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- *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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

Throughout, let k be an algebraically closed eld of characteristic 0. The classication 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 denitions 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
Def_{M} is a candidate for the completion of the local k-algebra 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 (denition 1.11) of see [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:

Denition 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 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 ). The category is the category of r-pointed Artinian k-
algebras S together with morphisms of r-pointed algebras.

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

and such that for some n > 0: ker(ρ) is called the radical of S and denoted ker(ρ) = rad(S): The procategory is the full subcategory of (r-pointed) k-algebras such that is an object of 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 of pointed Artinian
k-algebras.

**Remark 1.4.** Consider the r × r-matrices *e _{ii}*, the matrix with 1 at place

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

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

**Denition 1.5. **Let *k _{i} = k.e_{ii}*; i.e. the matrix algebra with k at the i`th place on the diagonal
and 0 elsewhere. The deformation functor is dened by

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

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

We use the notation 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 is an object such that there exists a proversal family with the property that the corresponding morphism of functors on is
smooth and an isomorphism when restricted to a morphism of functors on

**Definition 1.7.** A surjective morphism between two r-pointed Artinian k-algebras
is called small if ker If there exists a deformation such thatc is 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 for k-algebras A and (right) A-modules M and N).

**Theorem 1.8.** Given a small morphism with kernel in and There exists an obstruction such that o(π,MS) = 0 if and only if there exists a lifting is a torsor under

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

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

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

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

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

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

is an isomorphism.

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

Denition 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

for every subfamily with corresponding subfamily of maximal ideals 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 A nitely generated k-algebra R is called an algebraization of if R has r maximal (left) ideals such that the formal moduli of the simple left R-modules is isomorphic to that is

The paper is organized as follows: Section 2 and 3 contains the classication 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.

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

Writing up the Jacobi identity

we can rewrite for every the Jacobi identity as follows:

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

Thus and We choose the basis for in lexicographic order. Then the matrix of b with respect to these bases is We get

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

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

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

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

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

Fulton and Harris gives the following classication in [4]:

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

and the only pairs of isomorphic Lie-algebras are Moreover, the Heisenberg
Lie-algebra n_{3} is the only nilpotent one.

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

Put

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,

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

Thus the Jacobi identity is

and the components are given accordingly by dening ideals in

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 action. This is because the dierent orbits have dierent closures. Thus we start by nding dening ideals of the orbit-closures. We use the notatation o(x) for the and for the closure of this orbit.

First of all, because the group action is continuous, it follows that the orbits are irreducible, i.e. the surjectionis continuous and is a linear irreducible group, see e.g. [10]. Secondly, the dimensions of the orbits are given by their isotropy groups. The isotropy group of is the vector space This 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:*

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

**Remark 5.2.** We do not prove radicality of the ideals. Gerhard Pster has implemented the
algorithm of Krick-Logar and Kemper for computing radicals in Singular [5]. This algorithm
proves that that the dening ideals below for the closures of the orbits of sl_{2}, and 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.

**sl _{2}**

is an element in which is of dimension 6. Thus This means that a dening ideal for is

**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 becauseand because the dimensions
coincide. Letting

a dening ideal of the closure of the orbit of *l*_{-1} is

**n _{3}**

n3 is an element in and is characterized by the fact that its rank is 1. Letting *s _{ij}*be
the ij-minor of the matrix (xij) we nd that a dening ideal of the closure of the orbit of n3 is

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

takes the same value on the isomorphism classes of We compute and
nd that GL3(k) takes any Jij into any other and any Jii into any other Jll:
We also check for one choice that As our dening ideal of the closure of the orbit of is invariant, we see that forcontains Lie-algebras from only
one orbit, the orbit of *l _{a}*, where and thus this algebraic set must be the closure of
the orbit. The dierent expressions for J are the following fractions:

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

**l _{1}**

We see that both *r _{3}* and

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

we see that this ideal contains *l _{1}*, but not r3 because x22 6= 0 there. This implies that

and it follows that

*r3*

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

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

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 exists, the tangent space in a point is the Chevalley-Eilenberg-MacLane cohomology 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 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 . where V;W runs through all possible selections of pairs of A - G-modules. In our situation, with the given action on Spec(A) = The A - G-modules we consider are

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 be two G-invariant ideals in A. Then

and the action of g ∈ G is given by

Also notice the following:

**Lemma 6.3.** For all selections of pairs of ideals

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

With these preliminaries, we are left with some straight forward calculations, all based on the same technique: Consider the standard elementary 3 × 3-matrices (interchange rows i and j), Ei(c); (multiply the i`th. row with c), Eij(c), (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

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.

We recall that Assume that is invariant.

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

Thus, if φ is invariant, the composition

leaves φ unaltered. In particular which implies that h is homogeneous of degree 3. Furthermore, we nd 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

fullls 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 that is h = 0: Using similar techniques, we nd that

**Example. Computation of**

Recall that for We nd the following actions on (j_{1}; : : : ; j_{6}):

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

for all g ∈ G, that is for all the generators of G. First, let g = c Id. Then the invariance of φ means that This forces each *h _{i}* to be homogeneous of degree
2. Then tells us which degree 2 monomials that are possible. Finally,
investigating the action of gives

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

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

The result of this computation is the following:

**section 6.4. **For For all other possible selections of pairs of orbits

**The formal noncommutative moduli**

In the present situation we consider the closures of the orbits

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

and we use the notation

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

where

and where and are given by the condition

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

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 innitesimal family denes a lifting to

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 where

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

Given a k-algebra A. An A-module V is given by a morphism φ: 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 HH^{1} 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 of

This can not be the case because then

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

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

**section 7.1.** The noncommutative moduli Lie(3) is

where the two rst rows correspond to the Lie-algebras and *l _{1}* respectively, and where the
four last rows corresponds to (respectively).

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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