We discuss several partial solutions to the so-called “coquecigrue problem” of Loday; these solutions parallel, but also generalize in several directions, the classical Lie group-Lie algebra correspondence. Our study highlights some clear similarities between the split and nonsplit cases and leads us to a general unifying scheme that provides an answer to the problem of the algebraic structure of a coquecigrue.
As is now well known, Leibniz algebras are a noncommutative, or rather, non-anti-symmetric, generalization of Lie algebras. The so-called “coquecigrue problem” was proposed by J. L. Loday as an analogue for these algebras of Lie’s third fundamental theorem: given a finite dimensional Leibniz algebra, look for a manifold possessing an algebraic structure such that its linearization yields the original Leibniz algebra. Ideally, this algebraic structure would be a binary operation, but because there is no formal, precise, definition of what a coquecigrue is, this has proven to be a rather difficult question. To what extent this program can be fulfilled is, as of now, still not entirely clear, but we will describe here what we believe is the right approach.
A first hint at a general answer to this problem first appeared in [
In the same paper it was also argued that, again via an appropriate rack, an answer for
Nevertheless, because the splittings of a Leibniz algebra are not necessarily unique, this opens up questions of uniqueness of these integral manifolds that in a sense are even more delicate than the standard situation in the classical Lie theory, where groups that are locally but not globally diffeomorphic have the same algebra.
One of our main objectives here (Section
In another direction, since the digroup construction does not work for nonsplit Leibniz algebras, a different approach is required here. The best results to date for this case were given in the Ph.D. thesis of Covez, [
With this in mind, in Section
Then, on the basis of the different types of solutions thus far considered, we reach our final aim in this work, which is to propose what seems to be a good framework for a general geometrical solution to the coquecigrue problem. The construction, given in Section
All the integral manifolds for a Leibniz algebra start with the fact that these algebras have some natural quotients that are Lie algebras. Therefore, it is natural to suspect that the integral manifolds will have the structure of a manifold that projects onto a Lie group related to these algebras. Fiber bundles fulfill this requirement but do not posses in general an
In Section
Finally, since this work deals with both algebraic and geometric constructions, we tried to make our presentation rather self-contained, down-to-earth, and based considerably on the concrete examples of Section
The purpose of this introductory section is mostly to recall some standard and well-known facts about Leibniz algebras, but also to introduce some useful examples. The basic reference here is [
A (left)
Lie algebras are obtained if the bracket is antisymmetric; this condition being then equivalent to the Jacoby identity, but in general it is a strong generalization.
One can of course define morphisms, ideals, quotients, and so forth, for Leibniz algebras in the usual way, and the most important instance of a quotient in this category naturally occurs when this quotient is a Lie algebra. The minimal ideal for which this holds is the two-sided ideal
On the other hand, as in the Lie case one can define and adjoint mapping by the assignment. For any
A simple class of Leibniz algebras, which are not in general Lie algebras, can be constructed as follows (see e.g., [
Let
Let us next recall the notion of splitting of a Leibniz algebra. For convenience, let us start by describing the general framework, which is as follows.
Suppose
But now, an important fact about
Let
It should be noticed that in general
It is rather easy to see that the
Notice that
Moreover, the
On the other hand, a class of nonsplit Leibniz algebras, called derived algebras in [
Let
Verification that this does indeed define a Leibniz algebra is immediate, but the point is that derived algebras are in general neither Lie algebras nor split.
Straightforward computations show that for the two-dimensional non-abelian Lie algebra,
Indeed, here the ideal
As mentioned, the relation of the splitting of algebras to the coquecigrue problem was discussed in [
A
The digroup is a Lie digroup if
Associated with the
For any digroup
One can also define projections
Now, given a Lie digroup, a Leibniz algebra is obtained as follows. To begin with, define a conjugation or
A For all For all
Pointed racks are perhaps most naturally obtained by conjugation on groups, but the point is that, just as in the case of the standard conjugation in Lie groups, a Leibniz bracket is obtained at the tangent space of a Lie digroup at the bar unit,
More explicitly, if
That this is well defined follows from the fact that the conjugation given by (
Since this derivation imitates in every respect the construction of Lie algebras from Lie groups, it is naturally a strong indication that digroups should be considered as an acceptable solution to the coquecigrue problem. Nevertheless, as is also well known this works only in the split case; let us now briefly recall why this is so.
The first key step is to observe that the
The digroup
In fact, every digroup arises this way: from a group action,
At any rate, for Lie digroups, since as a manifold the tangent space at
Conversely, given a split Leibniz algebra, written as a direct sum
Up to this point we have discussed mostly well-known facts. But now a subtle, yet important, point arises. As mentioned, a split Leibniz algebra might split in different ways, depending on the chosen ideal
An abelian Leibniz algebra
Then, the digroup
Now, while the example of an abelian algebra might seem somewhat artificial, since there is little to check in this case, one can further elaborate on this point, to see that the problem runs deeper.
For this recall that any Lie algebra
Now, let
This observation then gives two things.
On the one hand, it indicates that, for a given Leibniz algebra
On the other hand, and more important for us here, this implies the following theorem, which is one of our main results.
Suppose
Indeed, let
The relevance of this theorem is that it shows that, even for Lie algebras, if we drop the
As a consequence, in our view these new and somewhat odd nonstandard integral manifolds for the type of Lie algebras considered in the previous theorem should be regarded as the first nontrivial examples of
In any case, explicit examples of these highly nonequivalent integral manifolds for non-abelian Lie algebras can be obtained from the Levi decomposition theorem, and this provides nontrivial examples of these coquecigrues; for instance, this is the case where
Consider the reductive Lie algebra
Moreover, since as manifolds both digroups are diffeomorphic, there are no hidden geometrical anomalies that would compel us to disregard the nonstandard solution either.
And this fact actually lies beneath the possibility of splitting a
A different kind of generalization of the construction of integral manifolds for split Leibniz algebras comes from the observation that the computation of Formula (
Let
A first problem is that, in contrast to the digroup case, for a general bundle
Let
Let
Thus, if an action has fixed points, as do the actions required for the digroup construction, the choice of one of them, say
Let
Let finally
Then
Before proceeding with the proof, let us make the following comment, regarding the choice of the operation. This comes of course from the fact that we want to generalize the construction for digroups and so we could also start with (
Let
Now, since
Finally, it is clear that upon projection the operation
Turning things around, we can now state the relation to the coquecigrue problem as follows.
Let
Then, the associated bundle has the structure of a pointed rack and upon differentiation this structure gives a Leibniz algebra structure to
The conditions on
Thus, we have to show that
Now, the conditions
Then, the second condition for a rack amounts to solving the equation
Finally,
Now, according to Kinyon’s result, the data needed to construct the bundle
In this subsection we analyze a rather restricted class of nonsplit algebras, but our objective here is mostly to exhibit a different kind of possible solution to the coquecigrue problem and investigate the common traits between this new construction and what we had in the split case.
So, let
Now assume that
The
That this definition makes sense is the content of the following.
Let
Before proving the theorem, let us establish the following easy lemma.
Let
We simply compute
First we notice that for any
Next, we verify the conditions for a pointed rack, taking as distinguished point the unit
On the one hand, using the lemma we have
Moreover, using the chain rule and the fact that
Let us now make some important observations regarding this construction.
First, notice that relative to the ideal
Moreover, the map
Thus, in a way similar to the split case, the integral manifold constructed for
The following example further illustrates this.
We let
The rack structure is then explicitly given by
Notice in particular that (
As mentioned in the introduction, in [
Somewhat more precisely, in Covez’s work coquecigrues are obtained as central extensions of Lie groups by modules. The modules are defined by an action
Leaving the more technical details aside, let us just add that a key feature of this approach is that, by means of an augmentation process, the theory makes it possible to discriminate among the full category of Lie racks those that allow a good recovery of the Lie algebra-Lie group correspondence, which, as mentioned, is normally considered an essential point for the solution of the coquecigrue problem. Nevertheless, and as the author readily acknowledges, this answer is not complete, because the requisite of simple connexity of the group
Now, the construction presented in the previous section can be related to Covez's as follows.
For any Lie group
This is the general idea of the relationship between the two approaches, but at this point we need of course to verify several things. In particular, regarding the exponential rack structure in
Now, to check the last assertion, recall first that the pull-back bundle is explicitly defined as
So let
Of course, this does not solve the problem for more general algebras, since if we start from a pullback bundle and rack structures, we would need to determine from this data the representation and
Nevertheless, we can show that this makes sense, by explicitly comparing to the Covez data in a simple case. Indeed, consider the exponential coquecigrue of Example
To begin with, we identify the action of
On the other hand, since the only nontrivial bracket is
Therefore, the local rack structure integrating this Leibniz algebra, according to Covez's method, is
Now we have to relate both constructions of the coquecigrue, but this amounts to showing that the map defined by Formula (
Drawing from the results discussed for the split case and the construction presented in the previous section, we now describe a possible integration scheme for general Leibniz algebras. Before going into the final definition, let us make some
Suppose we start with a Leibniz algebra
Then, collecting these data, we propose that a general scheme for the coquecigrue should not just be a manifold, but a commutative diagram of smooth manifolds and maps of the following type:
Now, to specify what conditions are to be imposed on the remaining objects that complete the diagram, we argue as follows.
First, by definition the manifold
To explain more precisely what we mean by this, let us consider, after the observation of [
That this is indeed a rack is essentially due to the relation
Now, the point to be made here is that a key difference between the rack of Fenn and Rourke and the rack obtained by conjugation in the Lie group is that the former does not arise from a “double integration” of the Leibniz algebra structure; indeed, to recover the original Leibniz algebra from this rack, all we need to do is differentiate with respect to the
Therefore, one natural way to codify the desired restriction is to ask that the map
Moreover, although the above algebraic condition is in a sense the main restriction, there are also topological conditions that are natural. On the one hand, the map
Thus, gathering all of the above, the definition of a coquecigrue that we propose is as follows.
Let
An
Of course, from a geometrical point of view, one would think of the manifold
Moreover, notice that the exponential coquecigrues discussed before, as well as digroups, fit well into this scheme and even Covez's construction has these properties too; the bundle being trivial in the last two cases.
Finally, to address the requirement of a good reduction to the classical construction for the case of Lie algebras, observe that when the Leibniz algebra is the Lie algebra
But for example, if we consider the construction of the rack of Fenn and Rourke for a general Lie algebra, in view of the relation (
Put another way, in the sense of our definition, in general a Lie algebra
It should be pointed out that there are other approaches to the coquecigrue problem that have appeared in the literature; for instance, an interesting one is given in [
Our construction is certainly closer in spirit to Covez’s, in that we do attempt to obtain such an explicit manifold, but there might also be a relationship to Mostovoy’s solution, for the following reason. As mentioned, the rack of Fenn and Rourke also gives a global and general, but equally “unsatisfactory,” solution to the coquecigrue problem. However, as can be seen from diagram (
In summary, in our view Definition
Granted, the manifold
Moreover, this also leads to a better understanding of something that was missing in the original coquecigrue problem, in order for it to be well posed; namely, some hypotheses about the structure of the solution, such as the ones in our construction, need to be added. In fact, as our discussion leading to Theorem
The above remarks also suggest some interesting problems for future work; for instance consider the following. What are the functorial properties of this construction? Since the admissible coquecigrues are given in terms of commutative diagrams, and in particular this already includes the analogue of the exponential map, it is not unreasonable to expect that a similar analysis to the one of the classical Lie theory can be carried out to a large extent. Finding, and if possible classifying, the different admissible coquecigrues associated with a given Lie algebra. As we saw, it should be expected a nonuniqueness problem not only at the level of the global topological structure of the manifold, but also at the level of the algebraic structure. In particular, it would be nice to have something like a `universal' coquecigrue for a given algebra.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Fausto Ongay wishes to thank the Departamento de Geometria y Topologia of the Universitat de València, for its kind hospitality while in a sabbatical stay—also supported by a CONACYT grant, when part of this work was done. This work was partially supported by Grant MTM2009-08933 from the Spanish Ministry of Science and Innovation and by CONACYT, Mexico, Project 106 923.