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Received Date: June 17, 2009; Revised Date: September 20, 2009
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This paper is concerned with explaining and further developing the rather technical denition of a hom-Lie algebra given in a previous paper which was an adaption of the ordinary denition to the language of number theory and arithmetic geometry. To do this we here introduce the notion of Witt-hom-Lie algebras and give interesting arithmetic applications, both in the Lie algebra case and in the hom-Lie algebra case. The paper ends with a discussion of a few possible applications of the developed hom-Lie language.
The main purpose of this note is to explain the rather technical denition of a hom-Lie algebra given in  (in addition to motivating why I made that denition) and to provide a few novel examples of distinct number theoretical avour.
Let me point out, however, that in order to draw any serious and deep conclusions of a number-theoretical nature from the association with hom-Lie algebras, one needs to know more of the ner structure of hom-Lie algebras, something that is yet to be investigated. On the other hand, a recent preprint  proves that hom-Lie algebras are actually Lie algebras in a suitably braided category1. This should certainly aid in the study of the structural characteristics of hom-Lie algebras. In fact, this result implies that every structural result on Lie algebras should have a precise hom-Lie analogue. Therefore, most results on Lie algebras should be transferrable (in principle, at least) to a hom-Lie version. This is something that certainly should be studied further.
The contents of the paper is as follows. Section 2 deals with the denition of hom-Lie algebra and hom-Lie structure, both as given in  and in a slightly dierent, but equivalent, way that might be more easily understood from a purely algebraic point of view. Section 3 introduces Witt-Lie algebras in a general way and shows that already this \un-twisted" case is full of potential number theory, such as Gauss sums and CM-elliptic curves. Section 4 generalizes the Witt-Lie algebras to Witt-hom-Lie algebras and this is then studied in some detail. Interspersed throughout the text are questions and suggestions for further study.
In the nal few subsections of the paper, we give some "teasers" how the constructions introduced might be used in arithmetic. I am rather condent that there are interesting structures here waiting to be unveiled, both with respect to the study of hom-Lie algebras and the study of arithmetic and geometry.
Finally let me issue a warning: Some parts of this paper require more background than others and I do not always indicate what this background is. However, in most instances, I give pointers to the relevant literature where details may be found.
Notations. The following notations will be adhered to throughout.
• Λ will denote a commutative, associative integral domain with unity.
• Com(Λ) (Com(B), etc.) denotes the category of commutative, associative Λ-algebras (B-algebras, etc) with unity. Morphisms of Λ-algebras (B-algebras, etc.) are always unital, i.e., Λ(1) = 1.
• AΧ is the set of units in A (i.e., the set of invertible elements).
• EndΛ(A) denotes the Λ-module of Λ-algebra morphisms on A.
• will mean cyclic addition of the expression in bracket.
• Sch denotes the category of schemes; Sch=S denotes the category of schemes over S (i.e., the category of S-schemes) where S is some base scheme.
• When writing actions of group elements, we will alternatively use σ(a) and aσ, depending on the context.
• The notation Ag, AG will denote the xed ring of g ε G and G, respectively, i.e.,
• A will always denote an abelian group.
Here we work "backwards" compared to the denition given in , as we feel that this might give a somewhat more natural (i.e., less "Bourbaki-ist") introduction going from the special to the general.
Let A be a Λ-algebra and L an A-module. (Those who prefer can think of Λ=A = F, a eld, and L, an F-vector space.) Assume that σ is a Λ-linear endomorphism on L.
Definition 2.1. A hom-Lie algebra on L is a tuple where is Λ-bilinear product satisfying
A morphism of hom-Lie algebras and is a Λ-module morphism such that
Remark 2.2. One could wonder why the Λ-module A is there at all since everything in the denition is Λ-linear. But this is to allow greater exibility as it will hopefully become clear later.
Let G be a group of endomorphisms on L. We dene a G-hom-Lie structure on L to be a collection of hom-Lie algebras parametrized by the elements of G. More to the point, a G-hom-Lie structure on L is a family of hom-Lie algebras
Morphisms of G-hom-Lie structures are bit more subtle to dene: let L(G) and L'(G') be two G-hom-Lie structures (with dierent G's and underlying A-modules). Then a morphism is a pair consisting of a Λ-module morphism and a group morphism such that
and such that
The fact that G is a group means that any G-hom-Lie structure includes a Lie algebra corresponding to the unit element of G. This Lie algebra may, or may not, be the abelian Lie algebra. In this way, the G-hom-Lie structure can be viewed as a family of "deformations" of the Lie algebra in L(G) (strictly speaking there could be Lie algebras in the family corresponding to group elements dierent from the unit), making the notion of G-hom-Lie structure a very pleasing and intuitive construction.
Let me also remark that the "deformations" in the hom-Lie families are not "quasideformations" in the sense of . To recall, the quasi-deformation concept, can loosely be thought of as "deformations" of certain representations of Lie algebras. On the other hand, here the Lie algebras themselves are "deformed" (while not in any continuous, at or other "geometric" ways) in the category of hom-Lie algebras.
Theorem 2.3. Let be a hom-Lie algebra over a ring A and let A → B be a morphism of Λ-algebras. Then
where ( o ) denotes the multiplication in B, is a hom-Lie algebra over B.
It is very easy to prove this directly; we leave this to the reader.
It is obvious what is to be meant by sub-hom-Lie algebra. However, there is another snotion that is natural to consider here, namely, sub-hom-Lie structures.
Let (L;G) be a G-hom-Lie structure, where L is an A-module and G a group acting on L (and possibly also A). Then an H-sub-hom-Lie structure of (L;G) is a pair (K;H) together with two injections and . Notice that this includes the cases where either of these is the identity.
Dualizing, let (L;G) be a G-hom-Lie structure over A; then a quotient hom-Lie structure is an H-hom-Lie structure (K;H) and a pair of surjections and
Let H → G be a group morphism and L a G-hom-Lie structure. Notice that this means, in particular, that we have a representation Clearly, restriction induces a H-hom-Lie structure on L via
Hence, in particular, this holds for H a subgroup of G, thus inducing a sub-hom-Lie structure.
Similarly, for a surjection and a G-hom-Lie structure L, we get an induced quotient H-hom-Lie structure on
Global Hom-Lie structures
The \global" denition given in  was a bit more restrictive than necessary. Therefore, there is some discrepancies in the wording of the one below and the one from . I think that the one given here should be inclusive enough for most arithmetic circumstances.
Fix a scheme S ε ob(Sch). Let (S)fl denote the (big) at site associated with S. To recall, this is the category of morphisms U → S (the "open sets" of S), locally of nite type, with the obvious morphisms, .The covering families are families of at morphisms where i ε I for some index set I. For more details on this see  for instance. By we denote a sheaf of groups on (S)fl . Let W be a sheaf of --sets over (S)fl i.e., a sheaf W over (S)fl together with an action of for U → S.
The reason for this generality is that for arithmetic and geometric purposes, it is advantageous to allow for more general topologies than the Zariski topology; for instance etale topology  or the positive topology . (Of course, we could use any topological space as base here, i.e., not necessarily restricting our discussion to schemes.)
Let be the structure sheaf on ( S)fl in the sense that for and let A be a sheaf of - algebras. We denote by F an A-module. Let be a sheaf of groups over ( S)fl acting -linearly on F.
Denition 2.4. Given the above data, a hom-Lie structure for ,or -hom-Lie structure, on ( S)fl is a - -sheaf of A-modules F together with, for each covering an bilinear product such that
A morphism of hom-Lie structures and is a pair consisting of a morphism of -modules and a morphism of group schemes such that and where we have put
We thus get a category HomLieStrucS of all hom-Lie structures on ( S)fl with morphisms given in the denition.
Hence, a hom-Lie structure is a family of (possibly isomorphic) products parametrized by A product for fixed is a hom-Lie algebra on F.
In all cases, is a constant sheaf of groups, i.e., for all where G is a group (or group scheme over S). We will assume this from now on.
When we only consider one open set U = S, Denition 2.4 specializes to
(only one product for each g ε G).
When we need to specify the dierence of the above case and Definition 2.4, we call this the special case and Denition 2.4 the global case (and so the global case includes the special).
We will now introduce the notion of \arithmetic Witt algebras". These are graded Lie algebras coming endowed with obvious number-theoretical content. The notion of \generalized Witt algebras" have been around for a few decades and there are several more or less equivalent ways to dene this; we have chosen one that is suitable for our present needs, namely, number theory. We have not been able to nd this arithmetic application anywhere in the literature. In the next section we will generalize this to \Witt-hom-Lie algebras", taking into account "Galois structures".
The classical Witt-Lie algebra (the reason for the unorthodox notation will be clear from the discussion below) is dened as the complexied polynomial vector elds on the unit circle. More to the point
induced from the commutator. It is clearly Z-graded.
This can be generalized as follows. Let A be an abelian group written additively and let where Λ is an integral domain (this assumption is kept throughout), be a 1-dimensional character, i.e., a group morphism into the additive group of Λ (the superscript "a" is there to remind us that the character is additive). Denote by the free Z-module spanned by the formal symbols Let be a Z-algebra morphism, i.e., First define
where is a Λ-valued, symmetric group 2-cocycle, i.e., a map satisfying
Then dene a Λ-linear product on the base extension to T,
where and similarly for . Notice that does not define a multiplicative map unless is multiplicative2.The above construction gives the structure constants
and it is easy to check that this denes an A-graded Lie algebra called the Witt algebra of .
As to not be drown in awkward notation we will most often, instead of the correct simply write g, remembering that g is strictly an element of a group and not of the ring T. This will certainly not cause any severe confusion. When confusion does lurk, and used. Consequently, we also drop from the notation
We will nd this description more suitable in what follows. Therefore, it is strongly graded and crystalline in the sense of .
Remark 3.1. The above can be considered the "rank one" case of the following construction. Let S be an A-set (i.e., a set together with an action of A). Form the set of formal symbols Define the multigraded Lie product
We will however stick to the \rank one" case.
Remark 3.2. We remark also that this construction is functorial. Indeed, dene the category Ab(3) as the category whose objects are triples and morphisms group morphisms such that and Then is a functor from Ab(3) to the category Lie of Lie algebras.
To show that this is actually a very general notion (although we will generalize further), let us give a few simple examples.
Example 3.3. We rst consider two simple examples.
(a) The classical case is the Witt algebra for with the unit (principal) character and α the constant 2-cocycle
(b) Take and Now is the algebra that E. Witt actually studied. This can be given the following more conventional description. Consider the following vector space:
This is a Lie algebra under the commutator. In fact, we have the following isomorphism of Lie algebras:
When p ≠ 2, these are simple (modular) Lie algebras.
Clearly, more elaborate versions of these examples can be constructed by varying A and/or .
We will now give two more sophisticated examples coming from arithmetic.
Witt algebras from elliptic curves
Let K be a eld and let E be an elliptic curve over K. Let D be a K-algebra.
By the Mordell-Weil theorem, the group of L-rational points for L/K a nite extension of number elds, is a nitely generated abelian group. In addition, the group of N-torsion points where denotes the algebraic closure of K, i.e., the group of all points such that is isomorphic to This holds in general when K is a eld of characteristic prime to N. The full group of torsion points is the union of all (we assume here that char(K) = 0).
Using these groups we can now form three natural Witt algebras. Namely
There are several interesting possibilities for the K-algebra D. For instance, (a) D = L, a eld over considered as an -algebra; (c) where where v is a (discrete) valuation of K, and Kv the completion of K with respect to v; of course, we could also consider the ring of integers
The case (c) is interesting only when End the complex multiplication (CM) case.
Remark 3.4. Assume that K is a eld. Since D is an --algebra, (for instance) is also an -module. This means that shea es to a quasi-coherent sheaf on the 1-dimensional arithmetic scheme Spec().Therefore, we get a sheaf of Lie algebras over Spec ()
This example will be even more interesting in the context of hom-Lie algebras.
A cyclotomic Witt-Lie algebra
Let where p is prime.We will also consider the multiplicative group of units. In this case, a natural choice of additive character will be
Furthermore, let be the group of nth roots of unity. Take the nth cyclotomic eld, and pick a Dirichlet character i.e., a character ϑ such that Extend ϑ to a character on the whole A by dening such that #(gh) = #(g)#(h). Extend # to a character on the whole A by dening #(0) = 0, unless ϑ is the principal (unit) character in which case we define =1 There is nothing stopping us from letting T ≠ below. In fact, convenience is the only reason for assuming equality here.
where α is a 2-cocycle, with the linear Lie algebra structure given by (3.2). Inside this vector space there is an element on the form
a so-called Gauss sum. Recall that (h - g) is actually Letting ϑ* be another character, we can compute
Introducing the anti-symmetric pairing
we have the following nice description of
Proposition 3.5. Given the above, we have
Proof. The proof is a simple computation and is omitted.
Example 3.6. For an explicit example consider the two Dirichlet characters on (extended to Z=5Z) dened by
Now we compute
and so using the proposition , we get
Notice that, even in the case is not a Gauss sum, since the coefficients are not the image of some Dirichlet character (the images of which are all roots of unity). But there is more: even though, by denition, and have no term, their product in general has. Hence, this product is in general not even alinear combination of Gauss sums.
Remark 3.7. Normally, when computing with Gauss sums, the symbols are powers of some primitive mth root of unity Therefore, the Gauss sums are elements in the relative extension These extensions are rather cumbersome to work with, even when gcd(m; n) = 1, so working with symbols, as we have done, claries computations in our opinion.
We will use a generalization of Gauss sums further on. Therefore, let L=K be a nite Galois extension. Choose a multiplicative character such that Then the so-called resolvent is the group algebra element
(where the last equality is as vector spaces). We consider as an operator on L:
In other words, acts on L as σ. The element satises the following property: for any and we have
where is the inverse character. As consequences we have
Gauss sums are the result of the above when in which case Gal Particularly interesting is the case when ϑ is the unique order-two character, namely the Legendre symbol or more generally of course, mth power residue symbols. We leave it to the reader to express the product of two resolvents in analogy with the Gauss sum above. Notice, however, that this is only possible for abelian extensions L=K.
For an idea of why Gauss sums (or resolvents) are of utmost importance in number theory, see  for instance.
These examples show that there is indeed interesting arithmetic for ordinary Lie algebras. Despite this we will up the stakes and give a hom-Lie algebra generalization of the above to show that we can capture signicantly more arithmetic by (for instance) explicitly involving the Galois structure (where present). Actually, this case will in some sense be more natural than the Lie algebra case given above.
Let as before A be an abelian group with a character which will be (almost) constantly suppressed. Furthermore, let T be a Λ-domain, a subgroup together with a representation where V is a vector space (where denotes eld of fractions of T). Let x be the character of x. If the image of x is not in T, make a base extension to
Consider as the group algebra T[A], spanned by formal symbols be an algebra endomorphism on T and extend to
and then linearly as
This denes an algebra endomorphism on we define i.e., we demand that σ acts as a group endomorphism on A. Notice that we have yet to decide how to interpret and Define once again
a symmetric 2-cocycle. We assume that Let E be a T-module and make the base extension acts on E(A) as
Extend the action of G on T to a semilinear action on E:
Now we twist the action of G on E (and thus on T) as
Then (we suppress the dependence on ` in the notation) is a twisted derivation on E, i.e., a linear map satisfying
and where δ is the induced twisted derivation We extend the action of via the Tate twist. Explicitly,
This is a semilinear action of G on E[A]. It follows that can be extended canonically to E[A]. Assume that where
We denote this by Letting σ vary over G, we get the G-hom-Lie structure
where we, as shown, often omit α and x from the notation since these are xed in the G-hom-Lie structure.
Using the twisted Leibniz rule we nd the structure constants:
Notice that, unless for all g ε A, this product is not graded. We refer to it as a "σ-twisted grading". In  there are explicit examples of this phenomenon.
Remark 4.1. The above constructions are all functorial on suitably dened categories, just as in Remark 3.2. However, the constructions involve a lot of notation so, since this will not be important for us, we omit it, but invite the interested reader to construct these possible categories for her- or himself.
We keep the notation and conventions from before (but here T does not have to be a domain) and introduce the product
on the algebra The G-hom-Lie structure thus constructed is denoted by
(compare with from the previous section). Now we compute
There are two important points to make here: (1) Notice that
from the above computation is the analogue of equation (3.2); (2) observe that (4.4) is exactly the \algebra factor" in the structure constant equation (4.2). This means that (4.2) and (4.4) dene isomorphic hom-Lie algebras (under suitable conditions). That (4.4) indeed denes a hom-Lie algebra follows as a special case of [4, Theorem 3.1] (or can be proven directly with a straight-forward, albeit tedious, computation).
We will now use the above construction to generalize the Gauss sum construction from a previous section.
Gauss sums in Witt-hom-Lie algebras
Let L=K be an abelian Galois extension and put we denote two multiplicative characters of order We will now consider the Gauss sums (resolvents)
and their products in where (not necessarily in Gal(L=K)), We dene the action of A on as
To confuse things we will sometimes write the product in A as "+" and sometimes as composition Also, we define Just as for the previous case we can compute
where we, in the last equality, used that A is abelian, α symmetric and that
We now have the obvious generalization of Proposition 3.5 with essentially the same proof.
Proposition 4.2. Given the above, we have
CM-Elliptic curves and Witt-hom-Lie algebras
In this subsection we freely use concepts from complex multiplication and class eld theory. Most (all?) of what is used here can be found in [9, Chapter 2].
Let K be an imaginary quadratic number eld, i.e., a square-free integer. Furthermore, let be an elliptic curve over a eld F with complex multiplication by the ring of integers
Then the theory of complex multiplication tells us that where is the j- invariant of is the Hilbert class eld of K. In addition, the eld is a nite abelian extension of for all N. The extension is not necessarily abelian. In fact, we have the following exact sequence:
implying that is the semi-direct product
Now , we have
where is the class number of K and the Artin (reciprocity) map induces an isomorphism
with denoting the class group of K.
Obviously, with this setup we have several interesting possibilities for constructing Witthom- Lie structures. Consider, for instance,
Then acts on both and the grading group via the restriction morphism
In this way we get a Witt-hom-Lie structure or, alternatively, a Gal -structure (as discussed before).
Remark 4.3. The actual structure of the hom-Lie algebras (structures) thus constructed remains to be investigated, but let me express some doubt as to whether such an investigation will have any deep implications for number theory.
We keep the assumptions from the previous subsection, except that is not necessarily a CM-curve.
Consider now the case when m = p a prime. Then
(the "Tate group algebra over K")
where is the ring of p-adic integers. Given a 2-cocycle, we can lift this to a 2-cocycle via the successive
Twisting the action of on to gives us a GK-Witt-hom-Lie structure
Of course, there is nothing to stop you from taking some K-algebra instead of K and some other 2-cocycle instead of the induced αp.
Ideally, this construction would give us alternative ways to study Galois representations, i.e., representations of (absolute) Galois groups, in the form of hom-Lie structures. Thus, in a sense, we are in this way constructing "wisted Galois representations".
We can sheafify to a sheaf over and in this way we get a family of modular twisted Galois representations. (The details here is yet to be worked out.)
Rational points on abelian varieties
The study of rational points on abelian varieties is of fundamental importance in arithmetic and Diophantine geometry. Here we barely indicate how hom-Lie methods might aid in this study, leaving a more detailed exposition to a later treatise.
Let be an abelian variety over a number eld K. Clearly, acts on the L-rational points on , where is a eld extension. The Mordell-Weil theorem says that, if is finite, then tors.It is well known that is nite for all nite extensions Equally well known is that the N-torsion points
Since is nite, we have that is nite. But so is also nite.
By denition, is an abelian group, so is a commutative group algebra and GK acts on this in the obvious fashion. Therefore, we can form
for a character and a 2-cocycle.
The action of GK on induces an action on the quotient
and so we get a surjective morphism of GK-hom-Lie structures
Fitting this into the fundamental sequence
where is the Nth Selmer group and the N-torsion part of the Tate-Shafarevich group, we get a sequence of group algebras
The question that arises is, can this be extended to a sequence of hom-Lie structures? The
answer is yes, but in general not G
-hom-Lie structures. The reason for this general 'failure" is that the action of GK does not in general lift to an action onand hence, in general, not to or . Therefore, one needs to restrict to certain subgroups of GK. Unfortunately, the details of this has to be postponed to another paper.
Remark 4.4. The above discussion gives us a way to see a hom-Lie structure as something that is canonically given by the abelian group structure of much like the Lie algebra structure of algebraic groups (in this case this is obviously abelian). In this sense, the hom- Lie structure captures signicantly more information than the Lie structure, since it involves the rational points and the Galois action on these in a very explicit manner.