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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On moduli spaces of 3d Lie algebras

Arvid SIQVELAND*

Buskerud University College, PoBox 235, 3603 Kongsberg, Norway

*Corresponding Author:
Arvid SIQVELAND
Buskerud University College, PoBox 235, 3603 Kongsberg, Norway
E-mail: [email protected]

Received date: May 12, 2008; Revised date: June 06, 2008

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Abstract

We consider Lie algebras of dimension 3 up to isomorphism. We construct a noncommutative ane spectrum of the isomorphism classes as a noncommutative k-algebra M, using noncommutative deformation theory. This k-algebra is an example of a noncommutative structure.

Introduction

Throughout, let k be an algebraically closed eld of characteristic 0. The classi cation of threedimensional Lie algebras is well known. Up to isomorphism they are ab, sl2, r3, la, and n3, ([4]). In this paper, we are going to construct a noncommutative algebraic moduli (i.e. an algebraic geometric classifying structure) having the isomorphism classes of 3-dimensional Lie algebras as geometric points, in this case represented by 1-dimensional simple modules, see [9] or the de nitions 1.10, 1.11 below. We obtain an example of noncommutative deformation theory that is special in that it shows two families of Lie-algebras meeting in only one point.

We will achieve this by using noncommutative deformation theory, see [9,3]. In commutative deformation theory, the local formal moduli (or prorepresenting hull) of the deformation functor DefM is a candidate for the completion of the local k-algebraimage of the moduli space in the point corresponding to M, see [11], and in a lot of examples the moduli space is given locally around M as the spectrum of a natural nitely generated algebraization (de nition 1.11) image of imagesee [13,12]. Moreover, [13] contains an algorithm for computing the local formal moduli of DefM. In noncommutative deformation theory as given in [9,3], this is generalized to the noncommutative situation:

De nition 1.1. A noncommutative k-algebra R is called r-pointed if there exists exactly r isomorphism classes of simple one-dimensional quotient modules of image A morphism of r-pointed k-algebras, is a k-algebra homomorphism inducing the identity on the r one-dimensional quotient modules.

Definition 1.2 (the category image ). The category image is the category of r-pointed Artinian k- algebras S together with morphisms of r-pointed algebras.

The category image is characterized by the following: An r-pointed Artinian k-algebra is a k-algebra S together with morphisms ,  commuting in the diagram

image

and such that image for some n > 0: ker(ρ) is called the radical of S and denoted ker(ρ) = rad(S): The procategoryimage is the full subcategory of (r-pointed) k-algebras such that image is an object ofimage for all n ≥ 1, and such that R is (separated) complete in the I = rad(R)-adic topology.

Remark 1.3. The commutative k-algebras in a1 makes up the category image of pointed Artinian k-algebras.

Remark 1.4. Consider the r × r-matrices eii, the matrix with 1 at place i; i and 0 elsewhere, and tij(l), the matrices with the indeterminates tij(l) at place i; j and 0 elsewhere,image Then we use the notationimage for the k-algebra generated by these matrices under ordinary matrix multiplication. This is the noncommutative counterpart of the free k-algebra image in the commutative situation.

Let image be the idempotents. Putimage Then it follows that every r-pointed k-algebra can be written as the matrix algebraimage

Now, let image be right A-modules. Letimage be an r-pointed Artinian k-algebra.

De nition 1.5. Let ki = k.eii; i.e. the matrix algebra with k at the i`th place on the diagonal and 0 elsewhere. The deformation functor image is de ned by

image

Notice that the condition S- at in the commutative case is replaced by image in the noncommutative case. (Hereimage means isomorphic as k-vector spaces, or equivalently, as left S-modules).

Any covariant functor image extends in a natural way to a functorimage on the procategory image by

image

We use the notation image for the r-pointed Artinian k-algebras S for which (rad(S))2 = 0:

Definition 1.6. A prorepresenting hull ( or a local formal moduli ) for a pointed covariant functor image is an objectimage such that there exists a proversal familyimage with the property that the corresponding morphismimage of functors onimage is smooth and an isomorphism when restricted to a morphism of functors on image

Definition 1.7. A surjective morphism imagebetween two r-pointed Artinian k-algebras is called small if ker image If there exists a deformation image such thatc imageis called a lifting of VS to R.

The obstruction theory for the noncommutative deformation functor is the obstruction theory in small lifting situations, and is given in the references [8,9,3,15]. The main results from these articles relevant for this work is the following (notice the fact that imageimage for k-algebras A and (right) A-modules M and N).

Theorem 1.8. Given a small morphism image with kernelimage inimage andimageimage There exists an obstructionimage such that o(π,MS) = 0 if and only if there exists a lifting is a torsor under

image

Proof. The proof can be found in [9]. The part essential for the computations in this paper is: Assume image Thenimageimage

In [9] it is proved that DefV has a prorepresenting hull image Also, in [9] (Sections 5 and 6), the construction of a noncommutative scheme theory and moduli of isomorphism classes of modules is given. Let imagebe the obvious forgetful functor. Then a subalgebra

image

is constructed, and the restriction of the canonical homomorphism η,

image

gives an action of imageextending the action of A. In this situation, this construction is a closure operation, i.e. image This O-construction is then extended to in nite families of isomorphism classes of modules by "shea fying", obtaining for every nite family V a smaller k-algebra,image containing the image ofimage The nal noncommutative structure sheaf image is then a certain quotient of thisimage

Definition 1.9. A family V of A-modules will be called a prescheme for A, if

image

is an isomorphism. Then (V;A) is called an ane prescheme. The family V will be called scheme for A, if

image

is an isomorphism.

In the situation of the present paper, the construction of noncommutative orbit spaces translates to the following simpli cation:

De nition 1.10. A nitely generated k-algebra R is called an ane moduli (or spectrum) for a family V of A-modules if the maximal ideals in R is in one to one correspondence with the family V , and if

image

for every subfamily image with corresponding subfamily of maximal idealsimage As such, R is a moduli (affine spectrum) for its simple modules.

In our situation, the results are achieved by the following:

Definition 1.11. Let image A nitely generated k-algebra R is called an algebraization ofimage if R has r maximal (left) idealsimage such that the formal moduli of the simple left R-modulesimage is isomorphic to imagethat is

image

The paper is organized as follows: Section 2 and 3 contains the classi cation of 3-dimensional Lie-algebras as given for example in [4]. Section 4 considers the ane space of 3-dimensional Lie algebras and its components. In section 5 we compute the closures of the orbits under the GL3(k)-action giving the the isomorphism classes. These closures are the geometric points in our noncommutative moduli, the objects of our study. In section 6 we compute the tangent space dimensions of the moduli, and in section 7 we state the main result and explain it geometrically.

Moduli of Lie algebras

An n-dimensional Lie algebra g over k is determined by its structure coecients image given by its bracket productimagewhere a k-basis image for image is chosen, and

image

Writing up the Jacobi identity

image

we can rewrite for every image the Jacobi identity as follows:

image

An isomorphism of Lie algebras is an isomorphism of k-vector spaces g commuting with the bracket:

image

Thus image andimage We choose the basisimage forimage in lexicographic order. Then the matrix of b with respect to these bases isimage We get

image

where Coef(g) = (C(l; m; i; j)); that is

image

for i < j, and C(l; m; i; j) is the determinant of g after removing all rows except the l`th and the m`th, and removing all columns except the i`th and the j`th.

We use the notation

image

Put Lie(n) image denote the ane variety of GLn(k); that is image as above, and the set of isomorphism classes of n-dimensional Lie algebras is in bijective correspondence with the orbits image

The case n=1 contains only one Lie algebra, the abelian one, and in the case n = 2, the situation is well known. It is known that there exists no orbit space image

Classi cation of 3-dimensional Lie algebras

As before, we choose a basis image Then the lexicographic ordering ofimageimage gives the basisimage The matrix of the bracketimage with respect to this basis is

image

Fulton and Harris gives the following classi cation in [4]:

Lemma 3.1. There exists a k-vector space basis image for a non abelian Lie algebra g such that the matrix of structure coecients of g is in one of the following forms:

image

and the only pairs of isomorphic Lie-algebras are image Moreover, the Heisenberg Lie-algebra n3 is the only nilpotent one.

The k-scheme Lie(3)

In the 3 dimensional case, the Jacobi identity is given by the 3 equations

image

Put

image

Then the Jacobi identity can be written S  A = 0: This gives the well known decomposition in [4] of Lie(3) in two 6-dimensional irreducible components,

image

where image is given by the three equations corresponding to A = 0 andimage is given by the four equations corresponding to det(S) = 0, S  A = 0, see [2, 7]. Of course, image also works and gives another description of image We shall give understandable descriptions of the orbits of the di erent Lie algebras inimage under the action ofimage In the following, we replace the coordinatesimage by ordinary matrix coordinates. That is, we make the following identi cations:

image

Thus the Jacobi identity is

image

and the components are given accordingly by de ning ideals in image

The closure of the orbits

To construct a classifying (not necessarily commutative) algebraic space for 3-dimensional Liealgebras, we can construct the space for the closures of the orbits under the given imageaction. This is because the di erent orbits have di erent closures. Thus we start by nding de ning ideals of the orbit-closures. We use the notatation o(x) for the imageand imagefor the closure of this orbit.

First of all, because the group action image is continuous, it follows that the orbits are irreducible, i.e. the surjectionimageis continuous and imageis a linear irreducible group, see e.g. [10]. Secondly, the dimensions of the orbits are given by their isotropy groups. The isotropy group of imageis the vector space imageimageThis is done in [2]. Letting ab denote the abelian Lie algebra, we have the following dimensions of the closures of the orbits:

Lemma 5.1. We have:

image

We use this information to nd de ning ideals of the closures of the orbits.

Remark 5.2. We do not prove radicality of the ideals. Gerhard P ster has implemented the algorithm of Krick-Logar and Kemper for computing radicals in Singular [5]. This algorithm proves that that the de ning ideals below for the closures of the orbits of sl2, image and image for several choices of a are radical, but it remains to prove this for general a. However, these ideals, or rather their quotients are the objects of our study. The main result then proves to give a reasonable algebraic classifying structure. Of course, this indicates that all ideals in question are radical.

sl2

imageis an element in image.which is of dimension 6. Thus imageThis means that a de ning ideal for imageis

image

l-1

It is well known ([7]) that the intersection of the two components of Lie(3) is the closure of the orbit of l-1 This is so becauseimageand because the dimensions coincide. Letting

image

a de ning ideal of the closure of the orbit of l-1 is

image

n3

n3 is an element in imageand is characterized by the fact that its rank is 1. Letting sijbe the ij-minor of the matrix (xij) we nd that a de ning ideal of the closure of the orbit of n3 is

image

image

From [4], it is well known that on the complement of the intersection of the two components, the expression

image

takes the same value imageon the isomorphism classes of image We compute and nd that GL3(k) takes any Jij into any other imageand any Jii into any other Jll: We also check for one choice that image As our de ning ideal of the closure of the orbit ofimage is invariant, we see that forimagecontains Lie-algebras from only one orbit, the orbit of la, whereimage and thus this algebraic set must be the closure of the orbit. The di erent expressions for J are the following fractions:

image

This results in the following: For image is given by the ideal generated by the following polynomials:

image

l1

We see that both r3 and l1 is contained in the ideal given by imagefor instance

image

Thus both of the closures of the orbits are in the zero set of the ideal (j1(1); : : : ; j6(1)): Considering the invariant ideal

image

we see that this ideal contains l1, but not r3 because x22 6= 0 there. This implies that

image

and it follows that

image

r3

Because the orbit of r3 has to be the open set image it follows that its closure is the zero set of image Thus the objects of our study are given asimage modules, whereimagewith its given action on A in this case, and where image is the skew group algebra.

Tangent space dimensions

Let A be a nitely generated k-algebra with an action of a linearly reductive group G. In this paper, the interest is classi cation of the orbits in Spec(A) under the action of G. If the closures of two di erent orbits are di erent, we can consider classi cation of the closures of the orbits as well as the orbits themselves. Ifimage is the action of G on A, and if the ideal of the closure of the orbit of image is imagethen imagehas an induced G-actionimage Thus the obstruction theory above has to be generalized to the category of image modules which is the category of A modules M with G-action such that the two operations commute, i.e. image This is just the theory in [9] onimage modules.

This obstruction theory makes it possible to generalize the algorithm from [12]. This is done in [14] and will be published elsewhere. In this paper it turns out that the liftings are unobstructed, and so we will only need the tangent spaces. We need the following fact from [6]:

Lemma 6.1. Let M, N be two A - G-modules where G is reductive. Then

image

If a moduli space for 3-dimensional Lie algebras exists, it should have the expected tangent space dimensions corresponding to the correct cohomology. If a commutative modulispace image exists, the tangent space in a pointimage is the Chevalley-Eilenberg-MacLane cohomology image The commutative theory is well known, see [1], and it has been proved that there exists no commutative algebraic moduli. In the construction of the noncommutative moduli, the tangent spaces between the modules are studied. For Lie-algebras the correct cohomology giving these tangent spaces is not known. However there exists such a cohomology in the category of A- modules, namely image. This is the reason for passing from Lie algebras to A-modules, or in fact A - G-modules in our situation.

As explained in [9] the tangent space of the non commutative moduli of a family of A - G- modules is given by image. where V;W runs through all possible selections of pairs of A - G-modules. In our situation, imagewith the given action on Spec(A) =image The A - G-modules we consider are

image

To be able to compute the tangent spaces above, we need the following general fact which was proved in [14]. We assume that G is a linear reductive group acting on a k-algebra A of nite type, k algebraically closed.

Lemma 6.2. Let image be two G-invariant ideals in A. Then

image

and the action of g ∈ G is given by

image

imageimage

Also notice the following:

Lemma 6.3. For all selections of pairs of ideals imageimage

Proof. Using Singular [5], we can nd free resolutions and compute that imageHence image

With these preliminaries, we are left with some straight forward calculations, all based on the same technique: Consider the standard elementary 3 × 3-matrices image(interchange rows i and j), Ei(c); image (multiply the i`th. row with c), Eij(c),image (add c times the i`th. row to the j`th). Then these elements generate GL3(k) = G and the invariants under G are the elements invariant under these generators. We compute the action of each elementary matrix in the generator set of G on the variables xij by the rule

image

Then we nd the induced action on the generators of the ideals and use lemma 6.2 to compute the invariant homomorphisms.

We will give two computations as examples, both of importance, and state the rest.

Example. Computation ofimage

We recall that image Assume that imageimage is invariant.

Forimage we nd Coef(g)image and so the action of this element ona general point is given by

image

Thus, if φ  is invariant, the composition

image

leaves φ unaltered. In particular image which implies that h is homogeneous of degree 3. Furthermore, we nd imageimage Thus we are looking for a homogeneous polynomial h of degree 3 with the same properties as s above. It is easy to check that

image

ful lls the conditions, and it is straight forward to check that it is the only such (in the same way as in the next example). We nd that imagethat is h = 0: Using similar techniques, we nd that image

Example. Computation ofimage

Recall that for imageWe nd the following actions on (j1; : : : ; j6):

image

Now imageis determined by its image of j1; : : : ; j6, that isimage That φ is invariant means that it is invariant under the composition

image

for all g ∈ G, that is for all the generators of G. First, let g = c  Id. Then the invariance of φ means thatimage This forces each hi to be homogeneous of degree 2. Then image tells us which degree 2 monomials that are possible. Finally, investigating the action of imagegives

image

Then we have to prove that this invariant morphism (with = 1) is an element in image which means that it really is an A-module homomorphismimage This follows from the fact that  φ.S = 0; where S is the syzygy-module ofimage. Thus {φ} is a k-vector space basis for the 1-dimensional space image

Notice that exactly the same computation yields the computation of image and forimage Thus {φ} is also a basis for the 1-dimensional spaceimage and forimage

The result of this computation is the following:

section 6.4. Forimage For all other possible selections of pairs of orbitsimage

Noncommutative moduli

The formal noncommutative moduli

In the present situation we consider the closures of the orbits

image

To ease the tracking of the algorithm for computing the local formal moduli, we set

image

and we use the notation

image

To compute the obstructions in the Yoneda complex we choose free resolutions of V1 and V2 in the following way, where we have added the basis for image

image

where

image

and where imageand image are given by the condition

image

Then the cup products (second order generalized Massey products) are given by their rst term in the Yoneda complex. That is

image

where the last equality is "strict", meaning that the product of the matrices are zero in A. Notice that the obstruction calculus for A - G-modules and A-modules are compatible. This tells us that the in nitesimal family de nes a lifting to

image

hence this is a pro-representing hull.

The noncommutative moduli Lie(3)

Consider the family of 3-dimensional Lie-algebras given as the zero zet of image where

image

whwre imageThis family contains each isomorphism classimage exactly once. Renaming this family to g(C) we have

image

Given a k-algebra A. An A-module V is given by a morphism φ: image Using this, the obstruction theory and formation of moduli can be done in the Hochschild cohomology. In particular the tangent space of the deformation functor in this case is HH1imageimage See ([15]) for a standard example. This is used to suggest an algebraization of the local formal moduli where the maximal ideals on the diagonal represents the geometric point, i.e. the quotients of the maximal ideals on the diagonal are the simple modules.

The tangent space dimensions then suggests an algebraization ofimage

image

This can not be the case because then

image

which is contradicting any possible de nition of moduli. This is explained the following way: image deforms toimage and the closure of the orbit of this Lie-algebra containsimage These points are however collapsed to one point when dividing out with the group action. Thus the correct algebraization is

image

where the entry imagemeans the cyclic left k[C]-module generated by t12 with coecients in imagethe localization of k[C] in the maximal idealimage We have proved following:

section 7.1. The noncommutative moduli Lie(3) is

image

where the two rst rows correspond to the Lie-algebras image and l1 respectively, and where the four last rows corresponds to image (respectively).

Acknowledgements

The computation of the tangent space dimensions was more or less done in a not published thesis by Sondre Tveit. I would like to thank him and our common mentor O. A. Laudal for stating the problem. Laudal also helped me to understand the geometry of the result under a fruitful stay at the Insitut Mittag-Leer in Stockholm. Also, I would like to thank Eivind Eriksen for carefully reading the manuscript. Finally, I thank the referee for constructive comments improving the paper.

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