ( r, δ ) ∈ R × Der k ( A ) (cid:12)(cid:12)(cid:12) h r, φ A ( a ) i = φ A (cid:16) δ ( a ) (cid:17) for all a ∈ A (cid:27) . The following proposition, analogue of Proposition 2.9, argues in favour of the fact that L A provides a right adjoint for the functor U A : LieRin A → Ring A of §1.2. Proposition 3.3.
Let ( R, φ A ) be an A -ring and let ( L, ω ) be a Lie-Rinehart algebra over A . Then the datum of a morphism ψ L : L → L A ( R ) of Lie-Rinehart algebras over A isequivalent to the datum of a morphism of Lie algebras φ L : L → R such that (3) hold.Proof. By the universal property of the pullback, the existence of a morphism of A -modulesand of Lie algebras ψ L : L → L A ( R ) such that p ◦ ψ L = ω is equivalent to the existence ofa morphism of A -modules and of Lie algebras φ L : L → R such that φ A ∗ ◦ ̟ R ◦ φ L = φ A ∗ ◦ ω ,which is exactly (3). (cid:3) EAS OF LRAS AS LEFT ADJOINT 13
Theorem 3.4.
The functor L A : Ring A → LieRin A is right adjoint to the universal envelop-ing algebra functor U A : LieRin A → Ring A .Proof. Let (
A, L, ω ) be a Lie-Rinehart algebra and consider the assignment η L : L → L A (cid:16) U A ( L ) (cid:17) , X (cid:16) ι L ( X ) , ω ( X ) (cid:17) , (18)induced by ι L and ω via the universal property of the pullback in view of (2). It is easyto check that η L is a morphism of Lie-Rinehart algebras over A . Moreover, if f : ( L, ω ) → ( L ′ , ω ′ ) is a morphism of Lie-Rinehart algebras over A and if F : L A (cid:16) U A ( L ) (cid:17) → L A (cid:16) U A ( L ′ ) (cid:17) denotes the morphism induced by U A ( f ), then the fact that F (cid:16) η L ( X ) (cid:17) (18) = F (cid:16) ι L ( X ) , ω ( X ) (cid:17) (10) = (cid:18) U A ( f ) (cid:16) ι L ( X ) (cid:17) , ω ( X ) (cid:19) = (cid:16) ι L ′ (cid:16) f ( X ) (cid:17) , ω ′ (cid:16) f ( X ) (cid:17)(cid:17) for all X ∈ L entails that F ◦ η L = η L ′ ◦ f and so the collection { η L | L ∈ LieRin A } definesa natural transformation η : Id → L A ◦ U A .In view of Proposition 3.3 and the universal property of U A ( L ), for any A -ring ( R, φ A )and any morphism ψ L : L → L A ( R ) of Lie-Rinehart algebras over A , there exists a uniquemorphism of A -rings Φ : U A ( L ) → R such that Φ ◦ ι L = p ◦ ψ L . By a direct check L A (Φ) (cid:16) η L ( X ) (cid:17) (18) = L A (Φ) (cid:16) ι L ( X ) , ω ( X ) (cid:17) (10) = (cid:18) Φ (cid:16) ι L ( X ) (cid:17) , ω ( X ) (cid:19) = ψ L ( X )for all X ∈ L and Φ is the unique morphism of A -rings satisfying L A (Φ) ◦ η L = ψ L . Itfollows that η L is a universal map from L to L A in the sense of [11, III.1, Definition],for every ( L, ω ) ∈ LieRin A , and hence U A , L A form an adjoint pair by [11, IV.1, Theorem2(i)]. (cid:3) On morphisms, modules and the infinitesimal gauge algebra
We conclude with a few remarks concerning morphisms between Lie-Rinehart algebrasover different bases, modules over Lie-Rinehart algebras and the infinitesimal gauge algebraDO(
A, L, M ) of an A -module M with respect to ( A, L, ω ) described in [8, page 72].4.1.
Morphisms over different bases.
By mimicking the arguments used to proveLemma 2.1 and Proposition 3.1, one shows that the following result holds.
Proposition 4.1. If ( L ′ , ω ′ ) is a Lie-Rinehart algebra over a commutative k -algebra A ′ and φ : A → A ′ is a morphism of commutative k -algebras, then A A ′ A ( L ′ , ω ′ ) is a Lie-Rinehartalgebra over A . Proposition 4.1 suggests the following definition.
Definition 4.2.
A morphism of Lie-Rinehart algebras from (
A, L, ω ) to ( A ′ , L ′ , ω ′ ) is a pair( φ, Φ) where φ : A → A ′ is a morphism of commutative k -algebras and Φ : L → A A ′ A ( L ′ , ω ′ )is a morphism of Lie-Rinehart algebras over A . Remark . If A = A ′ and φ = Id , then L ′ itself with p = Id and p = ω ′ is a pullbackof Der k ( A ) = Der k ( A ) ω ′ ←− L ′ . Therefore, a morphism ( Id , Φ) of Lie-Rinehart algebras from(
A, L, ω ) to (
A, L ′ , ω ′ ) is the same as a morphism of Lie-Rinehart algebras over A as in§1.2.Recall that in [8, page 61] a morphism of Lie-Rinehart algebras from ( A, L, ω ) to( A ′ , L ′ , ω ′ ) is defined as a pair ( φ, ψ ) where φ : A → A ′ is a morphism of k -algebrasand ψ : L → L ′ is a morphism of Lie algebras and of left A -modules such that for all a ∈ A , X ∈ L , φ (cid:16) ω ( X )( a ) (cid:17) = ω ′ (cid:16) ψ ( X ) (cid:17)(cid:16) φ ( a ) (cid:17) . (19) Proposition 4.4.
The datum of a morphism of Lie-Rinehart algebras as in Definition 4.2is equivalent to the datum of a morphism in the sense of [8, page 61] .Proof.
By the universal property of the pullback, giving a morphism Ψ : L → A A ′ A ( L ′ , ω ′ )of Lie-Rinehart algebras over A is equivalent to giving a morphism of Lie algebras and ofleft A -modules ψ : L → L ′ such that φ ∗ ◦ ω ′ ◦ ψ = φ ∗ ◦ ω , which is exactly (19). (cid:3) Modules over Lie-Rinehart algebras.
Recall that an (
A, L, ω )-module in the senseof [8, page 62] is a left A -module M together with a morphism of Lie algebras ρ : L → End k ( M ) such that ρ ( a · X )( m ) = a · ρ ( X )( m ) and ρ ( X )( a · m ) = a · ρ ( X )( m ) + ω ( X )( a ) · m for all X ∈ L , m ∈ M , a ∈ A . Proposition 4.5.
For a left A -module M , the datum of a left ( A, L, ω ) -module structure asin [8, page 62] is equivalent to the datum of a morphism L → L A (cid:16) End k ( M ) (cid:17) of Lie-Rinehartalgebras over A .Proof. Completely analogous to the proof of Proposition 2.10 and Proposition 4.4. (cid:3)
As a consequence, Theorem 3.4 and Proposition 4.5 provide a conceptual proof of thewell-known equivalence between the category of (
A, L, ω )-modules and the category of U A ( L )-modules (see [8, page 65]).4.3. The infinitesimal gauge algebra of a module.
Let (
L, ω ) be a Lie-Rinehart alge-bra over A and let M be an A -module. In [8, page 72], a Lie-Rinehart algebra DO( A, L, M )is introduced, which acts on M by the analogue of infinitesimal gauge transformations. Weshow how DO( A, L, M ) can naturally be obtained via the constructions we performed and,as a consequence, how it naturally inherits a universal property as well.The following is the analogue of Proposition 1.5 for Lie-Rinehart algebras.
Proposition 4.6.
Let ( L, ω ) and ( L ′ , ω ′ ) be Lie-rinehart algebras over A . The product of ( L, ω ) and ( L ′ , ω ′ ) in LieRin A exists and can be computed as the pullback (5) with component-wise bracket and A -action and with anchor ω × := ω ◦ q = ω ′ ◦ q . EAS OF LRAS AS LEFT ADJOINT 15
Proof.
We already know that, with the structures of the statement, L × Der k ( A ) L ′ is an A -anchored Lie algebra and it is clearly a left A -module via the component-wise A -action.The anchor ω × , being the composition of A -linear maps, is A -linear. We are left to checkthe Leibniz condition (1). Since for every a ∈ A and ( X, Y ) ∈ L × Der k ( A ) L ′ we have ω ( X )( a ) = ω ′ ( Y )( a ) = ω × ( X, Y )( a ), the following direct computation concludes the proof: h ( X, Y ) , a · ( X ′ , Y ′ ) i = (cid:16) [ X, a · X ′ ] , [ Y, a · Y ′ ] (cid:17) (1) = (cid:16) a · [ X, X ′ ] + ω ( X )( a ) · X ′ , a · [ Y, Y ′ ] + ω ′ ( Y )( a ) · Y ′ (cid:17) = a · h ( X, Y ) , ( X ′ , Y ′ ) i + ω × ( X, Y )( a ) · ( X ′ , Y ′ ) . (cid:3) Let (
A, L, ω ) be a Lie-Rinehart algebra. Recall from [8, page 72] that for a given A -module M , the Lie-Rinehart algebra DO( A, L, M ) of infinitesimal gauge transformationsof M with respect to L is the subspace of End k ( M ) × L composed by the elements ( f, X )such that f ( a · m ) = ω ( X )( a ) · m + a · f ( m )for all a ∈ A , m ∈ M . The bracket and the A -action are given component-wise, while theanchor ˜ ω is induced by the restriction of the projection on the second factor. Proposition 4.7.
Let M be an A -module and let ( A, L, ω ) be a Lie-Rinehart algebra. TheLie-Rinehart algebra (cid:16) A, DO(
A, L, M ) , ˜ ω (cid:17) is the product in LieRin A of the Lie-Rinehartalgebras ( A, L, ω ) and L A (cid:16) End k ( M ) (cid:17) .Proof. Set E := End k ( M ) and φ : A → E, a l a . By definition of L A ( E ) and by the con-struction of the product in LieRin A , ( A, L, ω ) × L A ( E ) is the following pasting of pullbacks: L A ( E ) × Der k ( A ) L y q / / q (cid:15) (cid:15) L A ( E ) y p (cid:15) (cid:15) p / / E ̟ E (cid:15) (cid:15) Der k ( E ) φ ∗ (cid:15) (cid:15) L ω / / Der k ( A ) φ ∗ / / Der k ( A, E ) . Concretely, L A ( E ) × Der k ( A ) L = n ( f, X ) ∈ E × L | [ f, l a ] = l ω ( X )( a ) for all a ∈ A o withcomponent-wise bracket and A -action and with anchor given by p ◦ q = ω ◦ q = ˜ ω . (cid:3) It follows from Proposition 4.5 that M is an (cid:16) A, DO(
A, L, M ) , ˜ ω (cid:17) -module via q . Namely, ̺ := ( p ◦ q ) : DO( A, L, M ) → End k ( M ) satisfies the conditions of §4.2. Furthermore,DO( A, L, M ) admits a canonical Lie-Rinehart algebra morphism q : DO( A, L, M ) → L and, in fact, (cid:16) A, DO(
A, L, M ) , ˜ ω (cid:17) is universal with respect to these properties. Theorem 4.8.
Let M be an A -module and let ( A, L, ω ) be a Lie-Rinehart algebra. Forevery Lie-Rinehart algebra ( A, L ′ , ω ′ ) acting on M via ρ : L ′ → End k ( M ) and any morphism of Lie-Rinehart algebras f : L ′ → L , there exists a unique morphism ˜ f : L ′ → DO(
A, L, M ) of Lie-Rinehart algebras over A such that ̺ ◦ ˜ f = ρ and q ◦ ˜ f = f .Proof. It follows from Proposition 4.5 and Proposition 4.7. (cid:3)
Example 4.9.
Among all infinitesimal gauge algebras DO(
A, L, M ) associated with an A -module M , there exists a universal one, which is the Atiyah algebra A M of M (see e.g. [10,(1.1.3) Examples (c)]). This is the Lie-Rinehart algebra L A ( End k ( M )) of infinitesimal gaugetransformations of M with respect to Der k ( A ). Concretely, A M = n ( f, δ ) ∈ End k ( M ) × Der k ( A ) (cid:12)(cid:12)(cid:12) f ( a · m ) = a · f ( m ) + δ ( a ) · m o . By Theorem 4.8, if (
A, L, ω ) is a Lie-Rinehart algebra acting on M via ρ : L → End k ( M ),then there exists a unique morphism τ L : L → A M of Lie-Rinehart algebras over A suchthat p ◦ τ L = ρ . In particular, the datum of a left ( A, L, ω )-module structure on M isequivalent to the datum of a morphism L → A M of Lie-Rinehart algebras over A (see [10,(1.1.4) Definition] and Proposition 4.5). References [1] A. Ardizzoni, L. El Kaoutit, P. Saracco,
Differentiation and integration between Hopf algebroids andLie algebroids . Preprint (2019). (arXiv: 1905.10288)[2] N. Bourbaki, ´El´ements de math´ematique. Groupes et alg`ebres de Lie. Chapitre 1. Reprint of the 1972original . Berlin: Springer. 148 p. (2007).[3] S. D˘asc˘alescu, C. N˘ast˘asescu, S¸. Raianu,
Hopf algebras. An introduction . Monographs and Textbooksin Pure and Applied Mathematics, . Marcel Dekker, Inc., New York, 2001.[4] L. El Kaoutit, P. Saracco,
The Hopf Algebroid Structure of Differentially Recursive Sequences . Preprint(2020). (arXiv: 2003.08180)[5] L. El Kaoutit, P. Saracco,
Topological tensor product of bimodules, complete Hopf algebroids andconvolution algebras . Commun. Contemp. Math. (2019), no. 6, 1850015, 53 pp.[6] J. Huebschmann, Lie-Rinehart algebras, descent, and quantization . Galois theory, Hopf algebras, andsemiabelian categories, 295–316, Fields Inst. Commun., , Amer. Math. Soc., Providence, RI, 2004.[7] J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras . Ann.Inst. Fourier (Grenoble) (1998), no. 2, 425–440.[8] J. Huebschmann, Poisson cohomology and quantization . J. Reine Angew. Math. (1990), 57–113.[9] N. Jacobson,
Lie algebras . Interscience Tracts in Pure and Applied Mathematics, No. IntersciencePublishers (a division of John Wiley & Sons), New York-London 1962.[10] M. Kapranov,
Free Lie algebroids and the space of paths . Selecta Math. (N.S.) (2007), no. 2,277–319.[11] S. Mac Lane, Categories for the Working Mathematician. Second edition . Graduate Texts in Mathe-matics, . Springer-Verlag, New York, 1998.[12] M.-P. Malliavin, Alg`ebre homologique et op´erateurs diff´erentiels . Ring theory (Granada, 1986), 173–186, Lecture Notes in Math., , Springer, Berlin, 1988.[13] I. Moerdijk, J. Mrˇcun,
On the universal enveloping algebra of a Lie algebroid . Proc. Amer. Math. Soc. (2010), no. 9, 3135–3145.[14] G. S. Rinehart,
Differential forms on general commutative algebras . Trans. Amer. Math. Soc. ,1963, 195–222.[15] P. Saracco,
On anchored Lie algebras and the Connes-Moscovici’s bialgebroid construction . Preprint(2020). (arXiv: 2009.14656)
EAS OF LRAS AS LEFT ADJOINT 17 [16] M. E. Sweedler,
Groups of simple algebras . Inst. Hautes ´Etudes Sci. Publ. Math. No. (1974),79–189. D´epartement de Math´ematique, Universit´e Libre de Bruxelles, Boulevard du Triomphe,B-1050 Brussels, Belgium.
URL : sites.google.com/view/paolo-saracco URL : homepages.ulb.ac.be/˜psaracco Email address ::