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Structure Theory of Rack-Bialgebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Structure Theory of Rack-Bialgebras

Alexandre C1, Bordemann M2, Rivière S3 and Wagemann F3*

1Department of Mathematics, Université de Strasbourg, 4 Rue Blaise Pascal, 67081 Strasbourg, France

2Laboratoire de Mathématiques, Informatique et Applications, Université de Mulhouse, France

3Department of Mathematics, Université de Nantes, 1 Quai de Tourville, 44035 Nantes Cedex 1, France

*Corresponding Author:
Wagemann F
Department of Mathematics
Université de Nantes
1 Quai de Tourville, 44035
Nantes Cedex 1, France
Tel: 0276645089
E-mail: [email protected]

Received Date: July 17, 2016; Accepted Date: November 19, 2016; Published Date: November 30, 2016

Citation: Alexandre C, Bordemann M, Rivière S, Wagemann F (2016) Structure Theory of Rack-Bialgebras. J Generalized Lie Theory Appl 10:244. doi:10.4172/1736-4337.1000244

Copyright: © 2016 Alexandre C, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

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 wellknown 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.

Keywords

Coalgebras; Cocommutative Hopf dialgebras; Canonical rack bialgebras; Manifolds; Drinfeld center

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 ALie which is the vector space A with the bracket [a, b] :=abba. 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. The theme of the present article is to investigate links of this kind for more general objects than groups, namely for racks and digroups.

Recall that a pointed rack is a pointed set (X, e) together with a binary operation Equation:X × X × → X such that for all xX, the map y x Equation y is bijective and such that for all x, y, zX, the self-distributivity and unit relations

Equation

are satisfied. Imitating the notion of a Lie group, the smooth version of a pointed rack is called Lie rack.

An important class of examples of racks are the so-called augmented racks [1]. An augmented rack is the data of a group G, a G – set X and a map p: XG such that for all xX and all gG,

p(gx)=gp(x)g-1.

The set X becomes then a rack by setting x Equation y:=p(x) y . In fact, augmented racks are the Drinfeld center (or the Yetter-Drinfeld modules) in the monoidal category of G-sets over the (set-theoretical) Hopf algebra G, see for example [2]. Any rack may be augmented in many ways, for example by using the canonical morphism to its associated group or to its group of bijections or to its group of automorphisms.

In order to formalize the notion of a rack, one needs the diagonal map diagx : XX × X given by x ?(x, x). Then the self-distributivity relation reads in terms of maps m ° (idM × m),

=m ° (m× m)°( idM τM, M × idM) ° (diagM idM idM )

Axiomatizing this kind of structure, one may start with a coalgebra C and look for rack operations on this fixed coalgebra [3,4].

A natural framework where this kind of structure arises (as we show in Section 3) is by taking point-distributions (i.e. applying the Serre functor) over (resp. to) the pointed manifold given by a Lie rack. We dub the arising structure as rack bialgebra.

Lie racks are intimately related to Leibniz algebra Equation, i.e. a vector space Equation with a bilinear bracket [,]:EquationEquationEquation such that for all X, Y, ZEquation, [X, −] acts as a derivation:

[X,[Y,Z]]=[[X,Y],Z] + [Y,[X,Z]].         (1)

Indeed, Kinyon showed that the tangent space at eH 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 [5]. Conversely, every (finite dimensional real or complex) Leibniz algebra h may be integrated into a Lie rack REquation (with underlying manifold h) using the rack product.

Equation        (2)

noting that the exponential of the inner derivation adX for each X Equation is an automorphism.

Another closely related algebraic structure is that of dialgebras. A dialgebra is a vector space D with two (bilinear) associative operations Equation : D × DD and Equation : D × DD which satisfy three compatibility relations, namely for a, b, cD:

Equation

A dialgebra D becomes a Leibniz algebras via the formula Equation.

In this sense Equation and Equation are two halves of a Leibniz bracket. Loday and Goichot have defined an enveloping dialgebra of a Leibniz algebra [6,7].

One main point 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 Equation decomposes as a tensor product Equation where Equation 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 EAHA of EA with HA which may be identified to the associative quotient Aass of A. This result permits to show that the Leibniz algebra of primitives in A is a hemisemi- direct product, 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 [8].

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 Equation an augmented rack bialgebra UAR (Equation) 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(Equation) 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:

Equation

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 Equation, the vector space Equation becomes a Lie rack Equation with the rack product.

Equation

In Theorem 3.8, we show that the rack bialgebra UAR (Equation) associated to Equation coincides with the rack bialgebra F(Equation).

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, Δ) over K, we shall use Sweedler’s notation Δ(a)=Σ(a)a(1)a(2) for any aA. See also Appendix 4 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, Δ,ε,1,μ) be a K-module such that (B, Δ,ε,1) is a coassociative counital coaugmented coalgebra (a C3- coalgebra), and such that the linear map μ: BBB (the multiplication) is a morphism of C3-coalgebras (it satisfies in particular μ(1 ⊗ 1)=1). We shall call this situation a nonassociative C3I-bialgebra (where I stands for 1 being an idempotent for the multiplication μ). For another nonassociative C3I –bialgebra (B′, Δ′, ′, 1′, μ′ ) a K-linear map φ: BB′ will be called a morphism of nonassociative C3I-bialgebras iff it is a morphism of C3-coalgebras and is multiplicative in the usual sense φ(μ(ab))=μ′(φ(a))⊗φ(b)) for all a, bB. The nonassociative C3I-bialgebra (B, Δ,ε,1) is called left-unital (resp. right-unital) iff for all aB μ(1⊗a)=a (resp. μ(a⊗1)=a).

Moreover, consider the associative algebra A?HomK(B,B) equipped with the composition of K-linear maps, and the identity map idB as the unit element. There is an associative convolution multiplication * in the K-module HomK(B,A) of all K-linear maps B→HomK(B,B), see Appendix 4, eqn (103) for a definition with idBε as the unit element. For a given nonassociative C3I-bialgebra (B, Δ,,1,μ) we can consider the map μ as a map B→HomK(B,B) in two ways: as left multiplication map Lμ: Equation or as right multiplication map Equation. We call (B, ε,1,μ) a left-regular (resp. right-regular) nonassociative C3I-bialgebra iff the map Lμ (resp. the map Rμ) has a convolution inverse, i.e. iff there is a K-linear map Equation such that Equation, or on elements a, bB for the left regular case:

Equation      (3)

Note that every associative unital Hopf algebra(H, Δ,ε,1,μ, S) (where S denotes the antipode, i.e. the convolution inverse of the identity map in HomK(H,H)) is right- and left-regular by setting Equation and Equation.

Lemma 2.1: Let (B, Δ,ε,1,μ) be a nonassociative C3I-bialgebra.

1. If B is left-regular (resp. right-regular), then the corresponding K-linear map μ′: BBB is unique, and in case Δ is cocommutative, μ′ is map of C3-coalgebras.

2. If (B, Δ,ε,1,μ) is left-unital (rep. right-unital) and its underlying C3-coalgebra is connected, then (B, Δ,ε,1,μ) is always left-regular (resp. right-regular).

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: BBB is a morphism of coalgebras iff for each bB.

Equation       (4)

(and analogously for right multiplications). Both sides of the preceding equation, seen as maps of b, are in HomK(B,HomK(B,BB)). Since HomK(B,BB) is an obvious right HomK(B,B)-module, the K-module HomK(B,HomK(B,BB)) is a right HomK(B,HomK(B,B))-module with respect to the convolution. Define the K-linear map Equation HomK(B,BB) by:

Equation

Using eqn (4), the fact that Lμ′ is a convolution inverse of Lμ, and the cocommutativity of Δ, we get:

Equation

and μ′ preserves comultiplications. A similar reasoning where BB is replaced by K shows that μ′ preserves counits. Finally, it is obvious that Lμ′ (1) is the inverse of the K-linear map Lμ (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 Lμ (1)=idB, and the generalized Takeuchi-Sweedler argument, see Appendix 4, shows that Lμ has a convolution inverse. Right-unital bialgebras are treated in an analogous manner.

Note that any C3-coalgebra (C, Δ,ε,1,) becomes a left-unital (resp. right-unital) associative C3I-bialgebra by equipping with the left-trivial (resp. right-trivial) multiplication.

Equation       (5)

We shall call an element cB a generalized idempotent iff Σ(c) c(1) c(2)=c . Moreover cB will be called a generalized left (resp. right) unit element iff for all bB we have cb=ε(c)b (resp. bc=ε(c)b).

Rack bialgebras, augmented rack bialgebras and Leibniz algebras

Definition 2.1: A rack bialgebra (B, Δ,ε,1,μ) is a nonassociative C3I-bialgebra (where we write for all Equation such that the following identities hold for all a, b,cB

Equation          (6)

Equation         (7)

Equation       (8)

The last condition (8) is called the self-distributivity condition.

Note that we do not demand that the C3-coalgebra B should be cocommutative nor connected.

Example 2.1: Any C3 coalgebra (C, Δ,ε,1) carries a trivial rack bialgebra structure defined by the left-trivial multiplicaton

Equation         (9)

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: Let (B, Δ,ε,1,μ) a rack bialgebra –where we write Equation for all a, bB –, and let f: BB a morphism of C3-coalgebras such that for all a, bB

Equation        (10)

i.e. f is μ-equivariant. It is a routine check that (B, Δ,ε,1,μf) is a rack bialgebra where for all a, bB the multiplication is defined by

Equation       (11)

We shall call (B, Δ,ε,1,μf) the f-gauge of (B, Δ,ε,1,μ).

Example 2.2: Let (H, Δ H,ε H,μ H,1H,S) be a cocommutative Hopf algebra over K. Then it is easy to see (cf. also the particular case B=H and Φ=id H of Proposition 2.1) that the new multiplication μ(HH)→H, written Equation, defined by the usual adjoint representation

Equation          (12)

equips the C4-coalgebra (H, Δ H,ε H,1H) with a rack bialgebra structure.

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 Equation:X × X × → X such that for all xX, the map Equation is bijective and such that for all x,y,zX, the selfdistributivity and unit relations:

Equation

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 xX are set-like: Δ(x)=xx for all xX. 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, Section 3.1.

More generally there is the following structure:

Definition 2.2: An augmented rack bialgebra over K is a quadruple (B,Φ,H,l) consisting of a C3-coalgebra (B, Δ,ε, 1), of a cocommutative (!) Hopf algebra (H, ΔH,εH, 1H, μH, S), of a morphism of C3-coalgebras Φ: BH, and of a left action l: HBB of H on B which is a morphism of C3-coalgebras (i.e. B is a H-module-coalgebra) such that for all hH and aB

Equation      (13)

Equation       (14)

Where ad denotes the usual adjoint representation for Hopf algebras, see e.g. eqn (12).

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: (B,Δ,ε,1)→(B′,Δ′,ε′,1′) is a morphism of C3-coalgebras, and ψ: HH′ is a morphism of Hopf algebras such that the obvious diagrams commute:

Equation       (15)

An immediate consequence of this definition is the following:

Proposition 2.1: Let (B,Φ,H, l) be an augmented rack bialgebra. Then the C3-coalgebra (B,ε,1) will become a left-regular rack bialgebra by means of the multiplication:

Equation        (16)

for all a,bB. In particular, each Hopf algebra H becomes an augmented rack bialgebra via (H,idH,H,ad). In general, for each augmented rack bialgebra the map Φ:BH is a morphism of rack bialgebras.

Proof: For a proof of this proposition [9].

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: XG gives rise to an augmented rack bialgebra p: K[X]→ K[G].

Remark 2.1: Motivated by the fact that the augmented racks p: XG are exactly the Yetter-Drinfeld modules over the (set-theoretical) Hopf algebra G, we may ask about the relation of augmented rack bialgebras to Yetter-Drinfeld modules, and more generally of rack bialgebras to the Yang-Baxter equation. For these subjects.

The link to Leibniz algebras is contained in the following:

Proposition 2.2: Let (B, Δ,ε, 1, μ) be a rack bialgebra over K.

1. Then its K-submodule of all primitive elements, Prim(B)=:h, (see eqn (101) of Appendix 4) is a subalgebra with respect to μ (written aEquationb) satisfying the (left) Leibniz identity:

Equation     (17)

for all x, y, zEquation=Prim(B). Hence the pair ( Equation,[,]) with [x, y]? xEquationy for all x, yEquation 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, Equation and each subcoalgebra of order Equation, B(k), (see eqn (102)) is stable by left -multiplications with every a B. In particular, each B(k) is a rack subbialgebra of (B, Δ,ε, 1, μ).

Proof: 2. Let xEquation and aB. Since μ is a morphism of C3-coalgebras and x is primitive, we get:

Equation

whence aEquationx 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 kEquation, and let xB(k+1). Then:

Equation

where we have used the extended multiplication (still denoted Equation ) Equation and set:

Equation

by the definition of B(k+1), see Appendix 4. By the induction hypothesis, all the terms Equation andEquation are in B(k), when EquationEquation is in B(k)B(k), implying that aEquationx is in B(k+1).

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 (8) that:

Equation

proving the left Leibniz identity. The morphism statement is clear, since each morphism of rack bialgebras is a morphism of C3-coalgebras and preserves primitives.

Leibniz algebras have been invented by A. M. Blokh in 1965, and then rediscovered by J.-L. Loday in 1992 in the search of an explanation for the absence of periodicity in algebraic K-Theory [10,11].

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 2.2: Define set-like elements to be elements a in a rack bialgebra B such that Δ(a)=aa. Thanks to the fact that Equation is a morphism of coalgebras, the set of set-like elements Slike(B)is closed under Equation. In fact, Slike(B) is a rack, and one obtains in this way a functor Slike: RackBialg→Racks.

Proposition 2.3: The functor of set-likes Slike: RackBialg→Racks has the functor K[−]: Racks→RackBialg (see Example 2.3) as its leftadjoint.

Proof: This follows from the adjointness of the same functors, seen as functors between the categories of pointed sets and of C4-coalgebras, observing that the C4-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 (12)) gives the usual functor of group-like elements.

(Augmented) rack bialgebras for any Leibniz algebra

Let (Equation,[,]) be a Leibniz algebra over K, i.e. Equation is a K-module equipped with a K-linear map Equation satisfying the (left) Leibniz identity (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:

Equation      (18)

Equation       (19)

It is well-known and not hard to deduce from the Leibniz identity that both Q(Equation) and z(Equation) are two-sided abelian ideals of (Equation,[,]), that Q(Equation) ⊂ Equation, and that the quotient Leibniz algebras:

Equation        (20)

are Lie algebras. Since the ideal Q(Equation) is clearly mapped into the ideal Q(Equation′) by any morphism of Leibniz algebras Equation (which is a priori not the case for Equation !), there is an obvious functor Equation from the category of all Leibniz algebras to the category of all Lie algebras.

In order to perform the following constructions of rack bialgebras for any given Leibniz algebra (Equation,[,]), choose first a two-sided idealEquation such that:

Equation (21)

Let Equation denote the quotient Lie algebra Equation and let Equation be the natural projection. The data of Equation, i.e. of a Leibniz algebra h together with an ideal Equation such that Equation 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 Equation 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]=:adx(y)       (22)

because Equation. Note that:

Equation       (23)

as Lie algebras.

Consider now the C5-coalgebra (B=S(Equation), Δ, , 1) which is actually a commutative cocommutative Hopf algebra over K with respect to the symmetric multiplication •. The linear mapEquation induces a unique morphism of Hopf algebras:

Equation     (24)

satisfying

Equation          (25)

for any nonnegative integer k and x1,…,xkEquation. 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 C5-coalgebras. Consider now the universal enveloping algebra U(Equation) of the Lie algebra g. Since Equation⊂ K by assumption, the Poincaré-Birkhoff-Witt Theorem (in short: PBW) holds [12]. More precisely, the symmetrisation map Equation, defined by:

Equation        (26)

is an isomorphism of C5-coalgebras (in general not of associative algebras) [13]. We now need an action of the Hopf algebra H=U(Equation) on B, and an intertwining map Φ: B→ U(Equation). In order to get this, we first look at g-modules: The K-module h is a g-module by means of eqn (22), the Lie algebra Equation 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(Equation) are g-modules in the usual way, i.e. for all Equation

Equation       (27)

Equation        (28)

and of course Equation. Recall that U(Equation) is a Equation-module via the adjoint representation adξ(u)=ξ.uuuξ (for all ξ∈Equation and all u∈ U(Equation)).

It is easy to see that the map Equation (25) is a morphism of g-modules, and it is well-known that the symmetrization map ω (26) is also a morphism of Equation-modules. Define the K-linear map Equation the composition:

Equation      (29)

Then Φ is a map of C5-coalgebras and a map of g-modules. Thanks to the universal property of the universal enveloping algebra, it follows that S(Equation) and U(Equation) are left U(Equation)-modules, via (for all ξ1,…, kEquation, and for all a∈S(h))

Equation         (30)

and the usual adjoint representation (12) (for all u∈ U(Equation))

Equation      (31)

and that Φ intertwines the U(Equation)-action on C=S(Equation) with the adjoint action of U(Equation) on itself.

Finally it is a routine check using the above identities (27) and (12) that S(h) becomes a module coalgebra.

We can resume the preceding considerations in the following;

Theorem 2.1: Let (Equation,[,]) be a Leibniz algebra over K, let Equation be a twosided ideal of Equation such that Q(Equation) ⊂ z ⊂ z(Equation), let g denote the quotient Lie algebra Equation by Equation, and let Equation be the canonical projection.

1. Then there is a canonical U(Equation)-action l on the C5-coalgebra B:=S(Equation) (making it into a module coalgebra leaving invariant 1) and a canonical lift of p to a map of C5-coalgebras, Φ: S(Equation)→ U(Equation) such that eqn (14) holds.

Hence the quadruple (S(Equation), Φ, U(Equation), l) is an augmented rack bialgebra whose associated Leibniz algebra is equal to (Equation,[,]) (independently of the choice of Equation).

The resulting rack multiplication μ of S(Equation) (written Equation) is also independent on the choice of z and is explicitly given as follows for all positive integers k,l and Equation

Equation     (32)

where Equation denotes the action of the Lie algebraEquation (see eqn (23)) on S(h) according to eqn (27).

2. In case z=Q(Equation), the construction mentioned in 1. is a functor h→UAR(Equation) from the category of all Leibniz algebras to the category of all augmented rack bialgebras associating to h the rack bialgebra:

Equation

and to each morphism f of Leibniz algebras the pair Equation where Equation is the induced Lie algebra morphism.

3. For each nonnegative integer k, the above construction restricts to each subcoalgebra of order k, Equation, define an augmented rack bialgebra Equation which in case z=Q(Equation) defines a functor Equation from the category of all Leibniz algebras to the category of all augmented rack bialgebras.

Remark 2.3: This theorem should be compared to Proposition 3.5. [3]. In [3], 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 Equation be a Leibniz algebra. We will call the augmented rack bialgebra Equation the enveloping algebra of infinite order of Equation. As such, it will be denoted by Equation.

This terminology is justified, for example, by the fact that h is identified to the primitives in S(Equation) (cf Proposition 2.2). This is also justified by the following theorem the goal of which is to show that the enveloping algebra Equation fits into the following diagram of functors:

Equation

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 (12)) as a rack product.

Theorem 2.2: Let Equation be a Lie algebra. The PBW isomorphism Equation induces an isomorphism of functors:

Equation

Proof: The enveloping algebra Equation is by definition the functorial version of the rack bialgebra S(Equation), i.e. associated to the ideal Q(Equation). But in case h is a Lie algebra, Q(Equation)={0}. Then the map p is simply the identity, and Equation.

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 functo Prim, seen as a functor from the category of all rack bialgebras to the category of all Leibniz algebras. For a given Leibniz algebra (Equation,[,]) define the subcoalgebra of order 1 of the first component of UAR(1)(Equation) (see the third statement of Theorem 2.1), i.e.

Equation      (33)

With 1?1=1K which is rack subbialgebra according to Proposition 2.2. Its structure reads for all Equation and for all x, yEquation

Equation        (34)

Equation             (35) 

Equation        (36)

For the particular case of a Lie algebra (Equation,[,]), the above construction can be found in [3]. Moreover, for any other Leibniz algebraEquation and any morphism of Leibniz algebrasEquation define the K-linear map Equation as the first component of UAR(1)(f) (cf. the third statement of Theorem 2.1) by

Equation (37)

which is clearly is a morphism of rack bialgebras. Hence UR is a functor from the category of all Leibniz algebras to the category of all rack bialgebras. Now let (C, ΔC,εC,1C,μC) be a rack bialgebra, and let f: hPrim(C) be a morphism of Leibniz algebras. Define the K-linear map Equation by

Equation          (38)

and it is again a routine check that it defines a morphism of rack bialgebras. Moreover, due to the almost trivial coalgebra structure of UR(Equation), it is clear that any morphism of rack bialgebras UR(Equation)→C is of the above form and is uniquely determined by Equation. Hence we have shown the following:

Theorem 2.3: There is a left adjoint functor, UR, for the functor Prim (associating to each Rack bialgebra its Leibniz algebra of all primitive elements). For a given Leibniz algebra (Equation,[,]), the object UR(Equation) –which we shall call the Universal Rack Bialgebra of the Leibniz algebra (Equation,[,])– has the usual universal properties.

Next we can refine the above universal construction by taking into account the augmented rack bialgebra structure of Equation 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 Equation is:

Equation (39)

The third statement of Theorem 2.1 tells us that this is a welldefined 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 Equation be an augmented rack bialgebra, and let Equation be a morphism of Leibniz algebras. Clearly, as has been shown in Theorem 2.3, the map Equation given by eqn (37) is a morphism of rack bialgebras. Observe that the morphism of C3-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 Equation we have

Equation

It follows that the two-sided ideal Q(Prim(B′)) of the Leibniz algebra Prim(B′) is in the kernel of the restriction of Φ′ to Prim(B′), whence the map ′ induces a well-defined K-linear morphism of Lie algebras Equation. It follows that the compositionEquation is a morphism of Lie algebras, and by the universal property of universal envelopping algebras there is a unique morphism of associative unital algebras Equation. But we have for allEquation

Equation

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 C5-Hopf-algebras. For all λ∈K and xEquation we get:

Equation

showing the first equation Equation of the morphism equation (15). Moreover for all λ∈K, x∈h, and Equation we get

Equation

showing the second equation Equation of the morphism equation (15). It follows that the pair Equation is a morphism of augmented rack bialgebras. We therefore have the following

Theorem 2.4: There is a left adjoint functor, UAR, for the functor Prim (associating to each augmented rack bialgebra its Leibniz algebra of all primitive elements). For a given Leibniz algebra (Equation,[,]), the object UAR(Equation) –which we shall call the Universal Augmented Rack Bialgebra of the Leibniz algebra (Equation,[,])– has the usual universal properties.

The relationship between the different notions (taking into account also Remark (2.2)) is resumed in the following diagram:

Equation

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. We shall show in this section that rack bialgebras and certain cocommutative Hopf dialgebras are strongly related.

Left-unital bialgebras and right Hopf algebras: Let (B, Δ,ε,1,μ) be a nonassociative left-unital C3-bialgebra. It will be called left-unital C3-bialgebra iff μ is associative. In general, (B, ,ε,1,) need not be unital, i.e. we do not have in general a1=a. However, it is easy to see that the sub-module B1 of B is a C3-subcoalgebra of (B, Δ,ε,1), and a subalgebra of (B, ) such that Equation is a unital (i.e. left-unital and rightunital) bialgebra. Here Equation denote the obvious restrictions and corestrictions.

In a completely analogous way right-unital C3-bialgebras are defined.

A left-unital (resp. right unital) cocommutative C3-bialgebra (B, Δ,ε,1,μ) will be called a cocommutative right Hopf algebra (resp. a cocommutative left Hopf algebra), (B, Δ,ε,1,μ, S), iff there is a right antipode S (resp. left antipode S), i.e. there is a K-linear map S: BB which is a morphism of C3-coalgebras (B, Δ,ε,1) to itself such that

id*S=1ε (resp. S*id=1)           (40)

where * denotes the convolution product (see Appendix 4 for definitions). It will become clear a posteriori that right or left antipodes are always unique, see Lemma 2.2.

A first class of examples is of course the well-known class of all cocommutative Hopf algebras (H, Δ,ε,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 C4-coalgebra (C, Δ,ε,1) equipped with the left-trivial multiplication (resp. right trivial multiplication) μ0 (see eqn (5)) and trivial right antipode (resp. trivial left antipode) S0 defined by S0(x)=ε(x)1for all xC (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 C4-coalgebra (C, Δ,ε,1).

We have the following elementary properties showing in particular that each right (resp. left) antipode is unique:

Lemma 2.2: Let Equation be a cocommutative right Hopf algebra.

1. S*(S°S)=1ε, S°(S°S)=id*1ε, S*1ε=S and S°S°S=S, which for each Equation impliesEquation It follows that right antipodes are unique.

2. For all Equation.

3. For any element Equation, c is a generalized idempotent if and only if c=(c)S(c(1))c(2) iff there is Equation with c=(x)S(x(1))x(2), and all these three statements imply that c is a generalized left unit element.

Proof: 1. Since S is a coalgebra morphism, it preserves convolutions when composing from the right. This gives the first equation from statement 2.2. Hence the elements id, S, and S°S satisfy the hypotheses of the elements a,b,c of Lemma 5.1 in the left-unital convolution semigroup Equation, whence the second and third equations of statement 2.2. 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 5.1.

2. Again the elements μ, S°μ and (id*1εμ satisfy the hypotheses on the elements a,b,c of Lemma 5.1 in the left-unital convolution semigroup Equation (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 ° (SS) is a right inverse of μ, whence we get statement 2.2. by uniqueness of right inverses (Lemma 5.1).

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. Conversely, if Equation is a generalized idempotent, i.e. c=(μ °Δ)(c), we get –since °Δ is a morphism of C3- coalgebras– that

Equation

and all the three statements are equivalent. In order to see that every such element c is a generalized left unit element pick Equation and

Equation

since obviously ε(c)=ε(x), so c is a generalized left unit element.

There is the following right Hopf algebra analogue of the Suschkewitsch decomposition theorem for right groups (see Appendix 5):

Theorem 2.5: Let Equation a cocommutative right Hopf algebra. Then the following holds:

1. The K-submodule Equation is a unital Hopf subalgebra of Equation.

2. The K –submodule Equation is ageneralized idempotent} is a right Hopf subalgebra of Equation equal to the left-trivial right Hopf algebra defined by the C4-coalgebra Equation.

3. The map

Equation

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 Equation1 equipped with all the restrictions is a unital bialgebra. Note that for all Equation

Equation

Equation

Whence Equation is also a left antipode. It follows that Equation1 is a Hopf algebra.

2. Since the property of being an generalized idempotent is a K-linear condition, it follows that Equation is a K-submodule. Moreover since each Equation is of the general formEquation, and since the map Equation defined byEquation is an idempotent morphism of C3-coalgebras, we get

Δ(c)=Δ(ι(c))=(ιι)(Δ(x))

showing that Equation is a C3-subcoalgebra of Equation. Furthermore, since every element of Equation is a generalized left unit element (Lemma 2.2, 3.), the restriction of the multiplication of μ of Equation toEquation is left trivial. Finally,

Equation

showing that the the restriction of S to Equation is the trivial right antipode.

3. It is clear from the two preceding statements that Ψ is a welldefined linear map into the tensor product of two cocommutative right Hopf algebras. We have for all Equation

Equation

because all the terms S(a(2))a(3) and the components c(1),… of iterated comultiplications of generalized idempotents can be chosen in Equation (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 C3-coalgebras.

Equation

showing that Ψ−1 and hence Ψ is a morphism of left-unital algebras. Finally we obtain

Equation

thanks to Lemma 2.2, and intertwines right antipodes.

Note that the K-submodule of all generalized left unit elements of a right Hopf algebra Equation is given byEquation and thus in general much bigger than the submodule Equation of all generalized idempotents.

As it is easy to see that every tensor product HC of a unital cocommutative Hopf algebra H and a C4-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 C4-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 CH.

Dialgebras and Rack Bialgebras: Recall that a dialgebra over K is a K –module D equipped with two associative multiplications Equation satisfying for all a,b,cA:

Equation        (41)

Equation        (42)

Equation        (43)

An element 1 of A is called a bar-unit element of the dialgebra (A,Equation,Equation) and (A,1,Equation,Equation) is called a bar-unital dialgebra iff in addition the following holds

1Equationa=a,             (44)

aEquation1=a,             (45)

for all aA. Moreover, we shall call a bar-unital dialgebra (A,1,Equation,Equation) balanced iff in addition for all aA

Equation           (46)

Clearly each associative algebra is a dialgebra upon setting Equation=Equation 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:

Proposition 2.4: Let (A,Equation,Equation) be a dialgebra. Then the K-module A equipped with the bracket [,]:AAA, written [a,b],

Equation         (47)

is a Leibniz algebra, denoted by A.

In fact, this construction is well-known to give rise to a functor AA 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:

Example 2.5: Let (B,1B) 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 BAA and ABAEquation equipping A with the structure of a left B-module and a right B-module such that (bx)b′=b(xb′) for all b,b′∈B and for all xB. Suppose in addition that there is a bimodule map Φ: AB, i.e. Φ(bxb′)=bΦ(x) b′ for all b,b′∈B and for all xA. Then it is not hard to check that the two multiplications Equation,Equation: AAA defined by

Equation       (48)

equip A with the structure of a dialgebra. If in addition there is an element 1∈A such that Φ(1)=1B, then (A,1,Equation,Equation) will be a bar-unital dialgebra. We shall call this structure (A, Φ, B) an augmented dialgebra.

In fact, every dialgebra (A,Equation,Equation) arises in that fashion: Consider the K –submodule IA whose elements are linear combinations of arbitrary product expressions

Equation

(where all reasonable parentheses and symbols Equation and Equation are allowed) for any two strictly positive integer rn, and Equation. It follows that the quotient module A/I is equipped with an associative multiplication induced by both Equation and Equation Let Equation 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 Equation be equal to A/IK (adjoining a unit element) in case A does not have a bar-unit. Thanks to the defining equations (41), (42), (43), it can be shown by induction that for any strictly positive integer n, any a1,…,an,aA, and any product expression made of the preceding elements upon using Equation or Equation

Equation

proving in particular that I acts trivially from the left (via ) and from the right (via Equation) on A such that there is a well-defined Equation -bimodule structure on A such that the natural map Equation is a bimodule morphism. Hence Equation is always an augmented dialgebra, and the assignment Equation is known to be a faithful functor.

Note also that this construction allows to adjoin a bar-unit to a dialgebra (A,Equation,Equation): Consider the K-module Equation with the obvious Equation -bimodule structure α.(b+β)=α.b+αβ and (b+β).α=b.α+βα for all Equation and bA. Observe that the obvious map Equation defined byEquation is an Equation -bimodule map, and thatEquation is a bar-unit. The bar-unital augmented dialgebraEquation is easily seen to be balanced. There are nonbalanced bar-unital dialgebras as can be seen from the augmented bar-unital dialgebra example (BB,1B⊗1BB,B) where (B,1,μB) is any unital associative algebra and the bimodule action is defined by b.(b1b2).b′?(bb1)⊗(b2b′) for all b, b′, b1, b2B.

Again, in case the dialgebra (A,Equation,Equation,1) is bar-unital and balanced, note that Equation is an associative unital subalgebra A′ of A whose multiplication is induced by both and Equation for all Equation. Since the K-linear map Equation descends to a surjective morphism of associative algebras Equation by the above, it is clear that the ideal I contains the kernel of πA. On the other hand, if a∈Ker(πA) then Equation, and obviouslyEquation, thus inducing a useful isomorphism Equation, and thus a subalgebra injection Equation which is a right inverse to the projection A, i.e. Equation.

In this work, we also have to take into account coalgebra structures and thus define the following:

Definition 2.4: Let (A,Δ,ε,1) be cocommutative C3-coalgebra (a C4- coalgebra) and two K-linear maps Equation,Equation: AAA. Then (A,Δ,ε,1,Equation,Equation) will be called a cocommutative bar-unital di-coalgebra if and only if

1. (A, 1,Equation,Equation) is a bar-unital balanced dialgebra.

2. Both Equation and Equation are morphisms of C3-coalgebras.

If in addition there is a morphism of C3-coalgebras S: AA such that (A,Δ,ε,1,Equation,S) is a cocommutative right Hopf algebra and (A,Δ,ε,1,Equation,S) is a cocommutative left Hopf algebra, then (A,Δ,ε,1,Equation,Equation,S) is called a cocommutative Hopf dialgebra.

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. Moreover, a slightly more general context would have been to demand the existence of two different antipodes, a right antipode S for Equation , and a left antipode S′ for Equation. 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:

Proposition 2.5: Let (A,Δ,ε,1,Equation,Equation,S) be cocommutative Hopf dialgebra. Then the submodule of all primitive elements of A, Prime(A), is a Leibniz subalgebra of A equipped with the bracket (47).

Proof: Let x,yA be primitive. Then, using that Equation and Equation are morphisms of coalgebras, we get

Equation

because A is balanced, and therefore Equation is primitive.

The first relationship with rack bialgebras is the following simple generalization of a cocommutative Hopf algebra equipped with the adjoint representation:

Proposition 2.5: Let (A,Δ,ε,1,Equation,Equation,S) be cocommutative Hopf dialgebra. Define the following multiplication

Equation          (49)

1. The map Equation defines on the K-module A two left module structures, one with respect to the algebra (A, 1,Equation), and one with respect to the algebra (A, 1,Equation), making the Hopf-dialgebra (A,Δ,ε,1,Equation,Equation,S) a module- Hopf dialgebra, i.e.

Equation           (50)

Equation         (51)

Equation        (52)

Equation         (53)

2. (A,Δ,ε,1,μ) is a cocommutative rack bialgebra.

Proof: 1. First of all we have

Equation

where μEquation and μEquation stand for the multiplication maps Equation and Equation, and this is clearly a composition of morphisms of C3-coalgebras whence μ is a morphism of C3-coalgebras proving eqn (51). Next, there is clearly Equation for all bA, and, since the dialgebra is balanced, we get for all aA

Equation

Next, let a,b,cA. Then

Equation       (54)

Proving eqs (50). Next, for all Equation, we get

Equation

where –in the second to last equation– we have used the left antipode identity for the case Equation and the fact that (a)S(a(1)) a(2) is a generalized left unit element for the case Equation. It follows thatEquation is an A-modulealgebra proving eqs (52) and (53).

2. It remains to prove self-distributivity: For all a,b,cA, we get

Equation

and in the end

Equation

proving the self-distributivity identity.

The next theorem relates augmented cocommutative rack bialgebras with cocommutative Hopf dialgebras:

Theorem 2.6: Let (BB,H,l) be a cocommutative augmented rack bialgebra. Then the K-module Equation 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 bB:

Equation

Moreover, the Leibniz bracket on the K-module of all primitive elements of BH, 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 x⊗1H and 1B⊗ξ)

[x+ξ,y+η]=([x,y]+ .y)+([ΦB(x),η]+[ ξ,η])        (55)

where each bracket is of the form (47)1. Note that this Leibniz algebra is split over the Lie subalgebra Prim(H), the complementary two-sided ideal {x−ΦB(x)x∈ Prim(B)} being in the left center of a.

Proof: It is clear from the definitions that condition 2.6 defines a H-bimodule structure on CH making it into a module C3-coalgebra. Moreover, we compute for all Equation

Equation

whence Φ is a morphism of H-bimodules. Next, we get for all bB and hH:

Equation

proving the right antipode identity, and

Equation

proving the left antipode identity. Finally for all hH we get

Equation

implying that the bar-unital dialgebra is balanced.

Formula (55) is straight-forward:

Equation

because primitives are killed by counits, and the formula is proved.

A lengthy, but straight-forward reasoning shows that the above construction assigning (BB,H,l)→(BH,Φ,H) defines a covariant functor from the category of all cocommutative rack bialgebras to the category of all cocommutative Hopf dialgebras.

A particular case of the preceding theorem is obtained by picking any C4 –coalgebra (BBB,1B) such that there is any H-module coalgebra structure l on B (such that h.1BH(h)1B) and by choosing the trivial map ΦB(b)=εB(b)1H: It follows that (BB,H,l) is an augmented rack bialgebra with left-trivial multiplication. It turns out that the Hopf dialgebra BH formed out of this is already isomorphic to the general cocommutative Hopf dialgebra:

Theorem 2.7: Let (A,Δ,ε,1,Equation,Equation,S) be a cocommutative Hopf dialgebra. Let EA be the C3-subcoalgebra of all generalized idempotent elements2 with respect to Equation, and let HA=1EquationA be the Hopf subalgebra according to the Suschkewitsch decomposition of the left Hopf algebra (A,Δ,ε,1,Equation,S), see Theorem 2.5. Then we have:

1. By means of the Suschkewitsch isomorphism AEAHA for the left Hopf algebra (A,Δ,ε,1,Equation,S), we can transfer the cocommutative dialgebra structure of A to EAHA: There is a well-defined left modulecoalgebra action l of HA on EA defined by (for all cEH, hHA, aA such that h=1Equationa)

Equation        (56)

and the transferred multiplications Equation and the antipode S′ on EAHA read Equation

Equation       (57)

Equation         (58)

Equation       (59)

2. The K-linear map Equation descends to an isomorphism of associative algebras Equation.

3. S.Covez, 2006: The Leibniz subalgebra Prim(A) of A (equipped with the Leibniz bracket (47) is a split semidirect sum out of the twosided ideal Prim(EA)⊂ z(Prim(A)) and the Lie subalgebra Prim(HA), i.e. for all z,z′∈Prim(EA) and ξ,ξ′∈Prim(HA), we have

[z+ξ,z′+ξ′]=ξ.z+ [ξ,ξ′].             (60)

4. Let (BB,H,l) be a cocommutative augmented rack bialgebra. Then for the Hopf dialgebra BH of Theorem 2.6, we get that the Hopf subalgebra HBH equals 1BHH, and

Equation

which is isomorphic to B as a C4-coalgebra of BH.

Proof: 1. Note first that the right hand side of eqn (56) is just aEquationc of Proposition 2.6 which had been shown to be a left module-Hopf dialgebra action of (A,Equation) and of (A,Equation) on A. Observe that for all a, a′∈ A

Equation

whence the HA-action l is well-defined on A. Moreover we compute for all h∈ HA and all Equation such that

Equation

whence (A,Equation) is also a HA-module-algebra. Now let cEH. By definition, c is a generalized idempotent (w.r.t.Equation) ), henceEquation, and thus for all hH

Equation

Whence h.c is also in EA, and EA is a HA-submodule of A.

Recall the Suschkewitsch decomposition of the left Hopf algebra (A,Δ,ε,1,Equation,S) where one can use Theorem 2.5 and dualize all the formulas:

Equation

Formulas (58) and (59) consequences of Theorem 2.5. The only formula which remains to be shown is eqn (57). Note first that every generalized idempotent Equation is also a generalized idempotent with respect to Equation. Indeed, since all the components c(1) and c(2) in Δ(c)=Σ(c)c(1)c(2) can be chosen in Equation, we get

Equation

Next for all Equation, and a,a′∈A such that 1Equationa=h andEquation we get –since c is a generalized left unit element (w.r.t. Equation) thanks to 3. in Lemma 2.2–

Equation

and this is equal to

Equation

proving eqn (57).

2. Clear for any bar-unital balanced dialgebra.

3. Straight-forward computation using Prim(Equation)=Prim(Equation)⊗ Prim(Equation) where the latter is well-known to be a Lie algebra and the former is abelian.

4. For each bb and hH, we get

Equation

Proving the first statement. Moreover

Equation

Proving the form of the generalized idempotents, and since the K-linear map BBH given by (idB⊗(SH°ΦB))°Δ is an injective morphism of C3-coalgebra, the statement is proved.

The third statement had been proved by Simon Covez in his Master thesis 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:XG is an augmented pointed rack, see Example 2.4. By the above theorem, part 4., its associated cocommutative augmented Hopf dialgebra decomposes as BH, where the Hopf algebra H=K[G] is the standard group algebra and B=K[X]. The generalized idempotents are in this case

Equation

We finish this section with a formula relating universal algebras: The functor associating to any dialgebra A its Leibniz algebra A via eqn (47) is well-known to have a left adjoint associating to any Leibniz algebra (Equation,[,]) its (in general non bar-unital) universal enveloping dialgebra Ud(h) associated to h defined by

Equation       (61)

But also in the category of bar-unital balanced dialgebras, there is such a left adjoint: To any Leibniz algebra (Equation,[,]), we associate its universal balanced bar-unital enveloping dialgebra Equation

Equation       (62)

Before proving the theorem, we note that Equation is obtained by adjoining a balanced bar-unit to Ud(Equation).

Theorem 2.8: For any Leibniz algebra (Equation,[,]), the assignment Equation defines a left adjoint functor to the functor associating to any bar-unital balanced dialgebra its commutator Leibniz algebra.

Proof: Clearly, Equation is the cocommutative Hopf dialgebra associated to the universal augmented rack bialgebra Equation (cf. Theorem 2.6) which in turn is associated to the Leibniz algebra (Equation,[,]) (cf. Theorem 2.4). Since both assignments are functorial, it follows that the assignment Equation is a functor. It remains to prove the universal property: Let (Equation,[,]) be a Leibniz algebra, let (A,1,Equation,Equation) a bar-unital balanced dialgebra, and let ?:h→A be a morphism of Leibniz algebras. It follows that the K-linear map Equation vanishes on the two-sided ideal Q(Equation) and descends to a morphism Equation of the quotient Lie algebra Equation to Equation with its commutator Lie bracket such that Equation. Hence there is a unique morphismEquation of unital associative algebras extending Equation. Define the K-linear map Equation by (for allEquation and xh):

Equation

where we recall the natural injection of unital algebras Equation given by Equation for all aA. We shall show that Equation is a morphism of augmented dialgebras: We compute for all Equation, using that iA and Equation are morphisms of unital associative algebras and that in the image of iA, we can use the multiplication symbols Equation and Equation arbitrarily:

Equation

Showing the fact that Equation preserves the bimodule structures on the first component of Equation. Next we have for all x1, xh

Equation

Equation

and by induction on k in Equation and x1,…, xkh, we prove

Equation

Showing the fact that Equation preserves the bimodule structures on the second component of Equation. Hence Equation is a morphism of bar-unital (augmented) dialgebras. The uniqueness of Equation follows from the universal property of Equation

Coalgebra Structures for Pointed Manifolds with Multiplication

In this section, the symbol Equation denotes either the field of all real numbers, Equation, or the field of all complex numbers, Equation. 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 first the category of all pointed manifolds Equation whose objects consist of pairs (M, e) where M is a non-empty differentiable manifold and e is an element of M and whose morphisms (M, e)→ (M′, e′) are given by all smooth maps φ: MM′ of the underlying manifolds such that φ(e)=e′. Recall that the cartesian product × makes Equation into a monoidal category by setting (M, e1)× (N, e2)? (M× N,(e1, e2)) with the one-point set ({pt},pt) as unit object and the usual associators, left-unit and right-unit identifications borrowed from the category of sets [14]. This monoidal category is symmetric by means of the usual (tensor) flip mapEquation where the pair of distinguished points is also interchanged.

By simply forgetting about the differentiable structure we get the category of pointed sets.

Recall that a pointed manifold with multiplication is a triple (M, e,m) where (M, e) is a pointed manifold, and m:(M,e)× (M, e) (M, e) is a smooth map of pointed manifolds, i.e. is a smooth map M× MM such that m(e,e)=e. Moreover, a pointed manifold with multiplication will be called left-regular (resp. right-regular) if all the left (resp. right) multiplication maps Equation are diffeomorphisms. Morphisms of pointed manifolds with multiplication (M,e,m)→Equation are smooth maps of pointed manifoldsEquation such that

φ°m=m °(φ×φ).             (63)

The obvious generalization are a finite number of maps M×nM with n ≥ 1) arguments.

Again by forgetting about differentiable structures, we get the category of pointed sets with multiplication.

Coalgebra Structure for distributions supported in one point

For any pointed manifold (M, e) recall the Equation-vector space

EquationEquation is a continuous linear map and supp(T)={e}}      (64)

of all distributions supported in the singleton {e} [15]. Now let φ:MM′ be a smooth map such that φ:(e)→ e′. For any distribution Equation and any smooth function Equation the well-known prescription

Equation          (65)

gives a well-defined distribution φ*S on the target manifold M′ supported in e′=φ(e), and the map Equation is a Equation-linear map (which is continuous). Clearly for three pointed manifolds (M, e), (M′, e′), and Equation with smooth mapsEquation and EquationEquation we get

Equation              (66)

This defines a covariant functor Equation Vect to the category of all Equation-vector spaces by associating to any pointed manifold (M, e) the Equation-vector space Equation and to any smooth map (M, e)→ (M, e′) the linear map Equation. We call this functor Serre functor in tribute to the predominant role it plays in [16]. It is one of the main objects of this article.

There is, however, much more structure in this functor: First any distribution space Equation contains a canonical linear form ε=εe:EquationEquation defined by

εe(T)?T(1),                   (67)

Where 1 denotes the constant function MEquation whose only value is equal to 1∈Equation. Moreover each space Equation contains a canonical element 1=1e defined by the well-known delta distribution

Equation             (68)

and we clearly have

εe(1e)=1.                (69)

Moreover, both εe and 1e are natural in the following sense: let φ:(M, e)→ (M′, e′) be a smooth map of pointed differentiable manifolds. Then it is straight-forward to check that

Equation             (70)

Recall the well-known tensor product or direct product of two distributions: More generally, let M and N be two differentiable manifolds, and let Equation andEquation be two distributions (where the symbol Equation denotes the continuous dual space of the test function space Equation of all smooth Equation-valued functions with compact support) [17]. Let f: M×NEquation be a smooth function with compact support KM × N, and let K1?prM(K) ⊂ M, K1?prN(K) ⊂ N, whence K is a subset of the compact set K1×K2. Let Equation be the following map: For each xM, let Equation be the partial functionEquation. Then we set

(T(2)(f))(x)=T(fx).

The superscript (2) means here that we see f as a function of its second variable only, when applying the distribution T.

Upon using the approximation theorem of any distribution by a sequence of regular distributions, one can show that (T(2)(f): MEquation is a smooth function having compact support in K1. It is clear that Equation is linear, and it can be shown by the same approximation theorem that T(2) is continuous. It follows that the map

Equation

is a well-defined Equation-bilinear map Equation, and there is thus a unique linear map (where ⊗ denotes the usual algebraic tensor product over Equation)

Equation                 (71)

such that for all Equation we have

Equation

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. Equation andEquation the distribution F2M,N (ST) is supported in (e1,e2), i.e. is an element of Equation. We shall denote the restriction of the map Equation by the same symbol F2M,N. For three pointed manifolds (M,e1), (N,e2), and (P,e3), let αM,N,P:M×(N×P)→(M×NP be the usual associator for the monoidal category of all sets, and for three vector spaces V, W, X over Equation, let V,W,X: V ⊗(WX)→(VW)⊗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

Equation             (72)

hence eqn (3) of [15]. In the same vein, the two diagrams in eqn (4) of [15] are satisfied upon setting F0=εpt and λ:EquationVV and ρ:VEquationV the usual left-unit and right-unit identifications in the monoidal category of vector spaces.

Let Equation andEquation two other pointed differentiable manifolds, and let Equation and Equation two smooth maps of pointed differentiable manifolds. It is a straight-forward check that the map F2M,N to is natural in the following sense

Equation             (73)

Moreover, note that the map F0=εpt (see eqn (67)) defines an isomorphism of Equation to Equation which had already been seen to be natural.

As a result, the functor F is a monoidal functor in the sense of [15]. Moreover, since the category Equation is even a symmetric monoidal category by means of the canonical flip map Equation , see e.g. [15], and the monoidal category Equation-vect is also symmetric, it is not hard to see that the monoidal functor is also symmetric, see e.g. [15] for definitions

We shall now show that the monoidal functor F is strong, i.e. that F0=εpt and F2M,N are isomorphisms. This is clear for εpt. Recall that for each distribution T in Equation (where V is a nonempty open set in Equation containing the point e), there is nonnegative integer l (called the order of the distribution) such that

Equation

Where Equation for each multi-index k. In a slightly more algebraic

manner we can express this as follows: let E be a finite-dimensional real vector space, let VE be an open set containing eE. Then we have the following linear isomorphism Equation given by ΦS(1)=δe and for any positive integer k and vectors w(1),…,w(k)E and Equation

Equation             (74)

where • denotes the commutative multiplication in the symmetric algebra, see Appendix A. Using the fact that the inclusion map Equation of any chart domain of M such that e∈Uα defines an isomorphism Equation, and that any chart Equation defines an isomorphismEquation, we can conclude that there is a linear isomorphism

Equation      (75)

with the symmetric coalgebra S(Equation) on Equation (see Appendix A) computed as follows

Equation           (76)

where we write Equation where all the w(j)i are real numbers and e1,…,em is the canonical base of Equation. Note that for the particular case of M being an open set V of Equation and the chart ?α being the identity map the map Φα (see eqs (75) and (76)) coincides with the canonical map ΦS, see eqn (??).

For two pointed manifolds (M, e1) and (N, e2) and given charts (Uα,?α) of M such that e1Uα and Equation of N such that Equation, we thus have linear isomorphisms Equation, Equation (upon using the product chart Equation). Using the above definitions, one can compute that

Equation

Where Equation denotes the natural isomorphism of commutative associative unital algebras induced by the obvious inclusions Equation (first m coordinates) and Equation (last n coordinates). It follows that the natural map F2M,N is equal to Equation 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 map diagM:MM×M defined by

diagM(x)=(x,x)

for all xM. Clearly, diagM is a smooth map of pointed manifolds (M, e)→(M×M, (e,e)). Moreover, the diagonal map is clearly natural in the sense that

Equation

for any smooth map Equation of pointed differentiable manifolds. In other words, the class of diagonal maps diagM constitutes a natural transformation from the identity functor to the diagonal functor (M,e)→(M×M,(e,e)) and Equation. Define the following linear map Equation by

Equation           (77)

This definition has avatars with more than two tensor factors. Indeed, observe that the naturality relation (73) implies for φ=idM and ψ=diagM that

Equation

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 Equation-vector space Equation equipped with the linear maps Δe (cf. eqn (77)), εe (cf. eqn (67), and 1e (cf. eqn (68) is a C5-coalgebra which is (non canonically) isomorphic to the standard symmetric coalgebra

Equation

2. The above strong symmetric monoidal functor F extends to a functor –also denoted by F – from Equation to the symmetric monoidal category of C5-coalgebras over Equation.

3. The subspace of all primitive elements of the coaugmented coalgebra Equation is natural isomorphic to the tangent space Te(M). Moreover for each smooth map Equation of pointed manifolds the coalgebra morphism Equation induces the tangent map Equation.

Proof: 1. Coassociativity of Δe: This follows from the coassociativitydiagram of diagM* by first taking the induced diagram between distribution spaces which reads then

Equation

Starting for example on the left hand side, one replaces the map diagM* by F2M,N ° Δe, and also the map (idM× diagM)* by

Equation

Now one observes that one may apply the relation (72) on the left hand side. One obtains

Equation

One deduces coassociativity.

2. Cocommutativity of Δe: This follows from the symmetry of F (already noted before) and the cocommutativity of diagM*.

3. Counitality of Δe: This follows from the counitality of diagM*, i.e.

Equation

Where Equation andEquation are the canonical maps. Indeed, these equations induce the corresponding equations between distribution spaces, and translating direct products into tensor products (and thus (proje)* into ε and diagM* into Δe), one obtains counitality.

4. Connectedness of Δe: The coalgebra Equation is isomorphic to the symmetric algebra S(Equation), 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 Equation (see Appendix A for the definition of the primitives Prim(C) of a coalgebra C) defined by

Equation

Indeed the right hand side is clearly in Equation, and the Leibniz rule for the derivative shows that this is in Prim(Equation).Moreover the above map is clearly injective, and since Equation and dim(TeM)=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 TeM .

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 Equation are isomorphic to the cofree S(Equation), 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 Equation 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 UEquation and VEquation are pointed open sets, the coalgebra morphism φ* of a smooth map φ:UV of pointed manifolds is isomorphic to the coalgebra morphism Equation induced by the jet of infinite order of φ at the distinguished point e of U, j(φ)e, for further information [18]. The functorial equation (φ °ψ)*=φ*°ψ* can be computed out of the chain rule for higher derivatives.

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:

Theorem 3.2: Let (M,e,m) be a pointed manifold with multiplication. Then the C5-coalgebra Equation carries a multiplication, i.e. a linear map Equation which is a morphism of C5-coalgebras.

In case m is left-unital (resp. right unital), the nonassociative C5I-algebra Equation is left regular (resp. right regular)

Proof: The map μ exists and is linear by functoriality. We have trivially

Equation

and this shows that μ is a morphism of coalgebras by translating diagM into Δe using as before the maps of type F2. The regularity statements are a consequence of the connectedness of the C3-coalgebra Equation, 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:

Theorem 3.3: 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 Equation.

We just indicate the isomorphism: For any Equation, let ξ+ denote the left invariant vector field ξ+(g)?TeLg(ξ) generated by its value Equation. Then the mapEquation is given by (for all Equation andEquation)

Equation                       (78)

Where Equation denotes the Lie derivative in the direction of the vector field X.

Note that the identities for the inverse map Equation can be written as

Equation           (79)

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 Equation.

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*°F2G,G. The fact that the coalgebra is always connected implies by the Takeuchi-Sweedler argument (see Appendix 4) that the identity map idC 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 that the Hopf algebra F(G) is isomorphic to the universal enveloping algebra over the Lie subalgebra Equation of its primitive elements which is equal to TeG.

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. Lod 2001) is a pointed differentiable manifold (D,e) equipped with two smooth associative multiplications D×DD, written (x,y)xEquationy and (x,y)xEquationy (and preserving points, i.e.eEquatione=e=eEquation), such that the dialgebra conditions eqs (41), (42), (43), (44), and (45) hold for all x,y,zD and e (replacing 1): Hence (D,Equation,e) is a left unital Lie semigroup and (D,Equation,e) is a right unital Lie semigroup, and as for dialgebras, we shall say barunital dimonoid to stress the fact that the bar-unit e is among the data for the dimonoid.

Let us call a Lie dimonoid (D,e,Equation,Equation) balanced iff in addition for all xD the analogue of eqn (46) holds, i.e. Equation. Any Lie monoid (G,e,m) is a Lie dimonoid by setting Equation.

Another class of examples is obtained by the following important augmented dimonoid construction (cf Example 2.5): Let G be a Lie group, let (D, eD) be a pointed differentiable manifold, let G smoothly act on the left and on the right of M Equation such that Equation and xD, and let f:(D,eD)→(G,e) be a smooth map of pointed manifolds such that for all Equation and xD

Equation           (80)

Then the pointed manifold Equation will be a (bar-unital) dimonoid by setting

Equation        (81)

A general Lie digroup is defined (according to Liu) to be a (barunital) dimonoid (D,e,Equation,Equation) such that the left unital Lie semigroup (D,e,Equation) is a right group and the right unital Lie semigroup (D,e,Equation) is a left group (see Appendix 5 for definitions): Here the right inverse of x with respect to Equation does in general not coincide with the left inverse of x with respect to Equation. For an example, take any Lie group G, set (D,eD)=((G×G,(e,e)), define the two canonical G –actions g(g1,g2)?(gg1,g2) and (g1,g2)g?(g1,g2g) (for all g,g1,g2,∈g), and let f: G×GG be the group multiplication. Then (D,eD,Equation,Equation) will be a general digroup with Equation.

In [13, 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 x) is equal to its left inverse (w.r.t.Equation). This can be shown to be equivalent to demanding that the general Lie digroup (D,e,Equation,Equation) 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 Equation-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 (eX,e) and the two multiplications Equation andEquation for all x1,x1X and g1,g1G.

The following theorem is a direct consequence of the functorial properties of the functor F:

Theorem 3.5: Let (D,e,Equation,Equation) be a bar-unital Lie dimonoid (D,e,Equation,Equation).

1. The underlying vector space of the associated coalgebra F(D) to the bar-unital Lie dimonoid (D,e,Equation,Equation) equipped with 1 and the multiplications μEquationEquation is an associative bar-unital dialgebra.

In case D is balanced, F(D) is a cocommutative Hopf dialgebra.

2. In case (D,e,Equation,Equation) is a Lie digroup (in the sense of Kinyon), F(D) is a cocommutative Hopf dialgebra.

(Lie) racks: Recall that a Lie rack is a pointed manifold with multiplication (M,e,m) satisfying the following identities for all x,y,zM where the standard notation is m(x,y)=xEquationy

Equation            (82)

Equation (83)

Equation  (84)

In addition, one demands that (M,e,m) be left-regular, i.e. for all xM the left multiplication maps Equation should be a diffeomorphism.

Note the following version of the self-distributivity identity (84) in terms of maps:

m°(idM×m) =m°(m×m) °(idM×τM,M×idM) °(diagM×idM×idM)          (85)

Example 3.1: Note that every pointed differentiable manifold (M,e) carries a trivial Lie rack structure defined for all x,yM by

Equation      (86)

and this assignment is functorial.

Example 3.2: Any Lie group G becomes a Lie rack upon setting for all Equation

Equation      (87)

again defining a functor from the category of Lie groups to the category of all Lie racks.

Example 3.3: Let G be a Lie group and V be a (smooth) G-module (supposed to be a real or complex vector space). On X?V×G, we define a binary operation Equation by

Equation

for all Equation and allEquation. X is a Lie rack with unit 1?(0,1) which is called a linear Lie rack.

Example 3.4: Let (D,e,Equation,Equation) be a (balanced) digroup. Then formula (13) of [5],

Equation

equips the pointed manifold (D,e, ) with the structure of a Lie rack.

Any Lie rack (M,e,Equation) can be gauged by any smooth map f: (M,e)(M,e) of pointed manifolds satisfying for all x,yM

Equation

A straight-forward computation shows that the pointed manifold (M,e) equipped with the gauged multiplication Equation defined by

Equation

is a Lie rack(M, e,Equation).

Furthermore, recall that an augmented Lie rack Equation consists of a pointed differentiable manifold (M,eM), of a Lie group G, of a smooth map φ:MG (of pointed manifolds), and of a smooth left G –action Equation Equation such that for all gG, xM

Equation     (88)

Equation    (89)

It is a routine check that the multiplication Equation on M defined for all x,yM by

Equation   (90)

satisfies all the axioms (82), (83), and (84) of a Lie rack, thus making (M,eM,Equation) into a Lie rack such that the map φ is a morphism of Lie racks, i.e. for all x,yM

Equation        (91)

A morphism Equation of augmented Lie racks is a pair of maps of pointed differentiable manifolds Ψ: MM′ and ψ: GG′ such that ψ is homomorphism of Lie groups and such that all reasonable diagrams commute, viz: for all gG

Equation             (92)

Equation          (93)

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,Equation) 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,cC, using the same notation aEquationb for μ(ab):

Equation       (94)

Equation      (95)

Equation        (96)

Proof: 1. By definition, Equation are sent by μ=Equation* to the distribution Equation. We evaluate this formula for S=1. This gives the distribution Equation. But T(2) means that the function is seen as function of its second variable, i.e. T(2) (f°Equation) (y)=T(f(yEquation−)). On the other hand, the delta distribution 1 evaluates a function in e, thus

Equation

Because eEquationy=y for all yM. This shows 1T=T.

2. Exchanging the roles of the two variables in the above computation, we obtain for TEquation1 the distribution

Equation or in other wordsEquation, i.e. the above element y is now in the second place. We obtain

Equation

This shows TEquation1=ε(T)1.

3. As remarked before, the definition of Δe, namely Equation, induces thanks to the naturality relation (73) relations like

Equation

Therefore, starting from the relation induced on Equation by relation (85), one replaces (diagM×idM×idM)* by the above and obtains finally an equation equivalent to equation (8).

Remark 3.4: This theorem should be compared to Proposition 3.1 in [3]. In [3], 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. Once again, 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 Equation is a cocommutative augmented rack bialgebra Equation

We shall close the subsection with a geometric explanation of some of the structures appearing in Subsection 2.1: Let (Equation,[,]) be a real finitedimensional Leibniz algebra. There is the following Lie rack structure on the manifold Equation defined by

Equation           (97)

Moreover, pick a two-sided ideal Equation withEquation so that the quotient algebra Equation is a Lie algebra. Let Equation 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 Equation of G on h by automorphisms of Leibniz algebras. Consider the smooth map

Equation             (98)

Clearly Equation for all xh and gG whence Equation is an augmented Lie rack, and it is not hard to see that the Lie rack structure coincides with (97).

Theorem 3.8: The C5-rack bialgebra associated to the augmented Lie rack Equation by means of the Serre functor is isomorphic to the universal envelopping algebra of infinite order, UAR(Equation), see Definition 2.3 and Theorem 2.1.

Proof: First we compute φ*exp*°p*. Since Equation is linear, it is easy to see using formula (??) that for all Equation

Equation

see (??) for a definition of ΦS. Next, for all Equation, we shall show the formula Equation

Equation

(see eqn (78) for a definition of ΦU). Both sides of this equation are symmetric k-linear maps in the arguments ξ1,…,ξk, hence by the Polarization Lemma, it suffices to check equality in case ξ1=…=ξk=ξ [19]. Since for each real number t the map Equation is the flow of the left invariant vector field ξ+, we get

Equation

proving the above formula. It follows that

Equation        (99)

Next, we compute Equation*. We get for positive integers EquationEquation

Equation

Equation

where in the last line we have used a basis of Equation, have written, y1,…,yn (n=dim(Equation)) for the components of each vector y∈Equation, and used the notation Equations for the linear mapEquation. By induction on k it is easy to prove that

Equation

and using again the Polarisation Lemma, we finally get for all uU(Equation) and α∈S(Equation)

Equation        (100)

and the isomorphism with the augmented rack bialgebra Equation 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. The composition of F with the functor of primitives gives then the tangent functor (see text before Remark 3.1). This gives a new and completely algebraic functorial way to associate to a Lie rack its tangent Leibniz algebra.

Some Definitions Around Coalgebras (Appendix)

Let C be a module over a commutative associative unital ring K (which we shall assume to contain Equation). Recall that a linear map Δ:CCKC=CC is called a coassociative comultiplication iff (Δ⊗idC)°Δ=(idC⊗Δ)°Δ, and the pair (C,Δ) is called a (coassociative) coalgebra over K. Let Equation be another coalgebra. Recall that a K-linear map Φ:CC′ is called a homomorphism of coalgebras iff Δ′°φ=(⊗φ)°Δ. The coalgebra (C,Δ) is called cocommutative iff τ°Δ=Δ where τ: CCCC denotes the canonical flip map. Recall furthermore that a linear map ε: CK is called a counit for the coalgebra (C,Δ) iff (ε⊗idC)Δ=(idCε)°idC. The triple (C,Δ,ε) is called a counital coalgebra. Moreover, a counital coalgebra (C,Δ,ε) equipped with an element 1 is called coaugmented iff Δ(1)=1⊗1 and ε(1)=1∈K. Let C+C denote the kernel of ε. Recall that a morphism Equation of counital coaugmented coalgebras over K is a K-linear map satifying Equation. Moreover, for any counital coaugmented coalgebra the K-submodule of all primitive elements is defined by

Prime(C)?{xC Δ(x)=x⊗1+1⊗x}.          (101)

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, we shall call a counital coaugmented coalgebra connected iff the following holds: The sequence of submodules Equation defined by C(0)=K1 and recursively by

Equation              (102)

is easily seen to be an ascending sequence of coaugmented counital subcoalgebras of (C,Δ,ε,1), and if the union of all the C(k) is equal to C, then (C,Δ,ε,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 CC(k) from the category of coaugmented counital coalgebras to itself. We shall use the following acronyms:

Definition 4.1: We call a coassociative, counital, coaugmented coalgebra a C3-coalgebra. In case the C3-coalgebra is in addition cocommutative, we shall speak of a C4-coalgebra. Finally, a connected C4-coalgebra will be coined a C5-coalgebra.

Recall also that the tensor product of two counital coaugmented coalgebras (C,Δ,ε,1) and Equation is given byEquationEquation. Tensor products of connected coalgebras are connected. Recall the standard example: Let V be a K-module and Equation 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 xyyx for all x, yV. Denoting the commutative associative multiplication in S(V) (which is induced by the free multiplication) by , i.e.

Equation

we have Δ(x)=x⊗1+1⊗x for all xV and Δ(x1•…•xk)=(x1⊗1+1⊗x1)•… •(xk⊗1+1⊗xk) for all positive integers k and x1xkV. Recall that S0(V) is the free K-module K1 and the counit is defined by (λ1)=λ for all λ∈K and by declaring that ε vanishes on Equation. Moreover, the submodules (S(V))(n) are given by Equation, whence S(V) is clearly connected, so it is a C5-coalgebra whose submodule of primitive elements equals V.

Moreover, for a given coalgebra (C,Δ)) and a given nonassociative algebra (A,μ) whereμ:AAA is a given K-linear map, recall the convolution multiplication in the K –module HomK(C,A) defined in the usual way for any two K-linear maps φ,ψ:cA by

Equation        (103)

In case Δ is coassociative and associative, * will be associative. The following fact is rather important: If C is connected and if the K-linear map ?:CA vanishes on 1C, then any convolution power series of ? converges, i.e. the evaluation of some formal series Equation (with λrK and ?*0?1AεC) on cC always reduces to a finite number of terms. In particular, let :CA be a K-linear map such that ψ(1C)=1A. Then –as has been observed by Takeuchi and Sweedler– ψ has always a convolution inverse, i.e. there is a unique K-linear map Equation such thatEquation, where ψ′ is defined by the geometric series Equation [20,21]

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 [22] 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 Equation. 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 wellknown that for each x there is exactly one inverse element, called x−1.

Note that by a Lemma by L. E. Dickson 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), 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 5.1: 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 Equation 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– that all the left multiplications Equation (resp. right multiplications Equation) 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:

Equation        (104)

Equation      (105)

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 Equation andEquation 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 for a proof [23]. Hence Γe is a connected Lie group, and the Suschkewitsch decomposition Γ≅Γe×E, see Appendix 5, 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.

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. came independently to similar results which we incorporated in this paper. M.B. thanks Nacer Makhlouf for his question about the relations of rack-bialgebras to dialgebras, and Gwénael Massuyeau for his question about universals.

1For an explicit formula, see the end of the proof of the theorem.

2Recall that this means c=(Equation°Δ)(c).

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