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Journal of Generalized Lie Theory and Applications
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Arithmetic Witt-hom-Lie algebras

Daniel LARSSON*

Hφgskolen i Oslo, Pb 4, St. Olavs plass, 0130 Oslo, Norway

*Corresponding Author:
Daniel LARSSON
Hφgskolen i Oslo, Pb 4, St. Olavs plass, 0130 Oslo, Norway
E-mail:
[email protected]

Received Date: June 17, 2009; Revised Date: September 20, 2009

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Abstract

This paper is concerned with explaining and further developing the rather technical de nition of a hom-Lie algebra given in a previous paper which was an adaption of the ordinary de nition 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.

Introduction

The main purpose of this note is to explain the rather technical de nition of a hom-Lie algebra given in [4] (in addition to motivating why I made that de nition) 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 [1] 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 de nition of hom-Lie algebra and hom-Lie structure, both as given in [4] and in a slightly di erent, 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 con dent 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.

image 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.,

image

• A will always denote an abelian group.

The basic constructions

Here we work "backwards" compared to the de nition given in [4], as we feel that this might give a somewhat more natural (i.e., less "Bourbaki-ist") introduction going from the special to the general.

Hom-Lie structures

Hom-Lie algebras

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 image whereimage is Λ-bilinear product satisfying

image for all image

image

A morphism of hom-Lie algebras image and image is a Λ-module morphism image such that image

Remark 2.2. One could wonder why the Λ-module A is there at all since everything in the de nition 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 de ne 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

image

Morphisms of G-hom-Lie structures are bit more subtle to de ne: let L(G) and L'(G') be two G-hom-Lie structures (with di erent G's and underlying A-modules). Then a morphism image is a pair image consisting of a Λ-module morphism image and a group morphism image such that image

and such that

image

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 di erent 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 [5]. 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 image be a hom-Lie algebra over a ring A and let A → B be a morphism of Λ-algebras. Then

image

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.

Substructures

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 image and image. Notice that this includes the cases where either of these is the identity.

Quotient structures

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 image and image

Changing groups

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 imageClearly, restriction induces a H-hom-Lie structure on L via

image

Hence, in particular, this holds for H a subgroup of G, thus inducing a sub-hom-Lie structure.

Similarly, for a surjection image and a G-hom-Lie structure L, we get an induced quotient H-hom-Lie structure onimage

Global Hom-Lie structures

The \global" de nition given in [4] was a bit more restrictive than necessary. Therefore, there is some discrepancies in the wording of the one below and the one from [4]. 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, image.The covering families are families of at morphisms image where i ε I for some index set I. For more details on this see [6] for instance. By image we denote a sheaf of groups on (S)fl . Let W be a sheaf of image--sets over (S)fl i.e., a sheaf W over (S)fl together with an action of image 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 [6] or the positive topology [8]. (Of course, we could use any topological space as base here, i.e., not necessarily restricting our discussion to schemes.)

Let image be the structure sheaf on ( S)fl in the sense that image for image and let A be a sheaf of image- algebras. We denote by F an A-module. Let image be a sheaf of groups over ( S)fl acting image-linearly on F.

De nition 2.4. Given the above data, a hom-Lie structure for image,or image-hom-Lie structure, on ( S)fl is a image- -sheaf of A-modules F together with, for each coveringimage an imagebilinear product image such that

image

image

A morphism of hom-Lie structures image and image is a pairimage consisting of a morphism of image-modulesimage and a morphism of group schemes image such that image and image where we have put image

We thus get a category HomLieStrucS of all hom-Lie structures on ( S)fl with morphisms given in the de nition.

Hence, a hom-Lie structure is a family of (possibly isomorphic) products parametrized by image A product image for fixed image is a hom-Lie algebra on F.

In all cases, image is a constant sheaf of groups, i.e.,image for all image where G is a group (or group scheme over S). We will assume this from now on.

Specialization

When we only consider one open set U = S, De nition 2.4 specializes to

image

image

image (only one product for each g ε G).

When we need to specify the di erence of the above case and Definition 2.4, we call this the special case and De nition 2.4 the global case (and so the global case includes the special).

Arithmetic Witt-Lie algebras

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 de ne 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".

Witt-Lie algebras

The classical Witt-Lie algebra image (the reason for the unorthodox notation will be clear from the discussion below) is de ned as the complexi ed polynomial vector elds on the unit circle. More to the point

image

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 image 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 image the free Z-module spanned by the formal symbols image Let image be a Z-algebra morphism, i.e.,image First define

image

where image is a Λ-valued, symmetric group 2-cocycle, i.e., a map image satisfying

image

Then de ne a Λ-linear product on the base extension to T,

image

as follows:

image (3.1)

where image and similarly for image. Notice that image does not define a multiplicative map unless image is multiplicative2.The above construction gives the structure constants

image (3.2)

and it is easy to check that this de nes an A-graded Lie algebra called the Witt algebra of image.

As to not be drown in awkward notation we will most often, instead of the correct image 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, image and used. Consequently, we also drop image from the notationimage

image

We will nd this description more suitable in what follows. Therefore, it is strongly graded and crystalline in the sense of [7].

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 symbolsimage Define the multigraded Lie product

image

We will however stick to the \rank one" case.

Remark 3.2. We remark also that this construction is functorial. Indeed, de ne the category Ab(3) as the category whose objects are triplesimage and morphisms group morphismsimage such that image and image Thenimage 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 image withimage the unit (principal) characterimage and α the constant 2-cocycle image

(b) Take image and image Now image is the algebra that E. Witt actually studied. This can be given the following more conventional description. Consider the following imagevector space:

image

This is a Lie algebra under the commutator. In fact, we have the following isomorphism of Lie algebras:

image

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 image for L/K a nite extension of number elds, is a nitely generated abelian group. In addition, the group of N-torsion points image where image denotes the algebraic closure of K, i.e., the group of all pointsimage such that image is isomorphic toimage This holds in general when K is a eld of characteristic prime to N. The full group of torsion pointsimage is the union of all image (we assume here that char(K) = 0).

Using these groups we can now form three natural Witt algebras. Namely

image

image

image

There are several interesting possibilities for the K-algebra D. For instance, (a) D = L, a eld over image considered as an image-algebra; (c)image where imageimage 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 image

The case (c) is interesting only when End image the complex multiplication (CM) case.

Remark 3.4. Assume that K is a eld. Since D is an image --algebra,image (for instance) is also an image-module. This means that image shea es to a quasi-coherent sheafimage on the 1-dimensional arithmetic scheme Spec(image).Therefore, we get a sheaf of Lie algebras over Spec (image)

This example will be even more interesting in the context of hom-Lie algebras.

A cyclotomic Witt-Lie algebra

Let image where p is prime.We will also consider the multiplicative groupimage of units. In this case, a natural choice of additive character will beimage

Furthermore, let image be the group of nth roots of unity. Take image the nth cyclotomic eld, and pick a Dirichlet characterimage i.e., a character ϑ such thatimage Extend ϑ to a character on the whole A by de ningimage such that #(gh) = #(g)#(h). Extend # to a character on the whole A by de ning #(0) = 0, unless ϑ is the principal (unit) characterimage in which case we define image=1 There is nothing stopping us from letting T ≠  below. In fact, convenience is the only reason for assuming equality here.

We get

image

where α is a 2-cocycle, with the image linear Lie algebra structure given by (3.2). Inside thisimage vector space there is an element on the form

image

a so-called Gauss sum. Recall that (h - g) is actually image Letting ϑ* be another character, we can compute

image

Introducing the anti-symmetric pairing

image

Defined by

image

we have the following nice description of image

Proposition 3.5. Given the above, we have

image

Proof. The proof is a simple computation and is omitted.

Example 3.6. For an explicit example consider the two Dirichlet characters onimage (extended to Z=5Z) de ned by

image

and

image

Now we compute

image

and so using the proposition , we get

image

image

image

image

Notice that, even in the case image 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 de nition,image and image have noimage term, their productimage in general has. Hence, this product is in general not even aimagelinear combination of Gauss sums.

Remark 3.7. Normally, when computing with Gauss sums, the symbols image are powers of some primitive mth root of unityimage Therefore, the Gauss sums are elements in the relative extensionimage These extensions are rather cumbersome to work with, even when gcd(m; n) = 1, so working with symbols, as we have done, clari es 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 characterimage such that imageimageThen the so-called resolvent is the group algebra element

image

(where the last equality is as vector spaces). We consider imageas an operator on L:

image

In other words, image acts on L as σ. The elementimage satis es the following property: for any image and image we have

image (3.3)

where image is the inverse character. As consequences we have

image

Gauss sums are the result of the above when image in which case Galimageimage Particularly interesting is the case when ϑ is the unique order-two character, namely the Legendre symbolimage 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 [2] 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 signi cantly 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.

Arithmetic Witt-hom-Lie structures

Let as before A be an abelian group with a character image which will be (almost) constantly suppressed. Furthermore, let T be a Λ-domain, image a subgroup together with a representation image where V is aimage vector space (whereimage 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 toimage

Consider image as the group algebra T[A], spanned by formal symbolsimage be an algebra endomorphism on T and extend toimage

image

and then linearly as

image

This de nes an algebra endomorphism on image we define image i.e., we demand that σ acts as a group endomorphism on A. Notice that we have yet to decide how to interpretimage and imageDefine once again

image

a symmetric 2-cocycle. We assume that image Let E be a T-module and make the base extensionimageimage acts on E(A) as

image for image

Extend the action of G on T to a semilinear action on E:

image

Now we twist the action of G on E (and thus on T) as

image (4.1)

Then image (we suppress the dependence on ` in the notation) is a twisted derivation on E, i.e., a linear map image satisfying

image

and where δ is the induced twisted derivation image We extend the action of image via the Tate twist. Explicitly,

image

This is a semilinear action of G on E[A]. It follows that image can be extended canonically to E[A]. Assume thatimage where

image

The left T[A]-module image is a hom-Lie algebra by [3,4] under the product

image

We denote this by image Letting σ vary over G, we get the G-hom-Lie structure

image

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:

image

image (4.2)

Notice that, unless image for all g ε A, this product is not graded. We refer to it as a "σ-twisted grading". In [3] there are explicit examples of this phenomenon.

Remark 4.1. The above constructions are all functorial on suitably de ned 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.

Alternative construction

We keep the notation and conventions from before (but here T does not have to be a domain) and introduce the product

image (4.3)

on the algebra image The G-hom-Lie structure thus constructed is denoted by

image

(compare with image from the previous section). Now we compute

image

image

image

There are two important points to make here: (1) Notice that

image (4.4)

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) de ne isomorphic hom-Lie algebras (under suitable conditions). That (4.4) indeed de nes 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 image we denote two multiplicative characters of orderimage We will now consider the Gauss sums (resolvents)

image

and their products in image whereimage (not necessarily in Gal(L=K)),image We de ne the action of A onimage as

image

To confuse things we will sometimes write the product in A as "+" and sometimes as composition image Also, we define image Just as for the previous case we can compute

imgae

Now

image

image

image

image

where we, in the last equality, used that A is abelian, α symmetric and that image

We now have the obvious generalization of Proposition 3.5 with essentially the same proof.

Proposition 4.2. Given the above, we have

image

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., image a square-free integer. Furthermore, let image be an elliptic curve over a eld F with complex multiplication by the ring of integersimage

Then the theory of complex multiplication tells us that image where image is the j- invariant of image is the Hilbert class eld of K. In addition, the eldimage is a nite abelian extension of image for all N. The extension image is not necessarily abelian. In fact, we have the following exact sequence:

image

implying that image is the semi-direct product

image

Now , we have

image

where image is the class number of K and the Artin (reciprocity) map induces an isomorphism

image

with image denoting the class group of K.

Obviously, with this setup we have several interesting possibilities for constructing Witthom- Lie structures. Consider, for instance,

image

Then image acts on bothimage and the grading group via the restriction morphism

res : image

In this way we get a iamge Witt-hom-Lie structureimage or, alternatively, a Gal image -structureimage (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.

Galois representations

We keep the assumptions from the previous subsection, except that image is not necessarily a CM-curve.

Consider now the case when m = p a prime. Then

image (the "Tate group algebra over K")

where image is the ring of p-adic integers. Given image a 2-cocycle, we can lift this to a 2-cocycleimage via the successive image

Twisting the action of image on image to image gives us a GK-Witt-hom-Lie structure

image

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 image to a sheaf overimage 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 image be an abelian variety over a number eld K. Clearly, image acts on the L-rational points image on image, where image is a eld extension. The Mordell-Weil theorem says that, ifimage is finite, then imagetors.It is well known that image is nite for all nite extensionsimage Equally well known is that the N-torsion points

image

Since image is nite, we have thatimage is nite. But image soimage is also nite.

By de nition, image is an abelian group, so image is a commutative group algebra and GK acts on this in the obvious fashion. Therefore, we can form

image

for image a characterimage and image a 2-cocycle.

The action of GK on image induces an action on the quotient

image

and so we get a surjective morphism of GK-hom-Lie structures

image

Fitting this into the fundamental sequence

image

where image is the Nth Selmer group and image the N-torsion part of the Tate-Shafarevich group, we get a sequence of group algebras

image

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 K

-hom-Lie structures. The reason for this general 'failure" is that the action of GK does not in general lift to an action on

image and hence, in general, not toimage or image. 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 image much like the Lie algebra structure of algebraic groups (in this case this is obviously abelian). In this sense, the hom- Lie structure captures signi cantly more information than the Lie structure, since it involves the rational points and the Galois action on these in a very explicit manner.

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