Structure theory of Rack-Bialgebras

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution space supported in the chosen point. For Lie groups, it is well-known that this leads to the universal enveloping algebra of the Lie algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain cocommutative Hopf dialgebras.

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions.
We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution space supported in the chosen point. For Lie groups, it is well-known that this leads to the universal enveloping algebra of the Lie algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain cocommutative Hopf dialgebras.
The canonical rack bialgebras we have constructed for any Leibniz algebra lead then to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in 2013. We clarify this framework doing some deformation theory.

Introduction
All manifolds considered in this manuscript are assumed to be Hausdorff and second countable.
Basic Lie theory relies heavily on the fundamental links between associative algebras, Lie algebras and groups. Some of these links are the passage from an associative algebra A to its underlying Lie algebra A Lie which is the vector space A with the bracket [a, b] := ab − ba. On the other hand, to any Lie algebra g one may associate its universal enveloping algebra U(g) which is associative. Groups arise as groups of units in associative algebras. To any group G, one may associate its group algebra KG which is associative.
Indeed, Kinyon showed in [14] that the tangent space at e ∈ H of a Lie rack H carries a natural structure of a Leibniz algebra, generalizing the relation between a Lie group and its tangent Lie algebra. Conversely, every (finite dimensional real or complex) Leibniz algebra h may be integrated into a Lie rack R h (with underlying manifold h) using the rack product X ◮ Y := e ad X (Y ), (1.2) noting that the exponential of the inner derivation ad X for each X ∈ h is an automorphism.
Another closely related algebraic structure is that of dialgebras. A dialgebra is a vector space D with two (bilinear) associative operations ⊢: D ×D → D and ⊣: D × D → D which satisfy three compatibility relations, namely for a, b, c ∈ D: A dialgebra D becomes a Leibniz algebras via the formula In this sense ⊢ and ⊣ are two halves of a Leibniz bracket. Loday and Goichot have defined an enveloping dialgebra of a Leibniz algebra, see [12], [20].
One main point of the first part of this paper is the link between rack bialgebras and cocommutative Hopf dialgebras. In Theorem 2.5, we adapt Suschkewitsch's Theorem in semi-group theory to the present context. The classical result (see Appendix B) treats semi-groups with a left unit e and right inverses (analoguous results in the left-context), called right groups. Suschkewitsch shows that such a right group Γ decomposes as a product Γ = Γe × E where E is the set of all idempotent elements.
Its incarnation here shows that a cocommutative right Hopf algebra H decomposes as a tensor product H1 ⊗ E H where E H is the subspace of generalized idempotents.
Furthermore, we will show in Theorem 2.6 how to associate to any augmented rack bialgebra an augmented cocommutative Hopf dialgebra. In Theorem 2.7, we investigate what Suschkewitsch's decomposition gives for a cocommutative Hopf dialgebra A. It turns out that A decomposes as a tensor product E A ⊗ H A of E A with H A which may be identified to the associative quotient A ass of A. This result permits to show that the Leibniz algebra of primitives in A is a hemi-semi-direct product (see [14]), and thus always split. In this way we arrive once again at the result that Lie digroups may serve only to integrate split Leibniz algebras which has already been observed by Covez in his master thesis [6].
Here df (ξ) and dg(ξ) are linear functionals on g * , identified with elements of g.
In the same way, a general finite dimensional Leibniz algebra h gives rise to a smooth manifold h * , which carries now some kind of generalized Poisson bracket, namely {f, g}(ξ) := − ξ, [df (0), dg(ξ)] , see [9] for an explanation why we believe that this is the correct bracket to consider. In particular, this generalized Poisson bracket need not be skewsymmetric.
The quantization procedure of this generalized Poisson bracket proposed in [9] proceeds as follows: The cotangent lift of the above rack product is interpreted as a symplectic micromorphism. The generating function of this micromorphism serves then as a phase function in a Fourier integral operator, whose asymptotic expansion gives rise to a star-product.
One main goal of the present article is to set up a purely algebraic framework in which one may deformation quantize the dual of a Leibniz algebra. The main feature will be to recover -in a rather explicit algebraic mannerthe star-product which has been constructed in [9] by analytic methods.
Our main result in the present paper (Corollary 4.1) is the purely algebraic construction of a star-product on h * based on formula (1.2), and the proof that it is identical to the star-product of [9].
Closing the paper, we set up a general deformation theory framework in which the above star-product appears as a formal deformation, its infinitesimal term defining a cohomology class.
Let us comment on the content of the paper: All our bialgebra notion are based on the standard theory of coalgebras, some features of which as well as our notions are recalled in Appendix A. Rack bialgebras and augmented rack bialgebras are studied in Section 2. Connected, cocommutative Hopf algebras give rise to a special case of rack bialgebras. In Section 2.2, we associate to any Leibniz algebra h an augmented rack bialgebra UAR ∞ (h) and study the functorial properties of this association. This rack bialgebra plays the role of an enveloping algebra in our context. It turns out that a truncated, non-augmented version UR(h) is a left adjoint of the functor of primitives Prim.
We also study the "group-algebra" functor associating to a rack X its rack bialgebra K [X]. Like in the classical framework, K[−] is left adjoint to the functor Slike associating to a track bialgebra its rack of set-like elements. The relation between rack bialgebras and the other algebraic notion discussed in this paper is summarized in the diagram (see the end of Section 2.2) of categories and functors: In Section 2.3, we develop the structure theory for rack bialgebras and cocommutative Hopf dialgebras, based on Suschkewitsch's Theorem. Section 2.3 contains Theorem 2.5, Theorem 2.6 and Theorem 2.7 whose content we have described above.
Recollecting basic knowledge about the Serre functor F is the subject of Section 3. In particular, we show in Section 3.2 that F is a strong monoidal functor from the category of pointed manifolds Mf * to the category of coalgebras, based on some standard material on coalgebras (Appendix A). In Section 3.3, we apply F to Lie groups, Lie semi-groups, Lie digroups, and to Lie racks and augmented Lie racks, and study the additional structure which we obtain on the coalgebra. The case of Lie racks motivates the notion of rack bialgebra.
Recall that for a Leibniz algebra h, the vector space h becomes a Lie rack R h with the rack product X ◮ Y = e ad X (Y ).
In Theorem 3.8, we show that the rack bialgebra UAR ∞ (h) associated to h coincides with the rack bialgebra F (R h ).
The quantization procedure in Section 4 relies on an explicit approach in terms of coordinates on the (finite dimensional real) Leibniz algebra h. The main theorem (Theorem 4.1) describes the induced rack structure on exponentials from the rack structure which is based on formula (1.2). In Section 4.2, we develop the basics of rack bialgebra deformation theory to show that infinitesimal deformations are classified by H 1 and that the star-product gives a formal deformation of the trivial rack bialgebra.
Acknowledgements: F.W. thanks Université de Haute Alsace for an invitation during which the shape of this research project was defined. At some point, the subject of this paper was joint work of S.R. and F.W. with Simon Covez, and we express our gratitude to him for his contributions. Starting from this, S.R. [23] came independently to similar results which we incorporated in this paper. Some part of the results of this paper constitute the master thesis of C.A.. M.B. thanks Nacer Makhlouf for his question about the relations of rack-bialgebras to dialgebras, and Gwénael Massuyeau for his question about universals.

Several bialgebras
In the following, let K be an associative commutative unital ring containing all the rational numbers. The symbol ⊗ will always denote the tensor product of K-modules over K. For any coalgebra (C, ∆) over K, we shall use Sweedler's notation ∆(a) = (a) a (1) ⊗ a (2) for any a ∈ A. We will feel free to suppress the sum-sign in Sweedler's notation in complicated formulas for typographical reasons. See also Appendix A for a survey on definitions and notations in coalgebra theory.
The following Sections will all deal with the following type of nonassociative bialgebra: Let (B, ∆, ǫ, 1, µ) be a K-module such that (B, ∆, ǫ, 1) is a coassociative counital coaugmented coalgebra (a C 3 -coalgebra), and such that the linear map µ : B ⊗ B → B (the multiplication) is a morphism of C 3 -coalgebras (it satisfies in particular µ(1 ⊗ 1) = 1). We shall call this situation a nonassociative C 3 I-bialgebra (where I stands for 1 being an idempotent for the multiplication µ). For another nonassociative C 3 I-bialgebra (B ′ , ∆ ′ , ǫ ′ , 1 ′ , µ ′ ) a K-linear map φ : B → B ′ will be called a morphism of nonassociative C 3 I-bialgebras iff it is a morphism of C 3 -coalgebras and is multiplicative in the usual sense φ µ(a ⊗ b) = µ ′ φ(a) ⊗ φ(b) for all a, b ∈ B. The nonassociative C 3 I-bialgebra (B, ∆, ǫ, 1) is called left-unital (resp. right-unital ) iff for all a ∈ B µ(1 ⊗ a) = a (resp. µ(a ⊗ 1) = a). Moreover, consider the associative algebra A := Hom K (B, B) equipped with the composition of K-linear maps, and the identity map id B as the unit element. There is an associative convolution multiplication * in the K-module Hom K (B, A) of all K-linear maps B → Hom K (B, B), see Appendix A, eqn (A.3) for a definition with id B ǫ as the unit element. For a given nonassociative C 3 I-bialgebra (B, ∆, ǫ, 1, µ) we can consider the map µ as a map B → Hom K (B, B) in two ways: as left multiplication map . We call (B, ∆, ǫ, 1, µ) a left-regular (resp. right-regular ) nonassociative C 3 I-bialgebra iff the map L µ (resp. the map R µ ) has a convolution inverse, i.e. iff there is a K-linear map µ ′ : B ⊗ B → B (resp. µ ′′ : (2.1) Note that every associative unital Hopf algebra (H, ∆, ǫ, 1, µ, S) (where S denotes the antipode, i.e. the convolution inverse of the identity map in Hom K (H, H)) is right-and left-regular by setting µ ′ = µ • (S ⊗ id H ) and Lemma 2.1 Let (B, ∆, ǫ, 1, µ) be a nonassociative C 3 I-bialgebra.
1. If B is left-regular (resp. right-regular), then the corresponding K-linear map µ ′ : B ⊗ B → B is unique, and in case ∆ is cocommutative, µ ′ is map of C 3 -coalgebras.
Proof: 1. In any monoid (in particular in the convolution monoid) two-sided inverses are always unique. Moreover, as can easily be checked, a K-linear map φ : Using eqn (2.2), the fact that L µ ′ is a convolution inverse of L µ , and the cocommutativity of ∆, we get and µ ′ preserves comultiplications. A similar reasoning where B ⊗ B is replaced by K shows that µ ′ preserves counits. Finally, it is obvious that L µ ′ (1) is the inverse of the K-linear map L µ (1), and since the latter fixes 1 so does the former. The reasoning for right-regular bialgebras is completely analogous.

2.
For left-unital bialgebras we get L µ (1) = id B , and the generalized Takeuchi-Sweedler argument, see Appendix A, shows that L µ has a convolution inverse. Right-unital bialgebras are treated in an analogous manner. ✷ Note that any C 3 -coalgebra (C, ∆, ǫ, 1) becomes a left-unital (resp. rightunital) associative C 3 I-bialgebra by equipping with the left-trivial (resp. righttrivial ) multiplication Moreover c ∈ B will be called a generalized left (resp. right) unit element iff for all b ∈ B we have cb = ǫ(c)b (resp. bc = ǫ(c)b).

Rack bialgebras, (left) Leibniz algebras and structure theory
The last condition (2.6) is called the self-distributivity condition.
Note that we do not demand that the C 3 -coalgebra B should be cocommutative nor connected.
is a solution of the Yang-Baxter equation. We draw our reader's attention to the fact that Carter-Crans-Elhamdadi-Saito work in [4] with right racks, while we work here with left racks. The statement of their theorem works also for left racks, but then one has to takẽ In particular for any rack bialgebra,R q is a solution of the Yang-Baxter equation. ✸ Example 2.1 Any C 3 coalgebra (C, ∆, ǫ, 1) carries a trivial rack bialgebra structure defined by the left-trivial multiplicaton which in addition is easily seen to be associative and left-unital, but in general not unital. ✸ Another method of constructing rack bialgebras is gauging: i.e. f is µ-equivariant. It is a routine check that (B, ∆, ǫ, 1, µ f ) is a rack bialgebra where for all a, b ∈ B the multiplication is defined by We shall call (B, ∆, ǫ, 1, µ f ) the f -gauge of (B, ∆, ǫ, 1, µ).  In general, the adjoint representation does not seem to preserve the coalgebra structure if no cocommutativity is assumed.

Example 2.3
Recall that a pointed set (X, e) is a pointed rack in case there is a binary operation ✄ : X × X → X such that for all x ∈ X, the map y → x ✄ y is bijective and such that for all x, y, z ∈ X, the self-distributivity and unit relations are satisfied. Then there is a natural rack bialgebra structure on the vector space K[X] which has the elements of X as a basis. K[X] carries the usual coalgebra structure such that all x ∈ X are set-like: △(x) = x ⊗ x for all x ∈ X. The product µ is then induced by the rack product. By functoriality, µ is compatible with △ and e.
Observe that this construction differs slightly from the construction in [4], Section 3.1. ✸ More generally there is the following structure: where ad denotes the usual adjoint representation for Hopf algebras, see e.g. eqn (2.10). We shall define a morphism (B, Φ, H, ℓ) → (B ′ , Φ ′ , H ′ , ℓ ′ ) of augmented rack bialgebras to be a pair (φ, ψ) of K-linear maps where φ : is a morphism of C 3 -coalgebras, and ψ : H → H ′ is a morphism of Hopf algebras such that the obvious diagrams commute: An immediate consequence of this definition is the following Proposition 2.1 Let (B, Φ, H, ℓ) be an augmented rack bialgebra. Then the C 3 -coalgebra (B, ǫ, 1) will become a left-regular rack bialgebra by means of the multiplication a ⊲ b := Φ(a).b (2.14) for all a, b ∈ B. In particular, each Hopf algebra H becomes an augmented rack bialgebra via (H, id H , H, ad). In general, for each augmented rack bialgebra the map Φ : B → H is a morphism of rack bialgebras.
Proof: We check first that ⊲ is a morphism of C 3 -coalgebras B ⊗ B → B: Let a, b ∈ B, then -thanks to the fact that the action ℓ and the maps Φ are coalgebra morphisms- whence µ is a morphism of coalgebras. Clearly whence µ preserves counits. We shall next compute both sides of the self-distributivity identity (2.6) to get an idea: (2) ) .c, and we compute, using the fact that Φ is a morphism of C 3 -coalgebras, which proves the self-distributivity identity. Moreover we have and a ⊲ 1 = Φ(a).1 (2.11) = ǫ H Φ(a) 1 = ǫ B (a)1, whence the C 3 -coalgebra becomes a rack bialgebra. Finally, if we set and the other equation of the left-regularity condition (2.1) is proved similarly. ✷

Example 2.4
Exactly in the same way as a pointed rack gives rise to a rack bialgebra K[X], an augmented pointed rack p : X → G gives rise to an augmented rack bialgebra p :

Remark 2.2
Motivated by the fact that the augmented racks p : X → G are exactly the Yetter-Drinfeld modules over the (set-theoretical) Hopf algebra G, we may ask whether augmented rack bialgebras are Yetter-Drinfeld modules. In fact, any cocommutative augmented rack bialgebra (B, Φ, H, ℓ) gives rise to a Yetter-Drinfeld module over the Hopf algebra H. Indeed, B is a left H-module via ℓ, and becomes a left H-comodule via Now, in Sweedler notation, the coaction is denoted for all b ∈ B by Then the Yetter-Drinfeld compatibility relation reads This relation is true in our case, because ℓ is a morphism of coalgebras and is sent to the adjoint action via Φ.
Conversely, given a Yetter-Drinfeld module C over a Hopf algebra H, together with a linear form ǫ C : The map Φ intertwines the left action on C and the adjoint action on H thanks to the Yetter-Drinfeld condition. Now define a rack product for all x, y ∈ C by then we obtain a Yetter-Drinfeld version of self-distributivity as there is no comultiplication on C.
The fact that Φ is a morphism of coalgebras is then replaced by the identity which one needs to demand. Finally, one needs a unit 1 C ∈ C such that for all h ∈ H, h.1 C = ǫ H (h)1 C , ǫ C (1 C ) = 1 K , ρ(1 C ) = 1 H ⊗1 C , and Φ(1 C ) = 1 H . This is somehow the closest one can get to a rack bialgebra without having a compatible C 3 coalgebra structure on C. ✸ The link to Leibniz algebras is contained in the following Proposition 2.2 Let (B, ∆, ǫ, 1, µ) be a rack bialgebra over K.
1. Then its K-submodule of all primitive elements, Prim(B) =: h, (see eqn (A.1) of Appendix A) is a subalgebra with respect to µ (written a ⊲ b) satisfying the (left) Leibniz identity for all x, y, z ∈ h = Prim(B). Hence the pair (h, [ , ]) with [x, y] := x⊲ y for all x, y ∈ h is a Leibniz algebra over K. Moreover, every morphism of rack bialgebras maps primitive elements to primitive elements and thus induces a morphism of Leibniz algebras.
2. More generally, h and each subcoalgebra of order k ∈ N, B (k) , (see eqn (A.2) of Appendix A) is stable by left ⊲-multiplications with every a ∈ B. In particular, each B (k) is a rack subbialgebra of (B, ∆, ǫ, 1, µ).
Proof: 2. Let x ∈ h and a ∈ B. Since µ is a morphism of C 3 -coalgebras and x is primitive, we get whence a ⊲ x is primitive. For the statement on the B (k) , we proceed by induction: For k = 0, this is clear. Suppose the statement is true until k ∈ N, and let x ∈ B (k+1) . Then where we have used the extended multiplication (still denoted ✄) ✄ : by the definition of B (k+1) , see Appendix A. By the induction hypothesis, all the terms a (1) ⊲x (1) ′ and a (2) 1. It follows from 2. that h is a subalgebra with respect to µ. Let x, y, z ∈ h. Then since x is primitive, it follows from ∆(x) = x ⊗ 1 + 1 ⊗ x and the self-distributivity identity (2.6) that proving the left Leibniz identity. The morphism statement is clear, since each morphism of rack bialgebras is a morphism of C 3 -coalgebras and preserves primitives.
As an immediate consequence, we get that the functor Prim induces a functor from the category of all rack bialgebras over K to the category of all Leibniz algebras over K.

Remark 2.3
Define set-like elements to be elements a in a rack bialgebra B such that ∆(a) = a ⊗ a. Thanks to the fact that ✄ is a morphism of coalgebras, the set of set-like elements Slike(B) is closed under ✄. In fact, Slike(B) is a rack, and one obtains in this way a functor Slike : RackBialg → Racks. Proof: This follows from the adjointness of the same functors, seen as functors between the categories of pointed sets and of C 4 -coalgebras, observing that the C 4 -coalgebra morphism induced by a morphism of racks respects the rack product. ✷ Observe that the restriction of Slike : RackBialg → Racks to the subcategory of connected, cocommutative Hopf algebras Hopf (where the Hopf algebra is given the rack product defined in eqn (2.10)) gives the usual functor of group-like elements.

Remark 2.4
Compare the self-distributivity identity discussed in [16] or [17]. While the first identity takes place in a coalgebra, the second one takes place in a vector space V with a braiding τ , i.e. a bilinear map τ : V ⊗V → V ⊗V whose components we denote by τ (a⊗b) = b 1 ⊗a 2 (in generalized Sweedler notation). The braided Leibniz identity gives back the (left) Leibniz identity for the tensor flip τ (a ⊗ b) = b ⊗ a. ✸

(Augmented) rack bialgebras for any Leibniz algebra
Let (h, [ , ]) be a Leibniz algebra over K, i.e. h is a K-module equipped with a K-linear map [ , ] : h ⊗ h → h satisfying the (left) Leibniz identity (1.1).
Recall first that each Lie algebra over K is a Leibniz algebra giving rise to a functor from the category of all Lie algebras to the category of all Leibniz algebras.
Furthermore, recall that each Leibniz algebra has two canonical K-submodules In order to perform the following constructions of rack bialgebras for any given Leibniz algebra (h, [ , ]), choose first a two-sided ideal z ⊂ h such that let g denote the quotient Lie algebra h/z, and let p : h → g be the natural projection. The data of z ⊂ h, i.e. of a Leibniz algebra h together with an ideal z such that Q(h) ⊂ z ⊂ z(h), could be called an augmented Leibniz algebra. Thus we are actually associating an augmented rack bialgebra to every augmented Leibniz algebra. In fact, we will see that this augmented rack bialgebra does not depend on the choice of the ideal z and therefore refrain from introducing augmented Leibniz algebras in a more formal way. The Lie algebra g naturally acts as derivations on h by means of (for all x, y ∈ h) p(x).y := [x, y] =: ad x (y) (2.20) because z ⊂ z(h). Note that as Lie algebras.
Consider now the C 5 -coalgebra (B = S(h), ∆, ǫ, 1) which is actually a commutative cocommutative Hopf algebra over K with respect to the symmetric multiplication •. The linear map p : h → g induces a unique morphism of Hopf algebrasΦ = S(p) : for any nonnegative integer k and x 1 , . . . , x k ∈ h. In other words, the association S : V → S(V ) is a functor from the category of all K-modules to the category of all commutative unital C 5 -coalgebras. Consider now the universal enveloping algebra U(g) of the Lie algebra g. Since Q ⊂ K by assumption, the Poincaré-Birkhoff-Witt Theorem (in short: PBW) holds (see e.g. [22,Appendix]). More precisely, the symmetrisation map ω : S(g) → U(g), defined by ω(1 S(g) ) = 1 U(g) , and ω(ξ 1 • · · · • ξ k ) = 1 k! σ∈S k ξ σ(1) · · · ξ σ(k) , (2.24) see e.g. [10, p.80, eqn (3)], is an isomorphism of C 5 -coalgebras (in general not of associative algebras). We now need an action of the Hopf algebra H = U(g) on B, and an intertwining map Φ : B → U(g). In order to get this, we first look at g-modules: The K-module h is a g-module by means of eqn (2.20), the Lie algebra g is a g-module via its adjoint representation, and the linear map p : h → g is a morphism of g-modules since p is a morphism of Leibniz algebras. Now S(h) and S(g) are g-modules in the usual way, i.e. for all k ∈ N \ {0}, ξ, ξ 1 , . . . , ξ k ∈ g, and x 1 . . . , and of course ξ.1 S(h) = 0 and ξ.1 S(g) = 0. Recall that U(g) is a g-module via the adjoint representation ad ξ (u) = ξ.u = ξu − uξ (for all ξ ∈ g and all u ∈ U(g)).
Finally it is a routine check using the above identities (2.25) and (2.10) that S(h) becomes a module coalgebra. We can resume the preceding considerations in the following ) be a Leibniz algebra over K, let z be a two-sided ideal of h such that Q(h) ⊂ z ⊂ z(h), let g denote the quotient Lie algebra h/z by g, and let p : h → g be the canonical projection.
1. Then there is a canonical U(g)-action ℓ on the C 5 -coalgebra B := S(h) (making it into a module coalgebra leaving invariant 1) and a canonical lift of p to a map of C 5 -coalgebras, Φ : S(h) → U(g) such that eqn (2.12) holds. Hence the quadruple (S(h), Φ, U(g), ℓ) is an augmented rack bialgebra whose associated Leibniz algebra is equal to (h, [ , ]) (independently of the choice of z). The resulting rack multiplication µ of S(h) (written µ(a ⊗ b) = a ⊲ b) is also independent on the choice of z and is explicitly given as follows for all positive integers k, l and x 1 , . . . , x k , y 1 , . . . , y l ∈ h: x denotes the action of the Lie algebra h/z(h) (see eqn (2.21)) on S(h) according to eqn (2.25).
2. In case z = Q(h), the construction mentioned in 1. is a functor h → UAR ∞ (h) from the category of all Leibniz algebras to the category of all augmented rack bialgebras associating to h the rack bialgebra and to each morphism f of Leibniz algebras the pair S(f ), U(f ) where f is the induced Lie algebra morphism.
3. For each nonnegative integer k, the above construction restricts to each subcoalgebra of order k, from the category of all Leibniz algebras to the category of all augmented rack bialgebras.
Proof: 1. All the statements except the last two ones have already been proven. Note that for all x, y ∈ h we have by definition independently of the chosen ideal z. Moreover we compute p(x σ(1) ) · · · p(x σ(k) ) . y 1 • · · · • y l ), which gives the desired formula since for all x ∈ h and a ∈ S(h), we have p(x).a = ad s x (a).
2. Let f : h → h ′ be a morphism of Leibniz algebras, and let f : h → h ′ be the induced morphism of Lie algebras. Hence we get , and U(f ) : U(h) → U(h ′ ) the induced maps of Hopf algebras, i.e. S(f ) (resp. S(f )) satisfies eqn (2.23) (with p replaced by f (resp. by f )), and U(f ) satisfies for all positive integers k and ξ 1 , . . . , ξ k ∈ h. If ω : S(h) → U(h) and ω ′ : S(h ′ ) → U(h ′ ) denote the corresponding symmetrisation maps (2.24) then it is easy to see from the definitions that and composing from the left with ω ′ yields the equation Moreover for all x, y ∈ h we have, since f is a morphism of Leibniz algebras, and upon using eqn (2.25) we get for all a ∈ S(h) showing finally for all u ∈ U(h) and all a ∈ S(h) Associating to every Leibniz algebra (h, [ , ]) the above defined augmented rack bialgebra (S(h), Φ, U(h), ℓ), and to every morphism ψ : h → h ′ of Leibniz algebras the pair of K-linear maps Ψ = S(ψ), Ψ = U(ψ) , we can easily check that Ψ is a morphism of C 5 -coalgebras, Ψ is a morphism of Hopf algebras, such that the two relevant diagrams (2.13) commute which easily follows from (2.32) and (2.33). The rest of the functorial properties is a routine check.
3. By definition, the U(g)-action on S(h) (cf. eqs (2.25) and (2.28)) leaves invariant each K-submodule S r (h) for each nonnegative integer r whence it leaves invariant each subcoalgebra of order k, S(h) (k) . It follows that the construction restricts well. ✷

Remark 2.5
This theorem should be compared to Proposition 3.5 in [4]. In [4], the authors work with the vector space N := K ⊕ h, while we work with the whole symmetric algebra on the Leibniz algebra. In some sense, we extend their Proposition 3.5 "to all orders". However, as we shall see below, N is already enough to obtain a left-adjoint to the functor of primitives. ✸ The above rack bialgebra associated to a Leibniz algebra h can be seen as one version of an enveloping algebra of h.

Definition 2.3
Let h be a Leibniz algebra. We will call the augmented rack bialgebra (S(h), Φ, U(g), ℓ) the enveloping algebra of infinite order of h. As such, it will be denoted by UAR ∞ (h).
This terminology is justified, for example, by the fact that h is identified to the primitives in S(h) (cf Proposition 2.2). This is also justified by the following theorem the goal of which is to show that the (functorial version of the above) enveloping algebra UAR ∞ (h) fits into the following diagram of functors: Here, i is the embedding functor of Lie algebras into Leibniz algebras, and j is the embedding functor of the category of connected, cocommutative Hopf algebras into the category of rack bialgebras, using the adjoint action (see eqn (2.10)) as a rack product.
Proof: The enveloping algebra UAR ∞ (h) is by definition the functorial version of the rack bialgebra S(h), i.e. associated to the ideal Q(h). But in case h is a Lie algebra, Q(h) = {0}. Then the map p is simply the identity, and UAR ∞ (h) = j U (h) . ✷ As a relatively easy corollary we obtain from the preceding construction the computation of universal rack bialgebras. More precisely, we look for a left adjoint functor for the functor Prim, seen as a functor from the category of all rack bialgebras to the category of all Leibniz algebras. For a given Leibniz algebra h, [ , ] define the subcoalgebra of order 1 of the first component of with 1 := 1 = 1 K which is rack subbialgebra according to Proposition 2.2. Its structure reads for all λ, λ ′ ∈ K and for all x, y ∈ h which is clearly is a morphism of rack bialgebras. Hence UR is a functor from the categry of all Leibniz algebras to the category of all rack bialgebras. Now let (C, ∆ C , ǫ C , 1 C , µ C ) be a rack bialgebra, and let f : h → Prim(C) be a morphism of Leibniz algebras. Define the K-linear mapf : Next we can refine the above universal construction by taking into account the augmented rack bialgebra structure of UAR ∞ (h) to define another universal object. Consider the more detailed category of all augmented rack bialgebras. Again, the functor Prim applied to the coalgebra B (and not to the Hopf algebra H) gives a functor from the first category to the category of all Leibniz algebras, and we seek again a left adjoint of this functor, called UAR. Hence, a natural candidate for a universal augmented rack bialgebra associated to a given Leibniz algebra h is (2.40) The third statement of Theorem 2.1 tells us that this is a well-defined augmented rack bialgebra, and that UAR is a functor from the category of all Leibniz algebras to the category of all augmented rack bialgebras. Now let (B ′ , Φ ′ , H ′ , ℓ ′ ) be an augmented rack bialgebra, and let f : h → Prim(B ′ ) be a morphism of Leibniz algebras. Clearly, as has been shown in Theorem 2.3, the mapf : UR(h) → B ′ given by eqn (2.38) is a morphism of rack bialgebras. Observe that the morphism of C 3 -coalgebras Φ ′ sends the Leibniz subalgebra Prim(B ′ ) of B ′ into K-submodule of all primitive elements of the Hopf algebra H ′ , Prim(H ′ ), which is known to be a Lie subalgebra of H ′ equipped with the commutator Lie bracket [ , ] H ′ . Moreover this restriction is a morphism of Leibniz algebras. Indeed, for any Lie algebras, and by the universal property of universal envelopping algebras there is a unique morphism of associative unital algebras ψ : since ψ maps primitives to primitives whence ψ is a morphism of coalgebras. It is easy to check that ψ preserves counits, whence ψ is a morphism of C 5 -Hopf-algebras. For all λ ∈ K and x ∈ h we get showing the first equation showing the second equation The relationship between the different notions (taking into account also Remark (2.3)) is resumed in the following diagram: where UAR ∞ is not left-adjoint to Prim, while UR is, but does not render the square commutative. There is a similar diagram for augmented notions.

Relation with bar-unital di(co)algebras
In the beginning of the nineties the 'enveloping structure' associated to Leibniz algebras has been the structure of dialgebras, see e.g. [20]. We shall show in this section that rack bialgebras and certain cocommutative Hopf dialgebras are strongly related.
A first class of examples is of course the well-known class of all cocommutative Hopf algebras (H, ∆, ǫ, 1, µ, S) for which 1 is a unit element, and S is a right and left antipode. Secondly it is easy to check that every C 4 -coalgebra (C, ∆, ǫ, 1) equipped with the left-trivial multiplication (resp. right trivial multiplication) µ 0 (see eqn (2.3)) and trivial right antipode (resp. trivial left antipode) S 0 defined by S 0 (x) = ǫ(x)1 for all x ∈ C (in both cases) is a cocommutative right Hopf algebra (resp. cocommutative left Hopf algebra) called the cocommutative left-trivial right Hopf algebra (resp. right-trivial left Hopf algebra) defined by the C 4 -coalgebra (C, ∆, ǫ, 1).
We have the following elementary properties showing in particular that each right (resp. left) antipode is unique: Lemma 2.2 Let (H, ∆, ǫ, 1, µ, S) be a cocommutative right Hopf algebra.
2. For all a, b ∈ H: S(ab) = S(b)S(a). (2) , and all these three statements imply that c is a generalized left unit element.

For any element c ∈ H, c is a generalized idempotent if and only if
Proof: 1. Since S is a coalgebra morphism, it preserves convolutions when composing from the right. This gives the first equation from statement 1. Hence the elements id, S, and S • S satisfy the hypotheses of the elements a, b, c of Lemma B.1 in the left-unital convolution semigroup Hom K (H, H), * , 1ǫ , whence the second and third equation of statement 1. are immediate, and the fourth follows from composing the second from the right with S and using the third. Clearly S is unique according to Lemma B.1.

2.
Again the elements µ, S • µ and (id * 1ǫ) • µ satisfy the hypotheses on the elements a, b, c of Lemma B.1 in the left-unital convolution semigroup Hom K (H ⊗ H, H), * , 1(ǫ ⊗ ǫ) (using the fact that µ is a morphism of coalgebras) whence S • µ is the unique right inverse of µ. A computation shows that also µ • τ • (S ⊗ S) is a right inverse of µ, whence we get statement 2. by uniqueness of right inverses (Lemma B.1). 3. The second statement obviously implies the third, and it is easy to see by straight-forward computations that the third statement implies the first and the second.
and all the three statements are equivalent. In order to see that every such element c is a generalized left unit element pick y ∈ H and There is the following right Hopf algebra analogue of the Suschkewitsch decomposition theorem for right groups (see Appendix B): Theorem 2.5 Let (H, ∆, ǫ, 1, µ, S) a cocommutative right Hopf algebra. Then the following holds: 1. The K-submodule (H1, ∆| H1 , ǫ| H1 , µ H1⊗H1 , S| H1 ) is a unital Hopf subalgebra of (H, ∆, ǫ, 1, µ, S).

The map
is an isomorphism of right Hopf algebras whose inverse Ψ −1 is the restriction of the multiplication map.
Proof: 1. It is easy to see that H1 equipped with all the restrictions is a unital bialgebra. Note that for all a ∈ H whence S| H1 is also a left antipode. It follows that H1 is a Hopf algebra.
2. Since the property of being an generalized idempotent is a K-linear condition, showing that the the restriction of S to E H is the trivial right antipode.
3. It is clear from the two preceding statements that Ψ is a well-defined linear map into the tensor product of two cocommutative right Hopf algebras. We have for all x ∈ H because all the terms S(a (2) )a (3) and the components c (1) , . . . of iterated comultiplications of generalized idempotents can be chosen in E H (since the latter has been shown to be a subcoalgebra), and are thus generalized left unit elements (Lemma 2.2, 3.). Hence Ψ is a K-linear isomorphism. Moreover, it is easy to see from its definition that Ψ is a morphism of C 3 -coalgebras.
Next we compute for all a, a ′ ∈ H and c, c ′ ∈ E H : and -since c is a generalized left unit element- showing that Ψ −1 and hence Ψ is a morphism of left-unital algebras. Finally we obtain thanks to Lemma 2.2, and Ψ intertwines right antipodes. ✷ Note that the K-submodule of all generalized left unit elements of a right Hopf algebra H is given by K1 ⊕ Φ −1 (H1 ⊗ E + H ) and thus in general much bigger than the submodule E H of all generalized idempotents.
As it is easy to see that every tensor product H ⊗ C of a unital cocommutative Hopf algebra H and a C 4 -coalgebra C (equipped with the left-trivial multiplication and the trivial right antipode) is a right Hopf algebra, it is a fairly routine check -using the preceding Theorem 2.5-that the category of all cocommutative right Hopf algebras is equivalent to the product category of all cocommutative Hopf algebras and of all C 4 -coalgebras.
In the sequel, we shall need the dual left Hopf algebra version where all the formulas have to be put in reverse order: Here every left Hopf algebra is isomorphic to C ⊗ H.

Dialgebras and Rack Bialgebras
Recall (cf. e.g. [20]) that a dialgebra over K is a K-module D equipped with two associative multiplications ⊢, ⊣: (2.44) An element 1 of A is called a bar-unit element of the dialgebra (A, ⊢, ⊣) and (A, 1, ⊢, ⊣) is called a bar-unital dialgebra iff in addition the following holds for all a ∈ A. Moreover, we shall call a bar-unital dialgebra (A, 1, ⊢, ⊣) balanced iff in addition for all a ∈ A Clearly each associative algebra is a dialgebra upon setting ⊢=⊣ equal to the given multiplication. The class of all (bar-unital and balanced) dialgebras forms a category where morphisms preserve both multiplications and map the initial bar-unit to the target bar-unit. These algebras had been introduced to have a sort of 'associative analogue' for Leibniz algebras. More precisely, there is the following important fact, which can easily be checked, see e.g. [20]: is a Leibniz algebra, denoted by A − .
In fact this construction is well-known to give rise to a functor A → A − from the category of all dialgebras to the category of all Leibniz algebras in complete analogy to the obvious functor from the category of all associative algebras to the category of all Lie algebras. An important construction of (bar-unital) dialgebras is the following: Let (B, 1 B ) be a unital associative algebra over K, and let A be a K-module which is a B-bimodule, i.e. there are K-linear maps B ⊗A → A and A⊗B → Then it is not hard to check that the two multiplications ⊢, ⊣: equip A with the structure of a dialgebra. If in addition there is an element 1 ∈ A such that Φ(1) = 1 B , then (A, 1, ⊢, ⊣) will be a bar-unital dialgebra. We shall call this structure (A, Φ, B) an augmented dialgebra. ✸ In fact, every dialgebra (A, ⊢, ⊣) arises in that fashion: Consider the K-submodule I ⊂ A whose elements are linear combinations of arbitrary product expressions p a 1 , . . . , a r−1 , (a r ⊢ b r − a r ⊣ b r ), a r+1 , . . . , a n (where all reasonable parentheses and symbols ⊢ and ⊣ are allowed) for any two strictly positive integer r ≤ n, and a , . . . , a n , b r ∈ A. It follows that the quotient module A/I is equipped with an associative multiplication induced by both ⊢ and ⊣. Let A 1 ass be equal to A/I if A is bar-unital: In that case, the bar-unit 1 of A projects on the unit element of A/I; and let A 1 ass be equal to A/I ⊕ K (adjoining a unit element) in case A does not have a bar-unit. Thanks to the defining equations (2.42), (2.43), (2.44), it can be shown by induction that for any strictly positive integer n, any a 1 , . . . , a n , a ∈ A, and any product expression made of the preceding elements upon using ⊢ or ⊣ p(a 1 , . . . , a n ) ⊢ a = a 1 ⊢ · · · ⊢ a n ⊢ a = (a 1 ⊣ · · · ⊣ a n ) ⊢ a, a ⊣ p(a 1 , . . . , a n ) = a ⊣ a 1 ⊣ · · · ⊣ a n = a ⊣ (a 1 ⊢ · · · ⊢ a n ), proving in particular that I acts trivially from the left (via ⊢) and from the right (via ⊣) on A such that there is a well-defined A 1 ass -bimodule structure on A such that the natural map Φ A : A → A 1 ass is a bimodule morphism. Hence (A, Φ A , A 1 ass ) is always an augmented dialgebra, and the assignment A → (A, Φ A , A 1 ass ) is known to be a faithful functor. Note also that this construction allows to adjoin a bar-unit to a dialgebra (A, ⊢, ⊣): Consider the K-moduleÃ := A ⊕ A 1 ass with the obvious A 1 assbimodule structure α.(b + β) = α.b + αβ and (b + β).α = b.α + βα for all α, β ∈ A 1 ass and b ∈ A. Observe that the obvious map ΦÃ :Ã → A 1 ass defined by ΦÃ(b + β) = Φ A (b) + β is an A 1 ass -bimodule map, and that 1 = 1 A 1 ass is a bar-unit. The bar-unital augmented dialgebra (Ã, ΦÃ, A 1 ass ) is easily seen to be balanced. There are nonbalanced bar-unital dialgebras as can be seen from the augmented bar-unital dialgebra example ( is any unital associative algebra and the bimodule action is Again, in case the dialgebra (A, ⊢, ⊣, 1) is bar-unital and balanced, note descends to a surjective morphism of associative algebras A 1 ass → A ′ by the above, it is clear that the ideal I contains the kernel of π A . On the other hand, if a ∈ Ker(π A ) then 0 = π(a) = 1 ⊢ a, and obviously a = 1 ⊢ a − 1 ⊣ a ∈ I, thus inducing a useful isomorphism A 1 ass ∼ = A ′ , and thus a subalgebra injection In this work, we also have to take into account coalgebra structures and thus define the following: We have used a relatively simple notion of one single compatible coalgebra structure motivated from differential geometry, see Section 3. In contrast to that, F. Goichot uses two a priori different coalgebra structures, see [12]. Moreover, a slightly more general context would have been to demand the existence of two different antipodes, a right antipode S for ⊢, and a left antipode S ′ for ⊣. The theory -including the classification in terms of ordinary Hopf algebras-could have been done as well, but we have refrained from doing so since it is not hard to see that such a more general Hopf dialgebra is balanced iff S = S ′ . This fact is crucial in the following refinement of Proposition 2.4: Proof: Let x, y ∈ A be primitive. Then, using that ⊢ and ⊣ are morphisms of coalgebras, we get The first relationship with rack bialgebras is the following simple generalization of a cocommutative Hopf algebra equipped with the adjoint representation: Proposition 2.6 Let (A, ∆, ǫ, 1, ⊢, ⊣, S) be cocommutative Hopf dialgebra. Define the following multiplication µ : (2) ) . (2.50) Then we have the following: 1. The map ✄ defines on the K-module A two left module structures, one with respect to the algebra (A, 1, ⊢), and one with respect to the algebra (A, 1, ⊣), making the Hopf-dialgebra (A, ∆, ǫ, 1, ⊢, ⊣, S) a module-Hopf dialgebra, i.e.
where µ ⊢ and µ ⊣ stand for the multiplication maps ⊢ and ⊣, and this is clearly a composition of morphisms of C 3 -coalgebras whence µ is a morphism of C 3coalgebras proving eqn (2.52). Next, there is clearly 1 ⊲ b = b for all b ∈ A, and, since the dialgebra is balanced, we get for all a ∈ A proving eqs (2.51). Next, for all a, a ′ , a ′′ ∈ A, we get where -in the second to last equation-we have used the left antipode identity for the case ⊣ and the fact that (a) S(a (1) ) ⊢ a (2) is a generalized left unit element for the case ⊢. It follows that (A, ⊣ ⊢ ) is an A-module-algebra proving eqs (2.53) and (2.54). 2. It remains to prove self-distributivity: For all a, b, c ∈ A, we get and in the end proving the self-distributivity identity.

✷
The next theorem relates augmented cocommutative rack bialgebras with cocommutative Hopf dialgebras: Theorem 2.6 Let (B, Φ B , H, ℓ) be a cocommutative augmented rack bialgebra. Then the K-module (B ⊗ H, ∆ B⊗H , ǫ B ⊗ ǫ H , 1 B ⊗ 1 H , Φ, H) will be an augmented cocommutative Hopf dialgebra by means of the following definitions. Here we use Example 2.5 and take h, h ′ ∈ H and b ∈ B: Moreover, the Leibniz bracket on the K-module of all primitive elements of B ⊗H, a := Prim(B)⊕Prim(H), is computed as follows for all x, y ∈ Prim(B) and all ξ, η in the Lie algebra Prim(H) (writing x and ξ for the more precise where each bracket is of the form (2.48). Note that this Leibniz algebra is split over the Lie subalgebra Prim(H), the complementary two-sided ideal {x − Φ B (x) | x ∈ Prim(B)} being in the left center of a.
Proof: It is clear from the definitions that condition 2 defines a H-bimodule structure on C ⊗ H making it into a module C 3 -coalgebra. Moreover, we compute for all h, h ′ , h ′′ ∈ H and b ∈ B whence Φ is a morphism of H-bimodules. Next, we get for all b ∈ B and h ∈ H: proving the right antipode identity, and proving the left antipode identity. Finally for all h ∈ H we get implying that the bar-unital dialgebra is balanced. Formula (2.57) is straight-forward:  (2) ), (2.58) and the transferred multiplications ⊢ ′ and ⊣ ′ and the antipode S ′ on which is isomorphic to B as a C 4 -coalgebra of B ⊗ H. Proof: 1. Note first that the right hand side of eqn (2.58) is just a ⊲ c of Proposition 2.6 which had been shown to be a left module-Hopf dialgebra action of (A, ⊣) and of (A, ⊢) on A. Observe that for all a, a ′ ∈ A whence the H A -action ℓ is well-defined on A. Moreover we compute for all h ∈ H A and all a, a ′ , a ′′ ∈ A such that h = 1 ⊣ a: whence (A, ⊣) is also a H A -module-algebra. Now let c ∈ E H . By definition, c is a generalized idempotent (w.r.t. ⊣), hence c = (c) c (1) ⊣ c (2) , and thus for all Recall the Suschkewitsch decomposition of the left Hopf algebra (A, ∆, ǫ, 1, ⊢, S) where one can use Theorem 2.5 and dualize all the formulas: a (3) ).
Formulas (2.60) and (2.61) consequences of Theorem 2.5. The only formula which remains to be shown is eqn (2.59). Note first that every generalized idempotent c ∈ E H (w.r.t. ⊣) is also a generalized idempotent with respect to ⊢. Indeed, since all the components c (1) and c (2) in ∆(c) = (c) c (1) ⊗ c (2) can be chosen in E H , we get Next for all c, c ′ ∈ E H , h, h ′ ∈ H H , and a, a ′ ∈ a such that 1 ⊣ a = h and 1 ⊣ a ′ = h ′ , we get -since c is a generalized left unit element (w.r.t. ⊢)- and this is equal to proving eqn (2.59).

Clear for any bar-unital balanced dialgebra.
3. Straight-forward computation using Prim(H) = Prim(E H ) ⊕ Prim(H H ) where the latter is well-known to be a Lie algebra and the former is abelian. 4. For each b ∈ b and h ∈ H, we get proving the first statement. Moreover proving the form of the generalized idempotents, and since the K-linear map The third statement had been proved by Simon Covez in his Master thesis [6] in the differential geometric context of digroups, compare with Section 3.

Example 2.6
As an example, let us compute the Suschkewitsch decomposition for the augmented rack bialgebra K[X] where p : X → G is an augmented pointed rack, see Example 2.4. By the above theorem, part 4., its associated cocommutative augmented Hopf dialgebra decomposes as B ⊗H, where the Hopf algebra H = K[G] is the standard group algebra and B = K[X]. The generalized idempotents are in this case ✸ We finish this section with a formula relating universal algebras: The functor associating to any dialgebra A its Leibniz algebra A − via eqn (2.48) is well-known to have a left adjoint (see e.g. [20]) associating to any Leibniz algebra (h, [ , ]) its (in general non bar-unital) universal enveloping dialgebra Ud(h) associated to h defined by where we recall the natural injection of unital algebras i A : A 1 ass → A given by i A Φ A (a) = 1 ⊣ a for all a ∈ A. We shall show thatφ is a morphism of augmented dialgebras: We compute for all u, u ′ , u ′′ ∈ U(h), using that i A and U(ϕ) are morphisms of unital associative algebras and that in the image of i A , we can use the multiplication symbols ⊢ and ⊣ arbitrarily: showing the fact thatφ preserves the bimodule structures on the first component of Ud(h). Next we have for all and by induction on k in u ′ = p(x 1 ) · · · p(x k ) ∈ U(h) and x 1 , . . . , x k ∈ h, we prove Now, for all u ′ , u ′′ , v ∈ U(h) and x ∈ h, we get: showing the fact thatφ preserves the bimodule structures on the second component of Ud(h). Henceφ is a morphism of bar-unital (augmented) dialgebras. The uniqueness ofφ follows from the universal property of U(h) ✷

Coalgebra Structures for pointed manifolds with multiplication
In this section, the symbol K denotes either the field of all real numbers, R, or the field of all complex numbers, C. We define here the monoidal category of pointed manifolds, and exhibit the Serre functor sending a pointed manifold to the coalgebra of point-distributions supported in the distiguished point.
We recall further that this is a strong monoidal functor. Further down, we will study Lie (semi) groups, Lie racks, and Lie digroups as examples of this construction, motivating geometrically the notions of a rack bialgebra and of a Hopf dialgebra.

Pointed manifolds with multiplication(s)
Recall (3.1) The obvious generalization are a finite number of maps M ×n → M with n ≥ 1) arguments. Again by forgetting about differentiable structures, we get the category of pointed sets with multiplication. such that for all f ∈ D(M × N) we have

Coalgebra Structure for distributions supported in one point
Note also that it can be shown that the right hand side is equal to T S (1) (f ) where the notation is self-explanatory. Furthermore, it is not hard to see that for two distributions supported in one point, i.e. S ∈ E ′ e 1 (M) and T ∈ E ′ e 2 (N) the distribution F  [21]), and for three vector spaces V, W, X over K, let β V,W,X : V ⊗ (W ⊗ X) → (V ⊗ W ) ⊗ X be the well-known associator for the monoidal category of all vector spaces. By using the definitions, it is not hard to see that the following identity holds (3.11) Moreover, note that the map F 0 = ǫ pt (see eqn (3.5)) defines an isomorphism of E ′ pt ({pt}) to K which had already been seen to be natural. As a result, the functor F is a monoidal functor in the sense of [21, p.255-257]. Moreover, since the category Mf * is even a symmetric monoidal category by means of the canonical flip map τ M,N : M ×N → N ×M : (x, y) → (y, x), see e.g. [21, p.252-253], and the monoidal category K-vect is also symmetric, it is not hard to see that the monoidal functor is also symmetric, see e.g. [21, p.257] for definitions.
We shall now show that the monoidal functor F is strong, i.e. that F 0 = ǫ pt and F 2 M,N are isomorphisms. This is clear for ǫ pt . Recall that for each distribution T in E ′ e (V ) (where V is a nonempty open set in R m containing the point e), there is nonnegative integer l (called the order of the distribution) such that where c k ∈ K for each multi-index k, see e.g. [24,p.150,Thm. 6.25]. In a slightly more algebraic manner we can express this as follows: let E be a finite-dimensional real vector space, let V ⊂ E be an open set containing e ∈ E. Then we have the following linear isomorphism Φ S : S(E) → E ′ e (V ) given by Φ S (1) = δ e and for any positive integer k and vectors w (1) , . . . , w (k) ∈ E and f ∈ C ∞ (V, K) where • denotes the commutative multiplication in the symmetric algebra, see Appendix A. Using the fact that the inclusion map ι Uα : U α → M of any chart domain of M such that e ∈ U α defines an isomorphism ι Uα * : E ′ e (U α ) → E ′ e (M), and that any chart ϕ α : U α → V α ⊂ R m defines an isomorphism ϕ α * : E ′ e (U α ) → E ′ ϕα(e) (V α ), we can conclude that there is a linear isomorphism with the symmetric coalgebra S(R m ) on R m (see Appendix A) computed as follows (3.14) where we write R m ∋ w (j) = m i=1 w (j)i e i where all the w (j)i are real numbers and e 1 , . . . , e m is the canonical base of R m . Note that for the particular case of M being an open set V of R m and the chart ϕ α being the identity map the map Φ α (see eqs (3.13) and (3.14)) coincides with the canonical map Φ S , see eqn (3.12).
For two pointed manifolds (M, e 1 ) and (N, e 2 ) and given charts (U α , ϕ α ) of M such that e 1 ∈ U α and (Ũ β ,φ β ) of N such that e 2 ∈Ũ β , we thus have linear isomorphisms Φ α : S(R m ) → E ′ e 1 (M),Φ β : S(R n ) → E ′ e 2 (N), and Φ α,β : S(R m+n ) → E ′ (e 1 ,e 2 ) (M ×N) (upon using the product chart (U α ×Ũ β , ϕ α ×φ β )). Using the above definitions, one can compute that where Θ m,n : S(R m ) ⊗ S(R n ) → S(R m+n ) denotes the natural isomorphism of commutative associative unital algebras induced by the obvious inclusions R m ֒→ R m+n (first m coordinates) and R n ֒→ R m+n (last n coordinates). It follows that the natural map and is thus a linear isomorphism, whence the functor F is a strong monoidal functor.
In order to define more structure, let us consider the well-known diagonal for all x ∈ M. Clearly, diag M is a smooth map of pointed manifolds (M, e) → M × M, (e, e) . Moreover, the diagonal map is clearly natural in the sense This definition has avatars with more than two tensor factors. Indeed, observe that the naturality relation (3.11) implies for φ = id M and ψ = diag M that Similarly, we have relations of this type for any number of tensor factors. In the following, we invite the reader to look again at Appendix A for definitions and notations about coalgebras. We have the following Theorem 3.1 With the above notations: 1. The K-vector space E ′ e (M) equipped with the linear maps ∆ e (cf. eqn (3.15)), ǫ e (cf. eqn (3.5), and 1 e (cf. eqn (3.6) is a C 5 -coalgebra which is (non canonically) isomorphic to the standard symmetric coalgebra S(R m ), ǫ, ∆, 1 .
2. The above strong symmetric monoidal functor F extends to a functor -also denoted by F -from Mf * to the symmetric monoidal category of C 5 -coalgebras over K.
Starting for example on the left hand side, one replaces the map diag M * by F 2 M,M • ∆ e , and also the map (id M × diag M ) * by . Now one observes that one may apply the relation (3.10) on the left hand side. One obtains (d) connectedness of ∆ e : The coalgebra E ′ e (M ) is isomorphic to the symmetric algebra S(R n ), and the latter is connected. This shows part (1) of the statement, as the isomorphy to the standard symmetric coalgebra has been shown above.
The only thing which has to be shown for the second statement is the preservation of the coalgebra structure on the level of morphisms, which is clear.
For the third part, consider the linear map T e M → Prim(E ′ e (M )) (see Appendix A for the definition of the primitives Prim(C) of a coalgebra C) defined by Indeed the right hand side is clearly in E ′ e (M ), and the Leibniz rule for the derivative shows that this is in Prim(E ′ e (M )). Moreover the above map is clearly injective, and since Prim(S(R n )) = R n and dim(T e M ) = n it follows that the above map is an isomorphism of real vector spaces. The naturality is a simple computation. ✷ The last statement means that the composed functor Prim • F of the Serre functor F and the functor associating to any coalgebra C its space of primitive elements, Prim(C), is naturally isomorphic to the tangent functor T * associating to any pointed differentiable manifold (M, e) its tangent space T e M.

Remark 3.1
There is neither a canonically defined (i.e. not depending on the choice of a chart) projection from the coalgebra to its primitives, so the coalgebras E ′ e (M) are isomorphic to the cofree S(R m ), but in general not naturally, nor a canonically defined commutative multiplication (the classical convolution of distributions of compact support which needs the additive vector space structure). ✸

Remark 3.2
Note also the disjoint union x∈M E ′ x (M) (k) carries the structure of a smooth vector bundle over M: Its smooth sections coincide with the space of all differential operators of order k. ✸

Remark 3.3
In case U ⊂ R m and V ⊂ R n are pointed open sets, the coalgebra morphism φ * of a smooth map φ : U → V of pointed manifolds is isomorphic to the coalgebra morphism S(R m ) → S(R n ) induced by the jet of infinite order of φ at the distinguished point e of U, j ∞ (φ) e , see e.g. [15] for further information.

Pointed manifolds with multiplication and their associated bialgebras
We can now apply the Serre functor defined in the preceding Section 3.2 to pointed manifolds with multiplication: which is a morphism of C 5 -coalgebras. In case m is left-unital (resp. right unital), the nonassociative C 3 I-algebra E ′ e (M) is left regular (resp. right regular) Proof: The map µ exists and is linear by functoriality. We have trivially and this shows that µ is a morphism of coalgebras by translating diag M into ∆ e using as before the maps of type F 2 . The regularity statements are a consequence of the connectedness of the C 3 -coalgebra E ′ e (M ), see Lemma 2.1. ✷ In the following, we shall enumerate some important (sub)categories of pointed differentiable manifolds with multiplications.

Lie groups and universal enveloping algebras
Let G, m, e, ( ) −1 a Lie group. The following theorem is well-known (see [26]): The associated coalgebra with multiplication µ of the Lie group G, m, e, ( ) −1 is an associative unital bialgebra (in fact, a Hopf algebra) isomorphic to the universal enveloping algebra of the Lie algebra g = T e G of G.
We just indicate the isomorphism: For any ξ ∈ g, let ξ + denote the left invariant vector field ξ + (g) := T e L g (ξ) generated by its value ξ ∈ g = T e G. Then the map Φ U : U(g) → F (G) is given by (for all k ∈ N, ξ 1 , . . . , ξ k ∈ g and f ′ ∈ C ∞ (G, K)) where L X denotes the Lie derivative in the direction of the vector field X. Note that the identities for the inverse map g → g −1 can be written as (3.17) and an application of the functor F gives the convolution identities for the antipode, defined by S = ( ) −1 * .

Lie semigroups and Lie monoids
It is easy to see but presumably less known that the result of the preceding subsection remains true for a Lie monoid G, m, e : Theorem 3.4 1. The associated coalgebra with multiplication µ of the Lie monoid G, m, e is an associative unital bialgebra (in fact a Hopf algebra) isomorphic to the universal enveloping algebra of a Lie algebra g ∼ = T e G.
2. The associated coalgebra with multiplication µ of the right Lie group (G, m, e, ( ) −1 ) is a right Hopf algebra.
In order to see the first statement note that it is clear that the associated coalgebra C := F (G) carries an associative unital multiplication µ = m * •F 2 G,G . The fact that the coalgebra is always connected implies by the Takeuchi-Sweedler argument (see Appendix A) that the identity map id C has a convolution inverse, and is thus a Hopf algebra. Since the coalgebra C is connected and cocommutative, it follows from the Cartier-Milnor-Moore Theorem (see e.g. [22]) that the Hopf algebra F (G) is isomorphic to the universal enveloping algebra over the Lie subalgebra g of its primitive elements which is equal to T e G. The second statement is an immediate consequence of the functorial properties of F .

(Lie) dimonoids and digroups
Recall that a Lie dimonoid (see e.g. [20]) is a pointed differentiable manifold (D, e) equipped with two smooth associative multiplications D × D → D, written (x, y) → x ⊢ y and (x, y) → x ⊣ y (and preserving points, i.e. e ⊢ e = e = e ⊣ e), such that the dialgebra conditions eqs ( for all g, g ′ ∈ g and x ∈ D, and let f : (D, e D ) → (G, e) be a smooth map of pointed manifolds such that for all g, g ′ ∈ G and x ∈ D Then the pointed manifold D, e D , ⊢, ⊣ will be a (bar-unital) dimonoid by setting x ⊢ y := f (x)y and x ⊣ y := xf (y). (3.19) A general Lie digroup is defined (according to Liu, [18, Definition 1.1]) to be a (bar-unital) dimonoid (D, e, ⊢, ⊣) such that the left unital Lie semigroup (D, e, ⊢) is a right group and the right unital Lie semigroup (D, e, ⊣) is a left group (see Appendix B for definitions): Here the right inverse of x with respect to ⊢ does in general not coincide with the left inverse of x with respect to ⊣. For an example, take any Lie group G, set (D, e D ) = (G×G, (e, e) , define the two canonical G-actions g(g 1 , g 2 ) := (gg 1 , g 2 ) and (g 1 , g 2 )g := (g 1 , g 2 g) (for all g, g 1 , g 2 ∈ g), and let f : G × G → G be the group multiplication. Then (D, e D , ⊢, ⊣) will be a general digroup with (g 1 , g 2 ) −1 ⊢ = (g −1 2 g −1 1 , e) and (g 1 , g 2 ) −1 ⊣ = (e, g −1 2 g −1 1 ). In [14, Definition 4.1] Kinyon defines a Lie digroup as a general Lie digroup such that in addition for each x its right inverse (w.r.t. to ⊢) is equal to its left inverse (w.r.t. ⊣). This can be shown to be equivalent to demanding that the general Lie digroup (D, e, ⊢, ⊣) be balanced. Again using the Suschkewitsch decomposition Theorem (which applies in case the underlying manifold is connected), it is not hard to see that the category of all connected Lie digroups (in the sense of Kinyon) is equivalent to the category of all left G-spaces, i.e. whose objects are pairs (G, X) where G is a connected Lie group and X is a pointed connected left G-space (i.e. the distinguished point of X is a fixed point of the G-action) with obvious morphisms. Recall that the Lie digroup is given by X × G equipped with the point (e X , e) and the two multiplications (x 1 , g 1 ) ⊢ (x 2 , g 2 ) = (g 1 x 2 , g 1 g 2 ) and (x 1 , g 1 ) ⊣ (x 2 , g 2 ) = (x 1 , g 1 g 2 ) for all x 1 , x 1 ∈ X and g 1 , g 2 ∈ G.
The following theorem is a direct consequence of the functorial properties of the functor F :

(Lie) racks
Recall that a Lie rack is a pointed manifold with multiplication (M, e, m) satisfying the following identities for all x, y, z ∈ M where the standard notation is m(x, y) = x ⊲ y In addition, one demands that (M, e, m) be left-regular, i.e. for all x ∈ M the left multiplication maps L x : y → x ⊲ y should be a diffeomorphism. Note the following version of the self-distributivity identity (3.22) in terms of maps: Note that every pointed differentiable manifold (M, e) carries a trivial Lie rack structure defined for all x, y ∈ M by x ⊲ 0 y := y, (3.24) and this assignment is functorial. ✸ Example 3.2 Any Lie group G becomes a Lie rack upon setting for all g, g ′ ∈ G g ⊲ g ′ := gg ′ g −1 , (3.25) again defining a functor from the category of Lie groups to the category of all Lie racks. Examples of racks which are not the conjugation rack underlying a group abound, for example, every conjugation class and every union of conjugation classes in a group (defining an immersed submanifold) in a Lie group is a Lie rack. ✸ A straight-forward computation shows that the pointed manifold (M, e) equipped with the gauged multiplication ⊲ f defined by Furthermore, recall that an augmented Lie rack (see [11]) (M, φ, G, ℓ) consists of a pointed differentiable manifold (M, e M ), of a Lie group G, of a smooth map φ : M → G (of pointed manifolds), and of a smooth left G-action ℓ : Note that the trivial Lie rack structure of a pointed manifold (M, e) comes from an augmented Lie rack over the trivial Lie group G = {e}. Let (M, e, ⊲) be a Lie rack. Applying the functor F we get the following Theorem 3.6 The associated coalgebra F (M) with multiplication µ of the Lie rack M, e, m is a rack bialgebra, i.e. satisfying for all a, b, c ∈ C, using the same notation a ⊲ b for µ(a ⊗ b): Proof: We evaluate this formula for S = 1. This gives the distribution f → 1(T (2) (f • ✄)). But T (2) means that the function is seen as function of its second variable, i.e. T (2) On the other hand, the delta distribution 1 evaluates a function in e, thus because e ✄ y = y for all y ∈ M . This shows 1 ✄ T = T .
(2.5) Exchanging the roles of the two variables in the above computation, we obtain for T ✄ 1 the distribution T (1 (2) (f • ✄)) or in other words 1(T (1) (f • ✄)), i.e. the above element y is now in the second place. We obtain This shows T ✄ 1 = ǫ(T )1.
(2.6) As remarked before, the definition of ∆ e , namely ∆ e = F −1 2 M,M • diag M * , induces thanks to the naturality relation (3.11) relations like Therefore, starting from the relation induced on E ′ e (M ) by relation (3.23), one replaces (diag M × id M × id M ) * by the above and obtains finally an equation equivalent to equation (2.6).

Remark 3.4
This theorem should be compared to Proposition 3.1 in [4]. In [4], the authors work with the vector space K[M] generated by the rack M, while we work with point-distributions on a Lie rack M. In some sense, we extend their Proposition 3.1 "to all orders". Observe however that their structure is slightly different (motivated in their Remark 7.2). ✸ We get a similar theorem for an augmented Lie rack: Let g denote the Lie algebra of the Lie group G, then we have the Theorem 3.7 The associated coalgebra C with multiplication µ of an augmented Lie rack (M, φ, G, ℓ) is a cocommutative augmented rack bialgebra (C, φ * , U(g), ℓ) We shall close the subsection with a geometric explanation of some of the structures appearing in Subsection 2.1: Let h, [ , ] be a real finitedimensional Leibniz algebra. Then for any real number , there is the following Lie rack structure on the manifold h defined by x ◮ y := e adx (y) (3.35) For later use we note that on the space h[[ ]] of all formal power series the above formula makes sense if x, y are also formal power series. Moreover, pick a two-sided ideal z ⊂ h with Q(h) ⊂ z ⊂ z(h) so that the quotient algebra g := h/z is a Lie algebra. Let p : h → g be the canoncial projection. Let G be the connected simply connnected Lie group having Lie algebra g. Since g acts on h as derivations, there is a unique Lie group action ℓ of G on h by automorphisms of Leibniz algebras. Consider the smooth map Clearly φ(g.x) = gφ(x)g −1 for all x ∈ h and g ∈ G whence (h, φ, G, ℓ) is an augmented Lie rack, and it is not hard to see that the Lie rack structure coincides with (3.35) for = 1. Proof: First we compute φ * = exp * •p * . Since p : h → g is linear, it is easy to see using formula (3.12) that for all k ∈ N and x 1 , . . . , see (3.12) for a definition of Φ S . Next, for all k ∈ N and ξ 1 , . . . , ξ k ∈ g, we shall show the formula (for all f ′ ∈ C ∞ (G, K)) 16 for a definition of Φ U ). Both sides of this equation are symmetric k-linear maps in the arguments ξ 1 , . . . , ξ k , hence by the polarization Lemma (see e.g. [29]), it suffices to check equality in case ξ 1 = · · · = ξ k = ξ. Since for each real number t the map g → F ξ t (g) := g exp(tξ) is the flow of the left invariant vector field ξ + , we get proving the above formula. It follows that Next, we compute ℓ * . We get for positive integers k, l, ξ 1 , . . . , ξ k ∈ g, x ∈ h, and f ∈ C ∞ (h, K): where in the last line we have used a basis of h, have written y 1 , . . . , y n (n = dim(h)) for the components of each vector y ∈ h, and used the notation ℓ s for the linear map ℓ exp(s 1 ξ 1 ) • · · · • ℓ exp(s k ξ k ) . By induction on k it is easy to prove that and using again the Polarisation Lemma, we finally get for all u ∈ U(g) and (3.38) and the isomorphism with the augmented rack bialgebra UAR ∞ (h) = S(h) is established. ✷

Remark 3.5
Observe that the Serre functor can be rendered completely algebraic, i.e. for example for an algebraic Lie rack R (meaning that the underlying pointed manifold is a smooth algebraic variety and the rack product is algebraic), one can take as its Serre functor image F (R) the space of derivations along the evaluation map in the distinguished point. This gives a new and completely algebraic way to associate to a Lie rack its tangent Leibniz algebra. ✸

Deformation quantization of rack bialgebras 4.1 Deformation quantization via an explicit formula
In this subsection, K denotes the field of real numbers R or the field of complex numbers C. Let (h, [ , ]) be a finite dimensional Leibniz algebra of dimension n, and denote by h * its linear dual. In order to make computations more elementary we shall use a fixed basis e 1 , . . . , e n of h, but it is a routine check that all the relevant formulas are invariant under a change of basis. Let e 1 , . . . , e n be the corresponding dual basis of h * , i.e. by definition e i (e j ) = δ ij , for all i, j = 1, . . . , n. Furthermore, let c k ij for i, j, k = 1, . . . , n be the structure constants of the Leibniz algebra h with respect to the basis e 1 , . . . , e n , i.e. c k ij = e i ([e j , e k ]) for all i, j, k = 1, . . . , n. We will denote by x, y, z, . . . elements of h, while α, β, γ, . . . will denote elements of h * . Denote by α 1 , . . . , α n the coordinates of α ∈ h * with respect to the basis e 1 , . . . , e n . For all x ∈ h, denote bŷ x ∈ C ∞ (h * , K) the linear function given bŷ for all α ∈ h * . In the same vein, let ex be the exponential function given by for all α ∈ h * . For all integers i = 1, . . . , n, define a first order differential operator ad i on smooth functions f : h * → K by The following star-product formula, where is a formal parameter (which may be replaced by a real number in situations where the formula is convergent), will render h * a quantum rack in the sense of [9]. Let f, g ∈ C ∞ (h * , K). Proof: The proof of the theorem relies on the following Lemmas: The map "hat"ˆ: h → C ∞ (h * , K) which sends x ∈ h to the linear functionx extends to an injective morphism of commutative associative unital algebras Ψ : for all integers k and all x 1 , . . . , x k ∈ h.
Proof: This follows immediately from the freeness property of the algebra S(h). ✷ for all a ∈ S(h).
Proof: Indeed, it is enough to show this for x ∈ h ⊂ S(h) as both adjoint actions are derivations. Now we have for α ∈ h * : where the left-hand ✄ is the rack multiplication in the rack bialgebra S(h).
Proof: First of all, note that by linearity it is enough to show this for x 1 , . . . , x r = e i 1 , . . . , e ir with i 1 , . . . , i r ∈ {1, . . . , n}. By eqn (2.30), we have Applying Ψ gives then by the previous lemma. Now compute This expression is non zero only if k = r and {i 1 , . . . , i k } = {j 1 , . . . , j k }. In this case, the result is 1. One deduces the asserted formula. ✷ Now we come back to the proof of the theorem. The assertion of the theorem is the equality: ex ✄ eŷ = e x◮ y .
Summing up the assertion of the previous lemma (taking x 1 = . . . = x r = x), we obtain: and thus (as the rack product in S(h) is given by the adjoint action, using also that Ψ is multiplicative) This extends then to the asserted formula using that e adx is an automorphism of S(h) (because it is the exponential of a derivation). ✷ Corollary 4.1 The above defined star-product induces the structure of a rack with respect to the product ✄ on the set of exponential functions on h * , and this star-product is opposite to the star-product found in [9].
Proof: Via the formula of the theorem, the self-distributivity property of the rack product ◮ in the rack bialgebra S(h) translates into the self-distributivity property of ✄ on the set of exponential functions. Since the star-product defined in [9] is a series of bidifferential operators, and since such a series is uniquely determined by its values on exponential functions, the present star-product coincides with the one found in [9] thanks to the statement of the preceding theorem. ✷

General deformation theory for rack bialgebras
In this section, (R, ∆, ǫ, µ, 1) is a cocommutative rack-bialgebra over a general commutative ring K, and we use the notation r ✄s to denote the rack product µ(r⊗s) of two elements r and s of R. In this subsection, we will often drop the symbol Σ in Sweedler's notation of (iterated) comultiplications, so that the n-iterated comultiplication of r in R reads r (1) ⊗ · · · ⊗ r (n) := (∆ ⊗ Id ⊗n−1 ) • · · · • ∆(r) denote the K-algebra of formal power series in the indeterminate with coefficients in K. If V is a vector space over K, V stands for V [[ ]]. Recall that if W is a K-module, a K -linear morphism from V to W is the same as a power series in with coefficients in Hom K (V, W ) via the canonical map This identification will be used without extra mention in the following.  The self-distributivity relation is shown in a way very similar to the proof of Theorem 4.1, see eqn (4.2). ✸ As in the classical setting of deformation theory of associatice products, we will relate our deformation theory of rack products to cohomology. For this, let us first examine an introductory example: Example 4.2 Let (R, ✄) be a rack bialgebra, and suppose there exists a deformation ✄ = ✄ + ω of ✄. The new rack product ✄ should satisfy the self-distributivity identity, i.e. for all a, b, c ∈ R To the order 0 , this is only the self-distributivity relation for ✄. But to order 1 (neglecting order 2 and higher), we obtain: ω(a, b✄c)+a✄ω(b, c) = ω(a (1) ✄b, a (2) ✄c)+ω(a (1) , b)✄(a (2) ✄c)+(a (1) ✄b)✄ω(a (2) , c).
It will turn out that this is the cocycle condition for ω in the deformation complex which we are going to define. More precisely, we will have Proof: • eqn (4.3): Let us show that the assertion of eqn (4.3) is true for all n and i with 1 ≤ i < n by induction over i. Suppose that the induction hypothesis is true and compute i , r i+2 ,· · ·, r n ) r i , r i+2 ,· · ·, r n ) , which gives, thanks to the self-distributivity relation in the rack algebra R, i , r i+2 ,· · ·, r n ) = r 1 ✄ µ n−1 (r 2 ,· · ·, r n ) = µ n (r 1 , · · · , r n ), where we have used the induction hypothesis. This proves the assertion.
• eqn (4.4): The assertion follows here again from an easy induction using the self-distributivity relation.
✷ If (C, ∆ C ) and (D, ∆ D ) are two coassociative coalgebras and φ : C → D is a morphism of coalgebras, we denote by Coder(C, V, φ) the vector space of coderivations from C to V along φ, i.e. the vector space of linear maps f : C → D such that Let us note the following permanence property of coderivations along a map under partial convolution which will be useful in the proof of the following theorem. For a coalgebra A, maps f : A ⊗ B → V and g : A ⊗ C → V and some product ✄ : V ⊗ V → V , the partial convolution of f and g is the map f ⋆ part g : Lemma 4.4 Let A, B, C and V be coalgebras, V carrying a product ✄ which is supposed to be a coalgebra morphism. Let f : A ⊗ B → V be a coderivation along φ and g : A ⊗ C → V be a coalgebra morphism. Then the partial convolution f ⋆ part g is a coderivation along φ ⋆ part g.
Theorem 4.2 d R is a well defined differential.
Proof: That d R is well defined means that it sends coderivations to coderivations. It suffices to show that this is already true for all maps d n i,1 , d n i,0 and d n n+1 , which is the case. For this, we use Lemma 4.4. Indeed, a cochain ω ∈ C n (R; R) is a coderivation along µ n . By Proposition 4.1, µ n is a coalgebra morphism. On the other hand, it is clear from the formula for d n i,1 that d n i,1 is a partial convolution with respect to the first i − 1 tensor labels of µ i and ω. Therefore the Lemma applies to give that the result is a coderivation along the partial convolution of µ i and µ n , which is just µ n+1 again by Proposition 4.1. This shows that d n i,1 ω belongs to Coder(R ⊗n , R, µ n+1 ) as expected. The maps d n i,0 and d n n+1 can be treated in a similar way.
The fact that d R squares to zero is related to the so-called cubical identities satisfied by the maps d i,1 and the maps d i,0 , namely and auxiliary identities which express the compatibility of the maps d i,1 and d i,0 with d n n+1 , and an identity involving d n n+1 and d n+1 n+2 . One could call this kind of object an augmented cubical vector space.
We will not show the usual cubical relations, i.e. those which do not refer to the auxiliary coboundary map d n n+1 , because these are well-known to hold for rack cohomology, see [8], Corollary 3.12, and our case is easily adapted from there. One possibility of adaptation (in case one works over the real or complex numbers) is to take a Lie rack, write its rack homology complex (with trivial coefficients in the real or complex numbers), and to apply the Serre functor.

Definition 4.4
An infinitesimal deformation of the rack product is a deformation of the rack product over the K-algebra of dual numbersK := K / ( 2 ) , i.e. a linear map µ 1 : R ⊗2 → R such thatR := R ⊗K is a rack bialgebra overK when equipped with µ 0 + µ 1 . Two infinitesimal deformations µ 0 + µ 1 and µ 0 + µ ′ 1 are said to be equivalent if there exists an automorphism φ :R →R of the coalgebra of (R , ∆, ǫ) of the form φ := id R + α such that As usual, being equivalent is an equivalence relation and one has the following cohomological interpretation of the set of equivalence classes of infinitesimal deformations, denoted Def (µ 0 ,K ): The identificaton is obtained by sending each equivalence class [µ 0 + µ 1 ] in Def (µ 0 ,K ) to the cohomology class [µ 1 ] in H 2 (R; R).
Proof: One checks easily that the correspondence is well defined (if µ 0 + µ 1 is an infinitesimal deformation, then µ 1 is a 2-cocycle, see Example 4.2) and that it is bijective when restricted to equivalence classes. ✷ Remark 4.1 (a) The choice of taking coderivations in the deformation complex is explained as follows: The rack product µ is a morphism of coalgebras, and we want to deform it as a morphism of coalgebras with respect to the fixed coalgebra structure we started with. Tangent vectors to µ in Hom coalg (C ⊗ C, C) are exactly coderivations along µ. This is the first step: Deformations as morphisms of coalgebras. Then as a second step, we look for 1-cocycles, meaning that we determine those morphisms of coalgebras which give rise to rack bialgebra structures. The deformation complex in [4] takes into account also the possibility of deforming the coalgebra structure, and we recover our complex by restriction.
(b) Given a Leibniz algebra h, there is a natural restriction map from the cohomology complex with adjoint coefficients of h to the deformation complex of its augmented enveloping rack bialgebra UAR(h). The induced map in cohomology is not necessarily an isomorphism, as the abelian case shows. Observe that the deformation complex of the rack bialgebra K[R] for a rack R does not contain the complex of rack cohomology for two reasons: First, this latter complex is ill-defined for adjoint coefficients, and second, there are not enough coderivations as all elements are set-like. A way out for this last problem would be to pass to completions.

A Some definitions around coalgebras
Let C be a module over a commutative associative unital ring K (which we shall assume to contain Q). Recall that a linear map ∆ : C → C ⊗ K C = C ⊗ C is called a coassociative comultiplication iff ∆ ⊗ id C • ∆ = id C ⊗ ∆ • ∆, and the pair (C, ∆) is called a (coassociative) coalgebra over K. Let (C ′ , ∆ ′ ) be another coalgebra. Recall that a K-linear map Φ : C → C ′ is called a homomorphism of coalgebras iff ∆ ′ • φ = (φ ⊗ φ) • ∆. The coalgebra (C, ∆) is called cocommutative iff τ • ∆ = ∆ where τ : C ⊗ C → C ⊗ C denotes the canonical flip map. Recall furthermore that a linear map ǫ : C → K is called a counit for the coalgebra (C, ∆) iff ǫ ⊗ id C • ∆ = id C ⊗ ǫ • ∆ = id C . The triple (C, ∆, ǫ) is called a counital coalgebra. Moreover, a counital coalgebra (C, ∆, ǫ) equipped with an element 1 is called coaugmented iff ∆(1) = 1 ⊗ 1 and ǫ(1) = 1 ∈ K. Let C + ⊂ C denote the kernel of ǫ. Recall that a morphism φ : (C, ∆, ǫ, 1) → (C ′ , ∆ ′ , ǫ ′ , 1 ′ ) of counital coaugmented coalgebras over K is a K-linear map satifying (φ ⊗ φ) • ∆ = ∆ ′ • φ, ǫ ′ • φ = ǫ, and φ(1) = 1 ′ . Moreover, for any counital coaugmented coalgebra the Ksubmodule of all primitive elements is defined by Every morphism of counital coaugmented coalgebra clearly maps primitive elements to primitive elements, thus defining a functor Prim from the category of counital coaugmented coalgebras to the category of K-modules. Finally, following Quillen [22], we shall call a counital coaugmented coalgebra connected iff the following holds: The sequence of submodules (C (r) ) r∈N defined by C (0) = K1 and recursively by is easily seen to be an ascending sequence of coaugmented counital subcoalgebras of (C, ∆, ǫ, 1), and if the union of all the C (k) is equal to C, then (C, ∆, ǫ, 1) is called connected. We refer to each C (k) as the subcoalgebra of order k. Clearly, each C (k) is connected, and C (1) = K1 ⊕ Prim(C). Moreover, each morphism of counital coaugmented coalgebras maps each subcoalgebra of order k to the subcoalgebra of order k of the target coalgebra thus defining a functor C → C (k) from the category of coaugmented counital coalgebras to itself. We shall use the following acronyms: Definition A. 1 We call a coassociative, counital, coaugmented coalgebra a C 3 -coalgebra. In case the C 3 -coalgebra is in addition cocommutative, we shall speak of a C 4 -coalgebra. Finally, a connected C 4 -coalgebra will be coined a C 5 -coalgebra.
Recall also that the tensor product of two counital coaugmented coalgebras (C, ∆, ǫ, 1) and (C ′ , ∆ ′ , ǫ ′ , 1 ′ ) is given by ( . Tensor products of connected coalgebras are connected. Recall the standard example: Let V be a K-module and S(V ) = ⊕ ∞ r=0 S r (V ) be the symmetric algebra generated by V , i.e. the free algebra T(V ) (for which we denote the tensor multiplication by suppressing the symbol) modulo the two-sided ideal I generated by xy − yx for all x, y ∈ V . Denoting the commutative associative multiplication in S(V ) (which is induced by the free multiplication) by •, i.e.
Moreover, for a given coalgebra (C, ∆) and a given nonassociative algebra (A, µ) where µ : A ⊗ A → A is a given K-linear map, recall the convolution multiplication in the K-module Hom K (C, A) defined in the usual way for any two K-linear maps φ, ψ : c → A by In case ∆ is coassociative and µ associative, * will be associative. The following fact is rather important: If C is connected and if the K-linear map ϕ : C → A vanishes on 1 C , then any convolution power series of ϕ converges, i.e. the evaluation of some formal series ∞ r=0 λ r ϕ * r (with λ r ∈ K and ϕ * 0 := 1 A ǫ C ) on c ∈ C always reduces to a finite number of terms. In particular, let ψ : C → A be a K-linear map such that ψ(1 C ) = 1 A . Then -as has been observed by Takeuchi and Sweedler (see [28,Lemma 14] or [27, Lemma 9.2.3]-ψ has always a convolution inverse, i.e. there is a unique Klinear map ψ ′ : C → A such that ψ * ψ ′ = 1 A ǫ C = ψ ′ * ψ, where ψ ′ is defined by the geometric series ψ ′ = ∞ r=0 (1 A ǫ C − ψ) * r .

B Semigroups
We collect some properties of semigroups which are very old, but a bit less well-known than properties of groups. The standard reference to these topics is the book [5] by A. H. Clifford and G. B. Preston.
Recall that a semigroup Γ is a set equipped with an associative multiplication Γ×Γ → Γ, written (x, y) → xy. An element e of Γ is called a left unit element (resp. a right unit element resp. a unit element) iff for all x ∈ Γ we have ex = x (resp. xe = x resp. iff e is both left and right unit element). A pair (Γ, e) of a semigroup Γ and an element e is called left unital (resp. right unital resp. unital ) iff e is a left unit element (resp. a right unit element resp. a unit element). A unital semigroup is also called a monoid. It is well-known that the unit element of a monoid is the unique unit element (unlike left or right unit elements in general). Let (Γ, e) be a right unital or a left unital semigroup. Recall that for a given element x ∈ Γ an element y ∈ Γ is called a left inverse of x (resp. a right inverse of x resp. an inverse of x) iff yx = e (resp. xy = e resp. iff y is both a left and a right inverse of x). Clearly, a unital semigroup (Γ, e) such that every element has an inverse is a group. In that case it is well-known that for each x there is exactly one inverse element, called x −1 . Note that by a Lemma by L. E. Dickson (1905, see [5, p.4] for the reference) every left unital semigroup such that each element has at least one left inverse is already a group which can be shown by just using the definitions. Dually, every right unital semigroup such that each element has at least one right inverse is also a group. More interesting is the case of a left (resp. right) unital semigroup (Γ, e) such that every element x has at least one right (resp. left) inverse element. In that case (which is an equivalent formulation of a so-called right group (resp. left group), see [5, p.37]), the conclusion of Dickson's Lemma does no longer hold. In order to see what is going on, there is first the following useful Lemma B.1 Let (Γ, e) be a left-unital semigroup, let a, b, c three elements of Γ such that ab = e and bc = e.
Then c = ae, be = b, and the left multiplications L a : x → ax and L b : x → bx are invertible. In particular, given the element a, its right inverse b is unique under the above hypotheses.
The proof is straight-forward. The structure of right (resp.left) groups is completely settled in the Suschkewitsch Decomposition Theorem, 1928: Given a right group (Γ, e), it can be shown -using the above Lemma and elementary manipulations, see also [5, p.38, Thm 1.27]-that all the left multiplications L x : y → xy (resp. right multiplications R x : y → yx) are invertible, that for each element there is exactly one right (resp. left) inverse (whence there is a map Γ → Γ assigning to each element x its right (resp. left) inverse x −1 ), that the image of this right (resp. left) inverse map is equal to Γe (resp. eΓ) (which turns out to be a subgroup of (Γ, e)), and that (Γ, e) is isomorphic to the cartesian product Γe×E, (e, e) (resp. E×eΓ, (e, e) where E is the set of all left (resp. right) unit elements in (Γ, e) (coinciding with the set of all idempotent elements). For right groups, the aforementioned isomorphism is given as follows: Note that both components of φ −1 are idempotent maps. There is a completely analogous statement for left groups.
Recall that a Lie semigroup is a differentiable manifold Γ equipped with a smooth associative multiplication m : Γ × Γ → Γ. All the other definitions of semigroups mentioned above (such as left unital, right unital semigroups, monoids, groups, right groups, left groups etc.) carry over to the Lie, i.e. differentiable, case. Moreover for right Lie groups, it is easy to see that all the left multiplications are diffeomorphisms (since their inverse maps are left multiplications with the inverse elements and therefore smooth). This fact and the regular value theorem applied to the equation xy = e imply that the right inverse map is smooth since its graph is a closed submanifold of Γ × Γ and the restriction of the projection on the first factor of the graph is a diffeomorphism. As the maps x → xe = (x −1 ) −1 and x → x −1 x are smooth and idempotent, it follows that their images, the subgroup Γe, and the semigroup of all left unit elements, E, are both smooth submanifolds of Γ and closed sets provided Γ is connected, see e.g. [3, p.54, Satz 5.13] for a proof. Hence Γe is a connected Lie group, and the Suschkewitsch decomposition Γ ∼ = Γe × E, see Appendix B, is a diffeomorphism. Conversely, any cartesian product of a Lie group G and a differentiable manifold E equipped with the multiplication (g, x)(h, y) := (gh, y) is easily seen to be a right Lie group. An analogous statement holds for left Lie groups. It is not hard to see that the category of all connected right Lie groups is equivalent to the product of category of all connected Lie groups and the category of all pointed connected manifolds.