On anchored Lie algebras and the Connes-Moscovici's bialgebroid construction

We show how the Connes-Moscovici's bialgebroid construction naturally provides universal objects for Lie algebras acting on non-commutative algebras.


Introduction
Given a Hopf algebra H (possibly with bijective antipode S) and a left H-module algebra A, one can turn the vector space A ⊗ H ⊗ A into a left bialgebroid H := A ⊙ H ⊙ A over A in a natural way. This procedure has been introduced independently, and under different forms, by Connes and Moscovici [7] in their study of the index theory of transversely elliptic operators, and by Kadison [19] in connection with his work on (pseudo-)Galois extensions. Later, Panaite and Van Oystaeyen proved in [31] that the two constructions were in fact equivalent (isomorphic as A-bialgebroids) and that, as algebras, they were particular instances of the L-R-smash product introduced in [30]. Nevertheless, by following [4], we will refer to the bialgebroid A ⊙ H ⊙ A as the Connes-Moscovici's bialgebroid.
Following the foregoing and, in particular, in view of the results in [31], two observations were made, that triggered the present investigation: (i)that whenever a Lie algebra L acts by derivations on an associative algebra A (for the sake of simplicity, let us call it an A-anchored Lie algebra), then A becomes naturally an U k (L)-module algebra and (ii)that the associated Connes-Moscovici's bialgebroid construction satisfies a universal property (both as A e -ring and as A-bialgebroid, see [31, Proposition 3.1 and Theorem 3.2]) which suggests the possibility that A ⊙ U k (L) ⊙ A plays for an A-anchored Lie algebra L the same role played by the universal enveloping algebra for a Lie algebra.
Among anchored Lie algebras we find the well-known Lie-Rinehart algebras, which are in particular Lie algebras acting on commutative algebras. As it can be inferred from the substantial literature on the topic, Lie-Rinehart algebras are a deeply investigated area, in particular for its connections with differential geometry (the global sections of a Lie algebroid L → M form a Lie-Rinehart algebra over C ∞ (M)). Rinehart himself gave an explicit construction of the universal enveloping algebra U(R, L) of a Lie-Rinehart algebra in [35] and proved a Poincaré-Birkhoff-Witt theorem for the latter. Other equivalent constructions are provided in [11, §3.2], [17, page 64], [37, §18]. The universal property of U(R, L) as an algebra is spelled out in [17, page 64] and [24, page 174] (where it is attributed to Feld'man). Its universal property as an A-bialgebroid is codified in the Cartier-Milnor-Moore Theorem for U(R, L) proved in [27, §3], where Moerdijk and Mrčun show that the construction of the universal enveloping algebra provides a left adjoint to the functor associating any cocommutative bialgebroid with its Lie-Rinehart algebra of primitive elements and they find natural conditions under which this adjunction becomes an equivalence (as it has been done in [26] for cocommutative bialgebras and Lie algebras). Further algebraic and categorical properties and applications are investigated in [2,10,15,16,17].
However, there are many important examples of Lie algebras acting by derivations on associative algebras which are not necessarily commutative (actually, any Lie algebra acts by derivations on its universal enveloping algebra and any associative algebra acts by inner derivations on itself). Furthermore, while the space of primitive elements of a bialgebra is always a Lie algebra and a primitively generated bialgebra is always cocommutative, the space of primitive elements of a bialgebroid is not, in general, a Lie-Rinehart algebra and not every primitively generated bialgebroid is necessarily cocommutative. A third observation that stood up for the present investigation is that, instead, the space of primitive elements of a bialgebroid is always a Lie algebra acting by derivations on the base algebra.
These facts, together with the two foregoing observations (i) and (ii), called for the study of Lie algebras L acting on non-commutative algebras A in their own right and, in particular, for the study of the associated Connes-Moscovici's bialgebroid A ⊙ U k (L) ⊙ A, as it has been done for Lie-Rinehart algebras and their universal enveloping algebras.
In the present paper, we are mainly concerned with two universal properties of A ⊙ U k (L)⊙A, as an A e -ring and as an A-bialgebroid, which reflect the two well-known universal properties of universal enveloping algebras reported above. The first one (Theorem 2.9) exhibits A ⊙ U k (L) ⊙ A as the universal A e -ring associated with the A-anchored Lie algebra L, similarly to what happens for U(R, L) in [17, page 64]. Namely, for any A e -ring φ A : A e → R and any k-Lie algebra morphism φ L : L → L(R) such that for all a, b ∈ A and all X ∈ L, there exists a unique morphism of A e -rings Φ : B L → R extending φ L . This naturally affects the study of the representations of L (see Corollary 2.10). The second universal property (Proposition 3.4) exhibits A ⊙ U k (L) ⊙ A as the universal A-bialgebroid associated with the A-anchored Lie algebra L, similarly to what happens for U(R, L) in [27,Theorem 3.1(i)]. Namely, for any A-bialgebroid B and any morphism of k-Lie algebras φ L : L → B which lands into the space Prim(B) of primitive elements of B and that is compatible with the anchors, there exists a unique morphism of A-bialgebroids Φ : B L → B that extends φ L .
Concretely, after a first section devoted to recall some definitions and some preliminary results, we introduce A-anchored Lie algebras in §2. 1 and we prove that the Connes-Moscovici's bialgebroid associated to an A-anchored Lie algebra satisfies the stated universal property as A e -ring in §2.2 (Theorem 2.9). In §3.1, we detail how taking the space of primitives of an A-bialgebroid induces a functor from the category of A-bialgebroids to the category of A-anchored Lie algebras and in §3.2 we show that the Connes-Moscovici's construction provides a natural left adjoint to this latter functor (Theorem 3.6) and, at the same time, we prove the second universal property of A ⊙ U k (L) ⊙ A (Proposition 3.4). At this point, by finding inspiration from the Milnor-Moore and the Moerdijk-Mrčun theorems, we look for intrinsic conditions on a bialgebroid that allow us to recognize it as a A ⊙ U k (L) ⊙ A for a certain A-anchored Lie algebra. Section 4 is devoted to find a first answer (Theorem 4.14). After studying in more detail the space of primitives of a Connes-Moscovici's bialgebroid in §4.1, we tackle the question in the general framework in §4.2 and in the particular case of bialgebroids over a commutative base in §4.3. Finally, we conclude with some final remarks about future lines of investigation in §4.4.
Notation. All over the paper, we assume a certain familiarity of the reader with the language of monoidal categories and of (co)monoids therein (see, for example, [22,VII]).
We work over a ground field k of characteristic 0. All vector spaces are assumed to be over k. The unadorned tensor product ⊗ stands for ⊗ k . All (co)algebras and bialgebras are intended to be k-(co)algebras and k-bialgebras, that is to say, (co)algebras and bialgebras in the symmetric monoidal category of vector spaces (Vect k , ⊗, k). Every (co)module has an underlying vector space structure. Identity morphisms Id V are often denoted simply by V .
In order to avoid confusion between indexes of elements and coproducts or coactions, we will adopt the following variant of the Heyneman-Sweedler's Sigma Notation. For c in a coalgebra C, m in a left C-comodule M and n in a right C-comodule N we write (1) .
Given an algebra A, we denote by A o its opposite algebra. We freely use the canonical isomorphism between the category of left A-module A Mod and that of right A o -modules Mod A o . We also set A e := A ⊗ A o and we identify the category of left A e -modules A e Mod with that of A-bimodules A Mod A . Recall that any morphism of algebras η : A e → R leads to two commuting algebra maps s : for all a, b ∈ A) and conversely. Given two R-bimodules M and N, this gives rise to several A-module structures on M and N and it leads to several ways of considering the tensor product over A between the underlying A-bimodules. In the present paper we focus on the A-bimodule structure induced by A e acting on the left via η and we denote by s M t o . We usually consider this bimodule structure when taking tensor products. If we want to stress the fact that M is considered as a left A e -module, we may also write η M. Therefore, given two R-bimodules M and N, we consider the tensor product A-bimodule Inside M ⊗ A N, we will also consider the distinguished subspace which is often called Takeuchi For the sake of clarity, it will be useful to set for m ∈ η M η and a, b ∈ A.

Preliminaries
We begin by collecting some facts about bimodules, corings and bialgebroids that will be needed in the sequel. The aim is that of keeping the exposition self-contained. Many results and definitions we will present herein hold in a more general context and under less restrictive hypotheses, but we preferred to limit ourselves to the essentials.
Given a (preferably, non-commutative) k-algebra A, the category of A-bimodules forms a non-strict monoidal category ( A Mod A , ⊗ A , A, a, l, r). Nevertheless, all over the paper we will behave as if the structural natural isomorphisms were "the identities", that is, as if A Mod A was a strict monoidal category.
1.1. Graded and filtered A-bimodules. As far as we are concerned, we assume A to be filtered over Z with filtration F n (A) = 0 for all n < 0 and F n (A) = A for all n ≥ 0 and we assume it to be graded over Z with graduation A 0 = A and A n = 0 for all n = 0.
By a graded A-bimodule we mean an A-bimodule M with a family of A-subbimodules In what follows we will be interested in positively filtered and graded bimodules, that is, those for which the negative terms are 0. Given two filtered bimodules M, N, we can perform their tensor product M ⊗ A N and this is still a filtered bimodule: the k-th term of the filtration on M ⊗ A N is the A-subbimodule generated by the elements m ⊗ A n such that m ∈ F s (M), n ∈ F t (N) and s + t = k. Analogously for two graded A-bimodules. With these tensor products, we have that the categories A FMod A , ⊗ A , A and A GMod A , ⊗ A , A of filtered and graded bimodules, respectively, are monoidal categories (it follows, for instance, from [28, Chapter A, Proposition I.2.14 and Chapter D, Lemma VIII.1]). Morphisms of filtered (respectively, graded) bimodules are A-bilinear maps that respect the filtration (respectively, graduation).
The result we are principally interested in is that the construction of the graded associated to a filtered bimodule is functorial (see, for example, [28,chapter D,§III]). Moreover, the natural surjection uniquely determined by for m ∈ F s (M) and n ∈ F t (N) (see [28, page 318]), and the isomorphism ϕ 0 : A ∼ = gr(A) endow the functor with a structure of lax monoidal functor (see [1,Definition 3.1]). For further details about filtered and graded bimodules, we refer the reader to [28].

A-corings.
Recall that an A-coring is a monoid in the monoidal category of Abimodules ( A Mod A , ⊗ A , A). More concretely, an A-coring is an A-bimodule C endowed with a comultiplication ∆ C : C → C ⊗ A C and a counit ε C : C → A such that For the general theory of corings and their comodules, we refer to [6]. Later on, we will be particularly interested in (exhaustively) filtered A-corings such that the associated graded components are projective as A-bimodules. These are A-corings C endowed with an increasing filtration {F n (C) | n ∈ N} as A-bimodules such that C = n F n (C), gr n (C) = F n (C)/F n−1 (C) is a projective A-bimodule and for all n ≥ 0 (that is to say, ∆ C is a morphism of filtered A-bimodules). We will refer to these A-corings as graded projective filtered A-corings. By convention, we put F −1 (C) = 0. Notice that the inclusion (7) makes sense in view of the following well-known result for filtered (bi)modules (see, for instance, [11, Lemma B.1]).
for all n ≥ 0 and so F n (C) is a projective A-bimodule. Moreover, the canonical map If, in addition, the filtration {F n (C) | n ∈ N} is exhaustive, then C ∼ = gr(C) as A-bimodules and, in particular, C is a projective A-bimodule.
Proof. By definition and projectivity of gr n (C), we have a split short exact sequence of A-bimodules, which implies that, as A-bimodules, By proceeding recursively, one reaches (8). The second claim follows from [23, Theorem C.24, p. 93]. About the last claim in the statement, saying that the filtration is exhaustive means that C ∼ = lim − →n (F n (C)) as A-bimodules. Since (8) means that F n (C) ∼ = F n (gr(C)) as A-bimodules, we have that C ∼ = lim − →n (F n (C)) ∼ = lim − →n (F n (gr(C))) ∼ = gr(C) as claimed.
Analogously to the theory of filtered coalgebras (see, for example, [38, §11.1]), the graded A-bimodule gr(C) associated to a graded projective filtered A-coring C becomes a graded A-coring in a natural way, as the subsequent Proposition 1.2 formalize.
For the sake of clarity, by a graded A-coring we mean an A-coring D endowed with a graduation {D n | n ∈ N} as A-bimodule such that every D n is projective as A-bimodule, for all n ∈ N and ε D (D n ) = 0 for all n ≥ 1.
It can be seen as a comonoid in the monoidal category of graded A-bimodules. Notice that ∆ D is uniquely determined by the A-bilinear maps obtained by (co)restriction of ∆ D to the graded components of D and D ⊗ A D and which, in turn, are uniquely determined by the A-bilinear maps Proof. The first claim follows from the functoriality of gr(−) and from the fact that ∆ C : C → C ⊗ A C and ε C : C → A are filtered morphisms of A-bimodules. In fact, they induce graded morphisms of A-bimodules and gr(C) which provides an A-coring structure on gr(C), since gr(−) is lax monoidal. Concerning the second claim, it is enough to apply gr(−) to the diagrams expressing the comultiplicativity and counitality of f .
commutes for all n ≥ 0.   where the algebra structure on B × A B is given by the component-wise product commutes. In this paper we will focus on left bialgebroids over a fixed base algebra A, that we call left A-bialgebroids. A morphism of left A-bialgebroids between B and B ′ is then an algebra map φ : The category of left A-bialgebroids will be denoted by Bialgd A .
We will often omit to specify the A-bialgebroid B in writing the comultiplication ∆ B or the counit ε B , when it is clear from the context. Remark 1.6. Let us make explicit some of the relations involved in the definition of a left bialgebroid and some of their consequences. In terms of elements of A and B, and by resorting to Sweedler Sigma Notation, relations (6) become In particular, for all a ∈ A. As a consequence, the multiplicativity of ∆ forces and since it also preserves the unit, we have that Therefore, in light of the character condition on ε, Henceforth, all bialgebroids will be left ones, whence we will omit to specify it.
In view of this, one can endow B with a structure of A-bialgebroid with source s, target t, ∆ given by (up to the latter isomorphism) and ε by evaluation at 1 A (see [21, page 56]). (d) Let R → S be a depth two ring extension (see [20,Definition 3.1]) and set A := C S (R), the centralizer of R in S. Then the ring of endomorphisms B := End R (S) of S as an R-bimodule satisfies as above (see [20,Proposition 3.11]) and we may endow it with an A-bialgebroid structure exactly as in (c) ([20, Theorem 4.1]). (e) Let (H, m, u, ∆, ε) be a bialgebra and let A be a braided commutative algebra in H H YD. This means that A is at the same time a left H-module algebra (that is, an algebra in the monoidal category Furthermore, these structures are required to satisfy and (a (−1) · b) a (0) = ab for all a, b ∈ A and h ∈ H (the latter expresses the braided commutativity). Under these conditions, the smash product algebra H#A is an Abialgebroid with and unit 1 A ⊗ 1 H ⊗ 1 A . It can be endowed with an A-bialgebroid structure as follows Following [31], we will denote this bialgebroid by A⊙H ⊙A, shunning the use of symbols like # or ⋉, ⋊ in order to avoid confusion with two-sided smash/crossed products in the sense of [12]. Notice that for all a, It is, in fact, a left B-module algebra (see [4, §3.7.1] for a right-handed analogue). In particular, for all a, b ∈ A and all ξ ∈ B we have

The Connes-Moscovici's bialgebroid as universal A e -ring
In this section we introduce A-anchored Lie algebras and we show that the Connes-Moscovici's bialgebroid A ⊙ U k (L) ⊙ A naturally associated to an A-anchored Lie algebra satisfies a universal property as A e -ring. In particular, unless stated otherwise, we assume to work over a fixed base algebra A, possibly non-commutative. We conclude the section with an extension of the PBW theorem to bialgebroids of the form A ⊙ U k (L) ⊙ A.

A-anchored Lie algebras.
Definition 2.1. We call A-anchored Lie algebra an ordinary Lie algebra L over k together with a Lie algebra morphism ω : L → Der k (A), called the anchor. We will often write X · a for ω(X)(a). A morphism of A-anchored Lie algebras between (L, ω) and (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 will be denoted by AnchLie A .
Remark 2.2. The reader needs to be warned that the terminology "A-anchored" used here is inspired from the literature, but it neither strictly coincides with the classical notion of A-anchored module, nor it properly extends it. In fact, in the literature, an "A-anchored module" [33, §1] (also called "A-module with arrow" [32, §3] or "A-module fléché" [34, §1]) is an A-module M over a commutative algebra A together with an A-linear map M → Der k (A). Since, in the present framework, A is assumed to be preferably noncommutative, the vector space Der k (A) does not carry any natural A-module structure and hence we do not have any reasonable way to speak about an A-linear anchor. In spite of this, in order to limit the proliferation of different terminology in the field and trusting that the non-commutative context will help in distinguishing between the two notions, we decided to adopt the term "A-anchored" in this framework as well.
(d) A Lie-Rinehart algebra over a commutative algebra R (called in this way in honour of G. S. Rinehart, who studied them in [35] under the name of (K, R)-Lie algebras) is a Lie algebra L endowed with a (left) R-module structure R ⊗ L → L, r ⊗ X → r · X, and with a Lie algebra morphism ω : L → Der k (R) such that, for all r ∈ R and X, Y ∈ L, Clearly, any Lie-Rinehart algebra over R is an R-anchored Lie algebra. (e) Let B be an A-bialgebroid and consider the vector space of primitive elements This is a Lie algebra with the commutator bracket. Assume that X ∈ Prim (B). In light of Equation (20), X acts on A by derivations, which means that the assignment is well-defined. Moreover, implies that ω B is a morphism of Lie algebras and hence it is an anchor for Prim (B). As a matter of notation, we write θ B : Prim(B) → B for the canonical inclusion.
with anchor ω ′′ given by the restriction of ω.
For the sake of brevity, from now on we will only speak about ideals and subalgebras without reporting the syntagma "A-anchored Lie" in front. Definition 2.4(L3) is consistent in view of the following results.
Proof. The fact that the semi-direct product is a Lie algebra follows from the fact that, as Lie algebras, it is the semi-direct product of L ′ and L ′′ . Thus we only need to check that ω δ is a Lie algebra morphism. To this aim, we compute directly The following lemma should not be surprising. Lemma 2.6. Let (L, ω) be an A-anchored Lie algebra and let (L ′ , ω ′ ) and (L ′′ , ω ′′ ) be subalgebras of (L, ω). Then there exists a semi-direct product If this is the case, then δ : Proof. In one direction, notice that (L ′ , ω ′ ) is an ideal in (L ′′ ⋉ δ L ′ , ω δ ) via the canonical morphism L ′ → L ′′ ⊕ L ′ , X ′ → (0, X ′ ), and that (L ′′ , ω ′′ ) is a subalgebra of (L ′′ ⋉ δ L ′ , ω δ ) via the canonical morphism L ′′ → L ′′ ⊕ L ′ , X ′′ → (X ′′ , 0). Furthermore, one recovers δ as for all X ′ ∈ L ′ , X ′′ ∈ L ′′ . In the other direction, assume that (L ′ , ω ′ ) is an ideal in (L, ω) and that L = L ′′ ⊕L ′ as k-vector spaces. Consider further the assignment δ : for all X ′ ∈ L ′ , X ′′ ∈ L ′′ , which is (21), and so we may perform the semi-direct product (L ′′ ⋉ δ L ′ , ω δ ). Then so that L ′′ ⊕ L ′ ∼ = L is a morphism of Lie algebras and moreover whence it is of A-anchored Lie algebras, too.
Remark 2.7. The reader has to be warned that, despite the definition of ideal and of semi-direct product in Definition 2.4 has been inspired by the subsequent Lemma 4.1 and Proposition 4.3 and by the results in §4.3, they may turn out to be improper terminologies in the future. In fact, it is not true in general that the quotient of an A-anchored Lie algebra by an ideal is an A-anchored Lie algebra (unless the ideal has the zero anchor) or that (L, ω) is a semi-direct product of (L ′ , ω ′ ) and (L ′′ , ω ′′ ) if and only if there is a short exact sequence of A-anchored Lie algebras such that g admits a section σ which is a morphism of A-anchored Lie algebras. On the one hand, the canonical projection L ′′ ⋉ δ L ′ → L ′′ is not a morphism of A-anchored Lie algebra because it is not compatible with the anchors. On the other hand, in order to have that ω ′′ • g = ω and that ω • f = ω ′ , we should have had that which is not the case in general. What one may observe is that (L, ω) is a semi-direct product of (L ′ , ω ′ ) and (L ′′ , ω ′′ ) if and only if there is a short exact sequence of Lie algebras (24) such that g admits a section σ and both f and σ are morphisms of A-anchored Lie algebras (but g is not, in general).

2.2.
A universal A e -ring construction. Assume that we are given an A-anchored Lie algebra (L, ω). Recall that we may consider the universal enveloping algebra U k (L) of L and that there is a canonical injective k-linear map which allows us to identify X with its image x in U k (L). The anchor ω makes of A a left representation of L with L acting as derivations, that is, we have a Lie algebra morphism where L (End k (A)) is the Lie algebra associated to the associative algebra (End k (A), •, id A ). By the universal property of the universal enveloping algebra, there is a unique algebra morphism Ω : U k (L) → End k (A) which extends ω.
Lemma 2.8. The base algebra A is naturally an U k (L)-module algebra.
By Lemma 2.8, we may consider the Connes-Moscovici's A-bialgebroid A ⊙ U k (L) ⊙ A. For the sake of simplicity, we will often denote it by B L . As an A e -ring, it comes endowed with a Lie algebra morphism for all a, b ∈ A and all X ∈ L, where o is the L-module structure on the tensor product of two L-modules. Equivalently, η B L : A e → B L is a morphism of L-modules, where B L has the L-module structure induced by J L .
The A e -ring B L with J L is universal among pairs (R, φ L ) satisfying these properties.

Theorem 2.9.
Given an A e -ring R with k-algebra morphism φ A : A e → R and given a Lie algebra morphism φ L : L → L(R) such that for all a, b ∈ A and all X ∈ L, there exists a unique morphism of A e -rings Φ : Proof. By the universal property of the universal enveloping k-algebra U k (L), there exists a unique morphism of k-algebras φ ′ : U k (L) → R such that φ ′ • j L = φ L . Now, set U := U k (L) and consider the k-linear map of (30). It follows immediately from the definition that Now, a straightforward check using (29) and induction on a PBW basis of U shows that for all u ∈ U and a, b ∈ A. In view of this, we have Thus, Φ is a morphism of A e -rings and it is clearly the unique satisfying Φ • J L = φ L .
for all a, b ∈ A, X ∈ L and m ∈ M, makes of M a left A ⊙ U k (L) ⊙ A-module, and conversely.
Proof. If M is an A-bimodule, then the assignments . If we consider the Lie algebra morphism φ L := ρ, then equation (32) is exactly condition (29) and hence there is a unique morphism of A e -rings R : The other way around, if we have a morphism of A e -rings R : A ⊙ U k (L) ⊙ A → End k (M) and we compose it with J L we get a Lie algebra morphism φ L : L → End k (M) such that Remark 2.11. Observe that the algebra maps φ ′ : U k (L) → R and φ A : A e → R satisfy (31) if and only if they satisfy for all u ∈ U, a, b ∈ A. This implies that the A e -ring morphism Φ :

The Connes-Moscovici's bialgebroid as universal enveloping bialgebroid
Our next aim is to prove that the Connes-Moscovici's bialgebroid B L = A ⊙ U k (L) ⊙ A satisfies a universal property as A-bialgebroid as well, in the form of an adjunction between the category of A-anchored Lie algebras and the category of A-bialgebroids.
3.1. The primitive functor. In light of Example 2.3(e), we may consider the assignment P : Bialgd A → AnchLie A given on objects by P (B) = (Prim (B) , ω B ) and on morphisms by simply (co)restricting any φ : B → B ′ to the primitive elements, that is φ • θ B = θ B ′ • P(φ). The latter gives a well-defined morphism of A-anchored Lie algebras because it is compatible with the commutator bracket and for all X ∈ Prim (B) and a ∈ A. Summing up, we have the following result.

Proposition 3.1.
There is a well-defined functor P : Bialgd A → AnchLie A which assigns to every A-bialgebroid B its Lie algebra of primitive elements Prim (B) with anchor ω B : Next lemma states a property of the primitive elements of an A-bialgebroid that we already observed for B L in (28) and that will be useful to prove the universal property in the forthcoming section.

Lemma 3.2. For B an A-bialgebroid, every primitive element X ∈ Prim (B) satisfies
Xs (a) − s (a) X = s ε Xs (a) (34) for all a ∈ A. In particular, for all X ∈ Prim(B) and all a, b ∈ A.
Proof. In view of the definition of B ⊗ A B and of B × A B we have that By resorting to the left-hand side identity in (3), this relation can be written equivalently as or By applying B ⊗ A ε to both sides of (36) and by recalling that ε(X) = 0, we get If we apply instead ε ⊗ A B to (37) then we get which gives the other relation in (34).

An adjunction between AnchLie A and Bialgd A .
We show now how the Connes-Moscovici's bialgebroid construction provides a left adjoint to the functor P in a way that mimics the well-known "universal enveloping algebra/space of primitives" adjunction Remark 3.3. Despite being well-known, it seems that no "classical" reference explicitly reports the adjunction (38) in the form we stated it here. Nevertheless, it is straightforward to check that the involved functors are well-defined (they are, in fact, slight adjustments of the functors considered in [26, page 239]) and that they form an adjoint pair. The unit L → Prim(U k (L)) (induced by the canonical map j L of (25)) and the counit U k (Prim(B)) → B (the unique algebra morphism extending the Lie algebra inclusion Prim(B) ⊆ L(B)) are the obvious natural morphism which are proved to be bijective in [26,Theorem 5.18].

Proposition 3.4. Let (L, ω) be an A-anchored Lie algebra. Given an A-bialgebroid B and given a Lie algebra morphism φ L : L → L(B) such that φ L (L) ⊆ Prim(B) and
φ L (X) a = ω(X)(a) (39) for all a ∈ A and for all X ∈ L, there exists a unique morphism of A-bialgebroids Φ : B L → B such that Φ • J L = φ L and it is explicitly given by (30).
Proof. Set U := U k (L) and let φ ′ : U → B be the unique k-algebra map extending φ L . In view of (35) and (39), η B and φ L satisfy (29). Thus, by Theorem 2.9, there exists a unique morphism of A e -rings Φ :  A, B) is a morphism of bialgebroids, then we can conclude that Φ is a morphism of A-bialgebroids and finish the proof. Equivalently, we need to check that and for all u ∈ U. Since, in view of the PBW theorem, U admits a k-basis of the form it is enough to check (40) and (41) on the elements of this basis. A direct computation for all n ≥ 1. Therefore, relation (41) holds. Concerning (40), we notice first of all that which shows that it is satisfied for u = 1 U , and we prove by induction on n ≥ 1 that it also holds for u = x 1 · · · x n , where X 1 , . . . , X n ∈ L. For n = 1 we have (2) .
Assume now that (40) holds for n ≥ 1, that is, that we have for all X 1 , . . . , X n ∈ L and hence, in particular, so that it holds for n + 1 and we may conclude that it holds for every n by induction. Theorem 3.6. The assignment

induces a well-defined functor which is left adjoint to the functor
where ω B : Prim (B) → Der k (A) is the anchor of equation (33). Write ϑ B : U k (Prim (B)) → B for the unique algebra map extending the inclusion θ B : Prim (B) ⊆ B. Then the unit and the counit of this adjunction are given by respectively, where x = j L (X), as usual. Furthermore, every component of the unit is a monomorphism and hence B is faithful.
Proof. We need to see how B operates on morphisms. Let f : (L, ω) → (L ′ , ω ′ ) be a morphism of A-anchored Lie algebras. In view of the fact that for all X ∈ L and a ∈ A, the morphism γ L : L → Prim(B L ) induced by J L (namely, we have is a morphism of A-anchored Lie algebras for every L in AnchLie A and hence, by Corollary 3.5, there exists a unique morphism of A-bialgebroids Φ : and it is explicitly given by (30). We set B(f ) = Φ. In order to conclude, consider the natural assignment Corollary 3.5 states that for every φ L in AnchLie A (L, Prim(B)) there exists a unique Φ in Remark 3.7. In the context of the proof above, let F : U → U ′ be the unique k-algebra In particular, the counit of the adjunction is not surjective in general. We will see with Corollary 4.4 why also the unit is not surjective.

An intrinsic description of
Inspired by the results of Milnor-Moore and Moerdijk-Mrčun, which give an intrinsic description of those bialgebras/bialgebroids that are universal enveloping algebras of Lie algebras/Lie-Rinehart algebras, we look for necessary and sufficient conditions on an Abialgebroid B in order to claim that it is a Connes-Moscovici's bialgebroid A ⊙ U k (L) ⊙ A for some A-anchored Lie algebra (L, ω).

The primitives of the Connes-Moscovici's bialgebroid.
To begin with, we need a more detailed analysis of the space of primitives of B L .

Lemma 4.1. Let B be an A-bialgebroid. Then the k-vector subspace
is an ideal in Prim(B). The Lie bracket is explicitly given by

and the anchor by
for all a, b ∈ A.
Proof. The fact that s − t is contained in Prim(B) follows from t(a o ) .
The fact that it is a Lie ideal in Prim(B) with respect to the commutator bracket follows because Now, a direct computation shows that for all a, b ∈ A as claimed. Furthermore, for all a, b ∈ A and hence the proof is concluded.  For H a Hopf algebra and A an H-module algebra, we will often write 1 given by the vector space sum.
Proof. The fact that s − t is an ideal in Prim(A ⊙ H ⊙ A) has been established in Lemma 4.1 for a general A-bialgebroid. The fact that 1 A ⊗ Prim(H) ⊗ 1 A is a subalgebra is a straightforward computation. Moreover, notice that if 1 A ⊗X ⊗1 A = a⊗1 H ⊗1 A −1 A ⊗1 H ⊗a for some X ∈ Prim(H) and some a ∈ A, then which implies that In view of Lemma 2.6, we are left to check that Let us consider a primitive element ξ ∈ A ⊙ H ⊙ A. Fix a basis {e i | i ∈ S} for A as a vector space, where S is some set of indexes with a distinguished index 0 and e 0 = 1 A .
where almost all the h ij are 0. Consider also the dual elements {e * i | i ∈ S} of the e i 's. Since ξ is primitive, the following relation holds For k = 0 = l, let us apply the k-linear morphism (e * k ⊗ H ⊗ A) ⊗ A (A ⊗ H ⊗ e * l ) to both sides of the identity (46). We find out that Consider again the identity (46), that now rewrites By resorting to the k-linear isomorphism ( By applying H ⊗ 1 * A ⊗ H to both sides of the identity (48) we get that whence h 00 is primitive in H, and by applying H ⊗ e * k ⊗ H for all k = 0 we get By applying further H ⊗ ε H we find that h 0k = −ε(h k0 )1 H and hence from (49) we deduce that which proves that the inclusion (45) holds.

Corollary 4.4. For any A-anchored Lie algebra (L, ω) we have
Proof. It follows from [26,Theorem 5.18] that Prim(U k (L)) = L. Moreover, it is clear that It is evident from Corollary 4.4 why, in general, the unit γ L : L → Prim(B L ) from Theorem 3.6 cannot be surjective.

Primitively generated bialgebroids. Let B be an A-bialgebroid and consider the
Definition 4.6. We say that an A-bialgebroid B is primitively generated if B = n≥0 F n (B).
Remark 4.7. Definition 4.6 is given in the same spirit of [26, page 239]. In particular, the Connes-Moscovici's bialgebroid A ⊙ U k (L) ⊙ A of an A-anchored Lie algebra (L, ω) is primitively generated. Notice also that B is primitively generated if and only if Φ B of (50) is surjective.

rings in a functorial way and the following diagram commutes
Proof. By the universal property of the tensor A e -ring and a standard argument, any Proof. The fact that any morphism of A-bialgebroids is filtered follows from the commutativity of (53) and the definition of the primitive filtration. Now, set U := U k (Prim(B)) and L := Prim(B). Since, by Proposition 4.3, is surjective as well and we know from Example 4.5 that Φ B L is surjective. Since, by naturality of Φ B , the following diagram commutes whence ǫ B is surjective and B is primitively generated.
Let B be an A-bialgebroid. Recall that, given the filtered left A e -module B with filtration {F n (B) | n ≥ 0} as in (51) The following lemma, which should be well-known, is implicitly needed in the proof of Theorem 4.13 below. Its statement resembles closely [27,Remark 2.4]. Its proof can be deduced from the results in [11,Appendix B] and it follows closely the argument reported in [38, page 229] for coalgebras over a field, but we sketch it here for the sake of the reader.  Proof. It follows from Lemma 1.1 and Proposition 1.2, once proved that B with the filtration {F n (B) | n ∈ N} is a filtered A-coring. To this aim observe that, as a left A e -submodule of B, F n (B) is generated by 1 B and by elements of the form X 1 · · · X k for 1 ≤ k ≤ n and X i ∈ Prim(B) for all i. By applying ∆ B we find that and that . . , k}, p 1 < p 2 < · · · < p t and q 1 < q 2 < · · · < q s . By left A e -linearity of ∆ B , we may conclude that it is filtered. On the other hand, ε B is obviously filtered (by definition of the filtration on A). Therefore, ( s B t o , ∆ B , ε B ) is in fact a filtered A-coring.
In view of Lemma 4.11 and by mimicking [27, page 3140] and §1.2, we give the following definition.  Proof. In order to prove that φ is injective, we are going to prove that gr (φ) is injective. In view of [28, Chapter D, Corollary III.6], the latter implies that φ is injective as well.
To this aim, let us prove that gr n (φ) is injective for every n ≥ 0. To begin with, let us prove that gr 0 (φ) is injective.
i ) as generic element in ker (gr 0 (φ)). Then By applying ε B ′ to both sides we find out that i a Being φ injective on A e · Prim (B), we conclude that i s B (a i ) t B (b o i ) = 0 and hence gr 0 (φ) is injective.
To prove that gr 1 (φ) is injective, notice that an element in is a generic element belonging to ker (gr 1 (φ)). This implies that and hence there exists in B ′ . By applying ε B ′ again we find out that j a ′ j b ′ j = 0 and hence Summing up, but, being φ injective on A e · Prim(B), this yields that Finally, let us prove that gr n (φ) is injective for all n ≥ 1 by induction. We just showed the case n = 1. Assume that gr 1 (φ), . . . , gr n (φ) are all injective for a certain n ≥ 1 and consider an element z ∈ ker gr n+1 (φ) . Consider also the canonical projections for h + k = n + 1, as in (9). For all p, q such that p + q = n + 1 we have that Therefore, for all 1 ≤ h ≤ n we have that gr(B) (z) .
By the inductive hypothesis and projectivity of gr s (B) and gr s (B ′ ) as left A e -modules for all s ≥ 0, we know that gr h (φ) ⊗ A gr k (φ) is injective and hence is injective and hence z = 0. (CM2) B is graded projective and primitively generated, (CM3) the left A e -submodule of B generated by L is 0 (in which case we require η B to be injective) or it is free and generated by a k-basis of L, (CM4) A e · s − t ∩ A e · L = 0 (in particular, A e · Prim(B) = (A e · L) ⊕ (A e · s − t )).
Proof. We want to apply Theorem 4.13 to show that the conditions listed are sufficient. First of all, let us prove that for any A-anchored Lie algebra (L, ω) the Connes-Moscovici's bialgebroid B L is graded projective and that gr (B L ) is strongly graded (we already know that B L is primitively generated from Example 4.5 and Remark 4.7).
In view of (52), we know that F n (B L ) = A ⊗ F n (U k (L)) ⊗ A. By exactness of the tensor product over a field, the short exact sequence of k-vector spaces Therefore, we have gr n (B L ) ∼ = A ⊗ gr n (U k (L)) ⊗ A as left A e -modules. In particular, gr n (B L ) is a free left A e -module.
To show that gr (B L ) is strongly graded, consider an element gr(B L ) (z) = 0 for some h, k satisfying h + k = n, where the elements {a α } α in A are linearly independent over k as well as the elements {b β } β . Now, consider the commutative diagram If we plug z in it and we recall that gr(U) is strongly graded (that is, p gr(U ) h,k •∆ [n] gr(U ) is injective for all n ≥ 0 and for all h + k = n) then we find u α,β = 0 for all α, β and hence z = 0. Now, the inclusion of A-anchored Lie algebras L → Prim(B) extends uniquely of to a morphism of A-bialgebroids Ψ : B L → B, in view of the universal property of B L (Corollary 3.5). Moreover, similarly to what we did in the proof of Corollary 4.9, one can show that Ψ is surjective (because B is primitively generated and P(Ψ) is surjective). Therefore, to conclude by applying Theorem 4.13 we are left to check that the candidate isomorphism Ψ is injective when restricted to A e · Prim(B L ). Since Prim(B L ) = (1 A ⊗ L ⊗ 1 A ) ⊕ s B L − t B L , a generic element in A e · Prim(B L ) is of the form where we may assume that the X k 's are elements of a k-basis of L, without loss of generality. Thus, By (CM4), this entails that By (CM3), relation (55) yields that j a i,j ⊗ b i,j = 0 in A e for all i. Relation (56), instead, implies that However, since A e · L is a free left A e -module with action given via η B (or η B is injective by hypothesis), η B itself has to be injective and hence which, in turn, yields Summing up, (CM1) ensures the existence of a morphism Ψ and (CM2) entails that Ψ is surjective. Conditions (CM3) and (CM4), instead, allow us to conclude that Ψ is injective on A e · Prim (B L ) and hence, by Theorem 4.13, Ψ is injective on B L . Thus, Ψ is an isomorphism. The fact that the conditions (CM1) -(CM4) are necessary is clear Remark 4.15. In the context of the proof above, observe that if L = 0, then Prim(B) = s B − t B . If moreover B is primitively generated, then Ψ : A ⊙ k ⊙ A → B is surjective and it coincides with η B : A e → B up to the isomorphism A ⊙ k ⊙ A ∼ = A e . This is the point where injectivity of η B enters the picture. Example 4.16. Let A = Mat n (k) for n ≥ 2 and let B = End k (A) as in Example 1.7(c). It follows from the Skolem-Noether theorem that every derivation of A is inner. In particular, Prim(B) = s − t and conditions (CM1) and (CM4) are satisfied. Furthermore, as we have seen in Example 4.10, B is also primitively generated and, in fact, gr n (B) = 0 for all n ≥ 1.
In order to apply Theorem 4.14, we are left to show that η B is injective (notice that we already know it is surjective), but a straightforward computation reveals that η B coincides with the composition of isomorphisms Therefore, (CM2) and (CM3) are satisfied as well and, by Theorem 4.14, B ∼ = A⊙U k (0)⊙A.

Bialgebroids over commutative algebras.
A slightly more favourable situation is provided by the case of bialgebroids over a commutative base. Let us assume henceforth that A is a commutative k-algebra. This implies that now we can consider the target t of an A-bialgebroid B as an algebra map t : A → B and hence we will omit the (−) o . By Lemma 4.1, we may consider the quotient Lie algebra which, however, is not injective in general (that is, B and P are not adjoint functors). Nevertheless, we may always consider a "preferred" morphism of A-bialgebroids induced by a chosen injection ι B : Prim(B) → Prim(B) and by the counit of the adjunction in Theorem 3.6. Since the hypothesis on Prim(B) ensures that the foregoing structures is the free left A e -module generated by {v k | k ≥ 0}, whence it is primitively generated and graded projective (even free). Thus, Theorem 4.17 ensures that The

Final
Remarks. An additional step which deserve to be taken is to restrict the attention further to those A-bialgebroids B over a commutative algebra A such that s B = t B (for example, cocommutative A-bialgebroids). However, in this case the Connes-Moscovici's construction is not the correct construction to look at. One may prove that the assignment provides a surjective homomorphism of A-bialgebroids with kernel the ideal generated by s B L − t B L in B L . The A-bialgebroid structure on A # U k (L) is that of extension of scalars with trivial coaction on A, that is to say, and semi-direct product algebra structure, that is, In view of the results from §2.2 and §3.2, the foregoing observations suggest that A # U k (L) would be the right A-bialgebroid construction to consider, in order to recover the universal property of Theorem 2.9 and an adjunction as in Theorem 3.6. Nevertheless, we keep this question for a future investigation.