Deformations of Complex 3-Dimensional Associative Algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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# Deformations of Complex 3-Dimensional Associative Algebras

Alice Fialowski1, Michael Penkava2 and Mitch Phillipson2

1Institute of Mathematics, E¨otv¨os Lor´and University, 1053 Budapest, Hungary. Email: [email protected]

2Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, WI 54702-4004, USA.

Received Date: 27 August 2010; Accepted Date: 26 January 2011

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#### Abstract

We study deformations and the moduli space of 3-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. The main purpose of this paper is to give a logically organized description of the moduli space, and to give an explicit description of how the moduli space is constructed by extensions.

#### Introduction

The classification of associative algebras was instituted by Benjamin Peirce in the 1870’s [11], who gave a partial classification of the complex associative algebras of dimension up to 6, although in some sense, one can deduce the complete classification from his results, with some additional work. The classification method relied on the following remarkable fact:

Theorem 1. Every finite dimensional algebra which is not nilpotent contains a nontrivial idempotent element.

A nilpotent algebra A is one which satisfies An = 0 for some n, while an idempotent element a satisfies a2 = a. This observation of Peirce eventually leads to two important theorems in the classification of finite dimensional associative algebras. Recall that an algebra is said to be simple if it has no nontrivial proper ideals, and it is not the 1-dimensional nilpotent algebra over , given by the trivial product.

Theorem 2 (fundamental theorem of finite dimensional associative algebras). Suppose that A is a finite dimensional algebra over a field . Then A has a maximal nilpotent ideal N, called its radical. If A is not nilpotent, then A/N is a semisimple algebra, that is, a direct sum of simple algebras.

In fact, in the literature, the definition of a semisimple algebra is often given as one whose radical is trivial, and then it is a theorem that semisimple algebras are direct sums of simple algebras. Moreover, when A/N satisfies a property called separability over , then A is a semidirect product of its radical and a semisimple algebra.

A simple algebra is separable if and only if its center is a separable field over , and a semisimple algebra is separable if and only if its simple components are separable. Therefore, over the complex numbers every semisimple algebra is separable. To apply the theorem above to construct algebras by extension, we will use the following characterization of simple algebras [13].

Theorem3 (Wedderburn). If A is a finite dimensional algebra over , then A is simple if and only if A is isomorphic to a tensor product M ⊗ D, where M = (n,) and D is a division algebra over .

One can also say that A is a matrix algebra with coefficients in a division algebra over . An associative division algebra is a unital associative algebra where every nonzero element has a multiplicative inverse. (One has to modify this definition in the case of graded algebras, but we will not address this issue in this paper.)

Over the complex numbers, the only division algebra is itself, so Wedderburn’s theorem says that the only simple algebras are the matrix algebras. In particular, there can be no 3-dimensional simple algebras, so the only semisimple 3-dimensional algebra is 3, the direct sum of three copies of the 1-dimensional simple algebra.

The main goal of this paper is to give a miniversal deformation of every element and give a complete description of the moduli space of 3-dimensional associative algebras. Moreover, we will give a canonical stratification of the moduli space into projective orbifolds of a very simple type, so that the strata are connected only by deformations factoring through jump deformations, and the elements of a particular stratum are given by neighborhoods determined by smooth deformations.

#### Preliminaries

Suppose that V is a vector space, defined over a field whose characteristic is not 2 or 3, equipped with an associative multiplication structure m : VVV . The associativity relation can be given in the form

m ◦ (m ⊗ 1) = m ◦ (1 ⊗ m).

The notion of isomorphism or equivalence of associative algebra structures is given as follows. If g is a linear automorphism of V , then define

Two algebra structures m and are equivalent if there is an automorphism g such that = g(m). The set of equivalence classes of algebra structures on V is called the moduli space of associative algebras on V .

We wish to transform this classical viewpoint into the more modern viewpoint of associative algebras as being given by codifferentials on a certain coalgebra. To do this, we first introduce the parity reversion ΠV of a 2-graded vector space V . If V = VeVo is the decomposition of V into its even and odd parts, then W = ΠV is the 2-graded vector space given by We = Vo and Wo = Ve. In other words, W is just the space V with the parity of elements reversed.

Given an ordinary associative algebra, we can view the underlying space V as being 2-graded, with V = Ve. Then its parity reversion W is again the same space, but now all elements are considered to be odd. One can avoid this gyration for ordinary spaces, by introducing a grading by exterior degree on the tensor coalgebra of V , but the idea of parity reversion works equally well when the algebra is 2-graded, whereas the method of grading by exterior degree does not.

Denote the tensor (co)-algebra of W by , where Wk is the kth tensor power of W and W0 = . For brevity, the element in Wk given by the tensor product of the elements wi in W will be denoted by w1 · · ·wk. The coalgebra structure on Τ (W) is given by

Define , where π : VW is the identity map, which is odd, because it reverses the parity of elements. Note that d is an odd map. The space C(W) = Hom(T (W),W) is naturally identifiable with the space of coderivations of T (W). In fact, if φCk(W) = Hom(Wk,W), then φ is extended to a coderivation of T (W) by

The space of coderivations of T (W) is equipped with a 2-graded Lie algebra structure given by

The reason that it is more convenient to work with the structure d on W rather than m on V is that the condition of associativity for m translates into the codifferential property [d, d] = 0.

#### Construction of algebras by extensions

The theory of extensions of an algebra W by an algebra M can be described in the language of codifferentials (see [4]), which we will use in our constructions here, but there is a long history of the study of extensions, which we do not mention. Consider the diagram

of associative -algebras, so that V = MW as a -vector space, M is an ideal in the algebra V , and W = V/M is the quotient algebra. Suppose that δC2(W) and μC2(M) represent the algebra structures on W and M, respectively. We can view μ and δ as elements of C2(V ). Let Tk,l be the subspace of Tk+l(V ) given recursively by

Let Ck,l = Hom(Tk,l,M) ⊆ Ck+l(V ). If we denote the algebra structure on V by d, we have

where λ ∈ C1,1 and ψC0,2. Note that in this notation, μC2,0. Then the condition that d is associative: [d, d] = 0, gives the following relations:

Since μ is an algebra structure, [μ, μ] = 0, so if we define Dμ by Dμ(φ) = [μ, φ], then = 0. Thus Dμ is a differential on C(V ). Moreover Dμ : Ck,lCk+1,l. Let

Then the compatibility condition means that λ ∈ Z1,1. If we define Dδ+λ(φ) = [δ+λ, φ], then it is not true that = 0, but Dδ+λDμ = −DμDδ+λ, so that Dδ+λ descends to a map , whose square is zero, giving rise to the Dδ+λ-cohomology . If the pair (λ, ψ) give rise to a codifferential d, and (λ, ) give rise to another codifferential , then if we express = ψ + τ , it is easy to see that [μ, τ] = 0, and [δ + λ, τ] = 0, so that the image of τ in is a Dδ-cocycle, and thus τ determines an element {} ∈ .

If βC0,1, then g = exp(β) : Τ (V) → Τ(V) is given by g(m,w) = (m + β(w), w). Furthermore g = exp(−adβ) : C(V ) → C(V ) satisfies g(d) = , where with and = . In this case, we say that d and are equivalent extensions in the restricted sense. Such equivalent extensions are also equivalent as codifferentials on Τ (V). Note that λ and differ by a Dμ-coboundary, so in . If λ satisfies the MC-equation for some ψ, then any element in also gives a solution of the MC equation, for the given above. The cohomology classes of those λ for which a solution of the MC equation exists determine distinct restricted equivalence classes of extensions.

Let , so are codifferentials on Τ (M) and Τ (W), respectively. The group Gδ,μ is the subgroup of GM,W consisting of those elements g such that g (δ) = δ and g (μ) = μ. Then Gδ,μ acts on the restricted equivalence classes of extensions, giving the equivalence classes of general extensions. Also Gδ,μ acts on , and induces an action on the classes of λ giving a solution to the MC equation.

Next, consider the group Gδ,μ,λ consisting of the automorphisms h of V of the form h = g exp(β), where gGδ,μ, βC0,1 and λ = g (λ) + [μ, β]. If d = δ + μ + λ + ψ + τ, then , where

Thus the group Gδ,μ,λ induces an action on .

The general group of equivalences of extensions of the algebra structure δ on W by the algebra structure μ on M is given by the group of automorphisms of V of the form h = exp(β)g, where βC0,1 and gGδ,μ. Then we have the following classification of such extensions up to equivalence.

Theorem 4. The equivalence classes of extensions of δ on W by μ on M are classified by the following:

(1) equivalence classes of which satisfy the MC equation

for some ψC0,2, under the action of the group Gδ,μ;

(2) equivalence classes of under the action of the group Gδ,μ,λ.

Equivalent extensions will give rise to equivalent codifferentials on V , but it may happen that two codifferentials arising from nonequivalent extensions are equivalent. This is because the group of equivalences of extensions is the group of invertible block upper triangular matrices on the space V = MW, whereas the equivalence classes of codifferentials on V are given by the group of all invertible matrices, which is larger.

The fundamental theorem of finite dimensional algebras allows us to restrict our consideration of extensions to two cases. First, we can consider those extensions where δ is a semisimple algebra structure on W, and μ is a nilpotent algebra structure on M. In this case, because we are working over , we can also assume that ψ = τ = 0. Thus the classification of the extension reduces to considering equivalence classes of λ.

Second, we can consider extensions of the trivial algebra structure δ = 0 on a 1-dimensional space W by a nilpotent algebra μ. This is because a nilpotent algebra has a codimension 1 ideal M, and the restriction of the algebra structure to M is nilpotent. However, in this case, we cannot assume that ψ or τ vanish, so we need to use the classification theorem above to determine the equivalence classes of extensions. In many cases, in solving the MC equation for a particular λ, if there is any φ yielding a solution, then ψ = 0 also gives a solution, so the action of Gδ,μ,λ on takes on a simpler form than the general action we described above.

Consider the general setup, where an n-dimensional space is extended by an m-dimensional space . Then the module structure is of the form

and we can consider Lk and Rk to be m × m matrices. Then we can express the bracket [λ, λ], which appears in the MC equation in terms of matrix multiplication:

(3.1)

where i, j = 1, . . . m and k, l = m + 1, . . . m + n.

Next, suppose that

(3.2)

Since δ is semisimple, one can ignore the cocycle ψ in constructing an extension, so the MC equation is completely determined by (3.2) and (3.1), so we obtain the conditions. Therefore, the MC-equation holds precisely when

As a consequence, both Lk and Rk must be commuting nondefective matrices whose eigenvalues are either 0 or 1, which limits the possibilities. Moreover, it can be shown that Gδ, the group of automorphisms of W preserving δ, is just the group of permutation matrices. Thus if G = diag(G1,G2) is a block diagonal element of Gδ,μ, the matrix G2 is a permutation matrix. The action of G on λ is given by simultaneous conjugation of the matrices Lk and Rk by G1, and a simultaneous permutation of the k-indices determined by the permutation associated to G2.

It is well known that a collection of commuting diagonalizable matrices can be simultaneously diagonalized. When μ = 0, this means we can assume that all of the matrices Lk and Rk are diagonal. When μ ≠ 0, the matrices G1 are required to preserve μ, and the compatibility condition [μ, λ] also complicates the picture.

It is important to note that given an m and a nilpotent element μ on an m-dimensional space M, there is an n beyond which the extensions of the semisimple codifferential on an N-dimensional space with N greater than n are simply direct sums of the extensions of the n-dimensional semisimple algebra n and the semisimple algebra N–n . We say that the extension theory becomes stable at n. Moreover the deformation picture also stabilizes as well.

When μ = 0, we can easily see that at most m matrices Lk could be nonzero since we can assume they are diagonal matrices. Similarly, at most m matrices Rk can be nonzero. By rearranging using the fact that G2 is an arbitrary permutation matrix, one can assume that all Lk matrices when k > 2m vanish, and then we can assume that all Rk matrices vanish if k > 3m. Thus the extension theory becomes stable when n = 2m.

When μ ≠ 0, the matrices Lk and Rk are still diagonalizable, so it also follows that the extension theory is stable for n ≥ 2m, but in fact it may become stable for a much smaller value of n.

In higher dimensions, there are semisimple algebras which are not of the form n. Also, as m increases, the complexity of the nontrivial nilpotent elements μ increases as well. However, all 3-dimensional non-nilpotent algebras are extensions of a semisimple algebra of the type n, for n = 1, 2, and there is only one nontrivial nilpotent algebra in dimension 2, so the discussion above will suffice to characterize the nonnilpotent 3-dimensional algebras in a straightforward manner.

#### Associative algebra structures on a 3-dimensional vector space

In our classification of the algebra structure on a 3-dimensional space, we recall the classification of algebras on a 2-dimensional space, which is given in Table 1.

In constructing the elements of the moduli space by extensions, we need to consider two possibilities, extensions of the semisimple algebra structure on a 2-dimensional space W by the trivial algebra structure on a 1-dimensional space M, or extensions of an algebra structure on a 1-dimensional space W by a nilpotent algebra structure on a 2-dimensional space M. In this second case, we need to study both extensions of the simple algebra structure on W and the trivial structure. Moreover, there are two nilpotent algebra structures on M.

#### Extensions of the 2-dimensional semisimple algebra ⊕ by the 1-dimensional trivial algebra 0

Let The group GM,W is given by matrices of the form

The semisimple 2-dimensional algebra structure δ on W is given by the codifferential . The trivial 1-dimensional algebra structure on M is just μ = 0.

The module structure λ on M is given by

In other words, the matrices Lk and Rk are just numbers. Since m = 1 and n = 2, this is the stable case. There are exactly 5 solutions as follows:

These extensions give the nonequivalent codifferentials d2, d3, d4, d5 and d6.

#### Extensions of a 1-dimensional algebra by a 2-dimensional nilpotent algebra

Let . The group GM,W is given by matrices of the form

The general form of the module structure λ is given by the matrices

Extensions of the 1-dimensional simple algebra

In this case, we have . There are two choices of μ in this case, depending on whether the algebra structure on M is the trivial or nontrivial nilpotent structure.

Extensions by the nontrivial nilpotent algebra

In this case, . The condition [μ, λ] = 0 forces λ to have the following form:

Since the matrices cannot be defective, it follows that either both matrices are the identity matrix or both are the zero matrix. This gives exactly two possible choices for λ, corresponding to the codifferentials d7 and d8. It is interesting to note that this is the stable case even though n < 2m.

Extensions by the trivial nilpotent algebra

In this case, L1 is given by the identity matrix, diag(1, 0), or the zero matrix. If L1 is the identity matrix or the zero matrix, then R1 has the same possibilities, but when L1 is given by the matrix diag(1, 0), we obtain one additional case, where R1 is given by the matrix diag(0, 1).

This gives a total of 10 possibilities for λ, which gives rise to the codifferentials d9 through d18. This is not the stable case.

Extensions of the 1-dimensional trivial algebra 0

Because any extension of the trivial algebra structure δ = 0 by a nilpotent algebra is nilpotent, we cannot assume that the “cocycle” vanishes. Since δ vanishes, the Maurer-Cartan equation takes the following form:

Extensions by the trivial nilpotent algebra

In this case, the group Gδ,μ coincides with GM,W. The MC condition becomes simply [λ, λ] = 0, which forces and L1R1 = R1L1. As a consequence, up to equivalence, we can assume that lambda has the following form:

Thus we have . Also, if , for some t ≠ 0, then is equivalent to λ under the action of Gδ,μ, so the λ’s form a projective family labeled by projective coordinates (p : q).

Any element of C0,1 is of the form . Therefore, unless p = −q, it does not matter, up to equivalence, what the value of c2 is. It may seem natural to choose c2 = 0 as the generic value, but our experience has shown that this is not the correct choice in order to correctly align the family.

The reason for this is as follows. Let us consider a neighborhood of the codifferentials given by λ labeled by (1 :  −1). For all nearby values, choosing c2 = 1 yields an equivalent codifferential to the one resulting from choosing c2 = 0. However, at exactly this point, the two codifferentials are not equivalent. Since every value of c2 except c2 = 0 yields an equivalent codifferential to the one with c2 = 1, the codifferential given by choosing c2 = 0 has a jump deformation to the one given by c2 = 1. Thus the element with c2 = 1 should belong to the family instead of the one with c2 = 0.

Moreover we have , and since [λ, ψ] = 0, this forces c1 = 0 unless p = q. In this case, when p = q = 1, we obtain a special case for c1 = 1. This gives the element d19, while the generic case c1 = 0 and p and q being arbitrary, with c2 = 1, gives a projective family d20(p : q). When p = 1, q = −1 and c2 = 0, we obtain the special element d21.

In the case when p = q = 0, we obtain no condition on either c2 or c1. The group Gμ,δ,λ coincides with the group GM,W, and the action of this group on C0,2 is just the multiplication of GL (2, ) on 2, which has exactly 2 orbits. Thus we can reduce to the case when c1 = 0 and c2 = 0 or c2 = 1. The case c2 = 0 gives the zero codifferential, while the case c2 = 1 corresponds to the element d20(0 : 0), which fits the generic pattern.

Extensions by the nontrivial nilpotent algebra

Suppose that . Then

as before.

From the MC equation, we obtain that = 0 and c2 = 0, Thus we can assume that and that . Moreover, if q ≠ 0, we can assume that q = 1. However, we cannot make such a simplifying assumption for c1. Nevertheless, the case when c1 = 0 will be treated separately.

If c1 = 0 and q = 1, then . It can be shown that dd20(1 : 0). In fact, the permutation (1, 2, 3) can be seen to give this transformation. Note that the permutation matrix for this transformation does not lie in GM,W. Thus the two codifferentials do not give equivalent extensions, but give isomorphic algebra structures on V .

If . It is convenient to express c1 = (1− u)u, where u ≠ 0 and u ≠ 1. Let q ≠ 0 and g be given by the matrix

Then g(d) = d20(p : q), where p = q(u − 1) / u.

If λ = 0 and if g is the automorphism given by the matrix

Finally, if λ = 0 and ψ = 0, then , and the permutation (1, 2, 3) transforms d into d20(0 : 0). Thus the extensions by the nontrivial nilpotent algebra just give some special cases of the family d20(p : q), which already occur as extensions of the trivial algebra.

This completes the classification of the moduli space of 3-dimensional complex associative algebras by means of extensions. We summarize the information in Table 2, including the special cases of the family d20(p : q).

#### Cohomology and deformation theory

Hochschild cohomology was introduced in [10] and used to classify infinitesimal deformations of associative algebras. Suppose that

is an infinitesimal deformation of m. By this we mean that the structure mt is associative up to first order. From an algebraic point of view, this means that we assume that t2 = 0, and then check whether associativity holds. It is not difficult to show that it is equivalent to the following:

where, for simplicity, we denote m(a, b) = ab. Moreover, if we let

gt = I + tλ

be an infinitesimal automorphism of V , where λ ∈ Hom(V, V ), then it is easily checked that

This naturally leads to a definition of the Hochschild coboundary operator D on Hom(Τ (V ), V ) by

If we set Cn(V) = Hom(Vn, V ), then D : Cn(V ) → Cn+1(V). One obtains the following classification theorem for infinitesimal deformations.

Theorem 5. The equivalence classes of infinitesimal deformations mt of an associative algebra structure m under the action of the group of infinitesimal automorphisms on the set of infinitesimal deformations are classified by the Hochschild cohomology group:

Moreover, the Hochschild coboundary operation translates into the coboundary operator D on C(W), given by

This point of view on Hochschild cohomology first appeared in [12]. The fact that the space of Hochschild cochains is equipped with a graded Lie algebra structure was noticed much earlier [5-9].

For notational purposes, we introduce a basis of Cn(W) as follows. Suppose that . Then if I = (i1, . . . , in) is a multi-index, where 1 ≤ ikm, denote . Define by

where is the Kronecker delta symbol. In order to emphasize the parity of the element, we will denote by when it is an odd coderivation.

For a multi-index I = (i1, . . . , ik), denote its length by . Then since W is a completely odd space, the parity of is given by || = k + 1 (mod 2). If K and L are multi-indices, then denote. Then

from which it follows that

(7.1)

where (I, J, k) is given by inserting J into I in place of the kth element of I; that is, (I, J, k) = .

Let us recast the notion of an infinitesimal deformation in terms of the language of coderivations. We say that

is an infinitesimal deformation of the codifferential d precisely when [dt, dt] = 0 mod t2. This condition immediately reduces to the cocycle condition D(ψ) = 0. Note that we require dt to be odd, so that ψ must be an odd coderivation. One can introduce a more general idea of parameters, allowing both even and odd parameters, in which case even coderivations play an equal role, but we will not adopt that point of view in this paper.

For associative algebras, we require that d and ψ lie in C2(W). Since in this paper our algebras are ordinary algebras, so that the parity of an element in Cn(W) is n + 1, thus elements of C2(W) are automatically odd. More generally, one could require that d be an arbitrary odd codifferential, in which case we would obtain an A algebra, a natural generalization of an associative algebra.

We need the notion of a versal deformation in order to understand how the moduli space is glued together. To explain versal deformations, we introduce the notion of a deformation with a local base. It was first introduced for Lie algebras (see [1,2]) and the same definition holds for other algebraic structures as well. Here we recall this notion.

A local base A is a 2-graded commutative, unital -algebra with an augmentation ε : A, whose kernel m is the unique maximal ideal in A, so that A is a local ring. It follows that A has a unique decomposition A = ⊕ m and ε is just the projection onto the first factor. Let WA = WA equipped with the usual structure of a right A-module. Let TA(WA) be the tensor algebra of WA over A, that is, , where and . It is a standard fact that in a natural manner, and thus .

Any A-linear map f : TA(W) → TA(W) is induced by its restriction to T(W)⊗ = T(W), so we can view an A-linear coderivation δA on TA(WA) as a map δA : T(W) → T(W) ⊗ A. A morphism f : AB induces a map

given by . Moreover if δA is a codifferential, then so is f(A). A codifferential dA on TA(WA) is said to be a deformation of the codifferential d on T(W) if ε∗(dA) = d.

If dA is a deformation of d with base A, then we can express

where . The condition for dA to be a codifferential is the Maurer-Cartan equation,

If m2 = 0, we say that A is an infinitesimal algebra and a deformation with base A is called infinitesimal.

A typical example of an infinitesimal base is ; moreover, the classical notion of an infinitesimal deformation: dt = d + , is precisely an infinitesimal deformation with base .

A local algebra A is complete if

A complete, local augmented - algebra will be called formal and a deformation with a formal base is called a formal deformation. An infinitesimal base is automatically formal, so every infinitesimal deformation is a formal deformation.

An example of a formal base is A = [[t]] and a deformation of d with base A can be expressed in the form

This is the classical notion of a formal deformation. It is easy to see that the condition for dt to be a formal deformation reduces to

An automorphism of WA over A is an A-linear isomorphism gA : WAWA making the diagram below commute:

The map gA is induced by its restriction to T(W) ⊗ so we can view gA as a map

so we can express gA in the form

where λ : T(W) → T(W) ⊗ m. If A is infinitesimal, then .

Two deformations dA and A are said to be equivalent over A if there is an automorphism gA of WA over A such that . In this case, we write AdA.

An infinitesimal deformation dA with base A is called universal if whenever dB is an infinitesimal deformation with base B, there is a unique morphism f : AB such that .

Theorem 6 (see [1,2]). If , then there is a universal infinitesimal deformation dinf of d, given by

where is the base of this deformation.

A formal deformation dA with base A is called versal if given any formal deformation of dB with base B there is a morphism f : AB such that (see [1]). Notice that the difference between the versal and the universal property of infinitesimal deformations is that f need not be unique. A versal deformation is called miniversal if f is unique whenever B is infinitesimal. The basic result about versal deformation is the following theorem.

Theorem 7 (see [1,2]). If , then a miniversal deformation of d exists.

In this paper, we will only need the following special result to compute the versal deformations.

Theorem 8. Suppose for all i, j, then the infinitesimal deformation

is miniversal, with base .

The construction of the moduli space as a geometric object is based on the idea that codifferentials which can be obtained by deformations with small parameters are “close” to each other. From the small deformations, we can construct 1-parameter families or even multi-parameter families, which are defined for small values of the parameters, except possibly when the parameters vanish.

If dt is a one parameter family of deformations, then two things can occur. First, it may happen that dt is equivalent to a certain codifferential for every small value of t except zero. Then we say that dt is a jump deformation from d to . It will never occur that is equivalent to d, so there are no jump deformations from a codifferential to itself. Otherwise, the codifferentials dt will all be nonequivalent if t is small enough. In this case, we say that dt is a smooth deformation.

In [3], it was proved for Lie algebras that given three codifferentials d, and , if there are jump deformations from d to and from to , then there is a jump deformation from d to . The proof of the corresponding statement for associative algebras is essentially the same.

Similarly, if there is a jump deformation from d to , and a family of smooth deformations t, then there is a family dt of smooth deformations of d, such that every deformation in the image of t lies in the image of dt, for sufficiently small values of t. In this case, we say that the smooth deformation of d factors through the jump deformation to .

In the examples of complex moduli spaces of Lie and associative algebras which we have studied, it turns out that there is a natural stratification of the moduli space of n-dimensional algebras by orbifolds, where the codifferentials on a given strata are connected by smooth deformations, which do not factor through jump deformations. These smooth deformations determine the local neighborhood structure.

The strata are connected by jump deformations, in the sense that any smooth deformation from a codifferential on one strata to another strata factors through a jump deformation. Moreover, all of the strata are given by projective orbifolds. In fact, in all the complex examples we have studied, the orbifolds either are single points, or quotiented out by either Σn+1 or a subgroup, acting on by permuting the coordinates.

We do not have a proof at this time, but we conjecture that this pattern holds in general. In other words, we believe the following conjecture.

Conjecture 9 (Fialowski-Penkava). The moduli space of Lie or associative algebras of a fixed finite dimension n are stratified by projective orbifolds, with jump deformations and smooth deformations factoring through jump deformations providing the only deformations between the strata.

One might say that in this paper, we prove this conjecture is true for the moduli space of 3-dimensional complex associative algebras.

#### Deformations of the elements in the moduli space

The matrix of this codifferential is

This is the only 3-dimensional complex semisimple algebra, and is the direct sum of three copies of . The algebra is both unital and commutative. We have h0 = 3 and hn = 0 otherwise, so this algebra is rigid, as is always the case with semisimple algebras. Recall that H0 is the center of an associative algebra, so the commutativity of the algebra given by d1 is reflected in the dimension of H0. It is a nice fact that one can detect the commutativity of an algebra in terms of its cohomology.

The matrix of this codifferential is

This algebra is an extension of the complex semisimple algebra 2 by the trivial algebra 0. It cannot be decomposed as a direct sum. We have h0 = 1, and its center is spanned by v1 + v2, which is also the identity of this algebra, so this algebra is unital. Since hn = 0 for n > 0, the algebra is rigid. This algebra is isomorphic to its opposite algebra.

The matrix of this codifferential is

The algebra is an extension of the complex semisimple algebra 2 by the trivial algebra 0. It also is the direct sum of the simple 1-dimensional algebra C with the rigid 2-dimensional algebra d3. We have h0 = 1 and its center is spanned by v3. Since this algebra is a direct sum of two rigid algebras, it is not surprising that hn = 0 for n > 0, so that this algebra is rigid. Its opposite algebra is d4.

The matrix of this codifferential is

The algebra is an extension of the complex semisimple algebra 2 by the trivial algebra 0. It also is the direct sum of the simple 1-dimensional algebra C with the rigid 2-dimensional algebra d2. We have h0 = 1 and its center is spanned by v3. Since this is the opposite algebra to d3, the dimensions of its cohomology spaces Hn are the same as for d3. In other words, hn = 0 for n > 0, so this algebra is rigid.

The matrix of this codifferential is

The algebra is an extension of the complex semisimple algebra 2 by the trivial algebra 0. It is also the direct sum of the simple 1-dimensional algebra C with the unital commutative 2-dimensional algebra d4, so the algebra d5 is both commutative and unital. Since the 2-dimensional algebra d4 deforms to the 2-dimensional semisimple algebra, it is obvious that the 3-dimensional algebra d5 deforms to the semisimple algebra d1. In fact, the versal deformation of d5 is given by

If g is given by the matrix

then , as long as t ≠ 0. In general, hn = 1 for n > 0. In fact, a basis of Hn is just

In particular, d5 will have nontrivial extensions as an A algebra.

The matrix of this codifferential is

The algebra is the direct sum of the complex semisimple algebra 2 and the trivial algebra 0. Since the algebra is commutative, h0 = 3. This algebra is not unital. The algebra can also be seen as the direct sum of the simple 1- dimensional algebra and the 2-dimensional algebra d5, and its cohomology coincides with that of the 2-dimensional algebra. Its versal deformation is given by

If t ≠ 0, then where g is given by the matrix

In general, hn = 1 for n > 0. In fact, a basis of Hn is just

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the nontrivial nilpotent 2-dimensional algebra. It is the unique such extension by this nilpotent Lie algebra as a unital algebra, and it is also commutative.

Its versal deformation is given by

Note that the versal deformation is given by the infinitesimal deformation, and that it remains both unital and commutative. Therefore, it is not surprising that this codifferential deforms to d1 and d5. In fact, since d5 deforms to d1, the fact that d7 deforms to d5 implies that it will deform to d1.

In fact, , on the surface given by , except when v = u, v = –u/2 and v = −2u, where it is equivalent to d5, unless u = v = 0.

In general, it appears that hn = 2 for n > 0, while h0 = 3, since the algebra is commutative. It might be difficult to compute a basis for the cohomology, but the pattern looks similar to the codifferential d6 of the 2-dimensional algebras (in which case it might be difficult to compute this cohomology).

The matrix of this codifferential is

The algebra is the direct sum of the simple 1-dimensional algebra with the nontrivial nilpotent 2-dimensional algebra. It is nonunital, but is commutative. Its versal deformation is given by

Note that the versal deformation is given by the infinitesimal deformation, and that it remains commutative. Moreover, the versal deformation is just the direct sum of the simple 1-dimensional algebra and the versal deformation of the nontrivial nilpotent 2-dimensional algebra.

Except on the curves t1 = 0 and . Along the curve t1 = 0 (except when t2 = 0), it is equivalent to d5, and on the punctured curve , the deformation is equivalent to d6. Note that all three of the codifferentials to which d8 deforms are in fact direct sums of the simple 1-dimensional algebra (given by the codifferential ) and a 2-dimensional algebra to which the nilpotent 2-dimensional algebra deforms.

We have hn = 2 when n > 1 and the cohomology is precisely the same cohomology as the cohomology of the nilpotent 2-dimensional algebra. Since d8 is commutative, h0 = 3.

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is neither unital nor commutative. Its opposite algebra is d10. It is one of a family of 3 rigid 2-dimensional algebras. In fact, in dimension n, there is always a family of n rigid algebras which are extensions of the simple 1-dimensional algebra.

This algebra has a trivial center, but h1 = 3, reflecting that the space of outer derivations is 3-dimensional. We have and hn = 0 unless n = 1.

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is neither unital nor commutative. Its opposite algebra is d9.

This algebra has a trivial center, but h1 = 3, reflecting that the space of outer derivations is 3-dimensional. We have and hn = 0 unless n = 1. This is the second element in the family of 3 rigid extensions of the simple 1-dimensional algebra.

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is neither unital nor commutative. It is isomorphic to its opposite algebra.

This algebra has a trivial center, so h0 = 0. In fact, the pattern for cohomology seems to be

This algebra is the third element in the family of 3 rigid extensions of the simple 1-dimensional algebra.

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is neither unital nor commutative, and its center is spanned by v3. Its opposite algebra is d13.

The versal deformation is given by

and the versal deformation is equivalent to d3 when t ≠ 0.

Its cohomology is given by

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is neither unital nor commutative, and its center is spanned by v3. Its opposite algebra is d12.

The versal deformation is given by

and the versal deformation is equivalent to d4 when t ≠ 0.

Its cohomology is given by

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is also a direct sum of the 2-dimensional rigid algebra d2 and the trivial 1-dimensional algebra. It is neither unital nor commutative, and its center is spanned by v1. Its opposite algebra is d15.

The versal deformation is given by

However, in this case, there are relations on the base of the versal deformation. The relation is t2(t2t1) = 0, which has solutions t2 = 0 or t2 = t1. For the first solution, t2 = 0, the deformation is equivalent to d4. For the second solution, t2 = t1, the deformation is equivalent to d2. Notice how different the deformation situation is when there are relations. Neither d2 nor d4 deform into the other, so the deformations given by the two different solutions to the relations are not “near to” each other.

The cohomology is given by

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is also a direct sum of the 2-dimensional rigid algebra d2 and the trivial 1-dimensional algebra. It is neither unital nor commutative, and its center is spanned by v1. Its opposite algebra is d14.

The versal deformation is given by

As in the previous case, there are relations on the base of the versal deformation. The relation is t2(t2t1) = 0, which has solutions t2 = 0 or t2 = t1. For the first solution, t2 = 0, the deformation is equivalent to d3. For the second solution, t2 = t1, the deformation is equivalent to d2.

The cohomology is given by

The matrix of this codifferential is

The algebra is an extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra. It is also a direct sum of the 2-dimensional rigid algebra d4 and the trivial 1-dimensional algebra. It is not unital, but it is commutative.

The versal deformation is given by

On the punctured curve t1 = 0, the deformation is equivalent to d6, while on the punctured curve t2 = 0, the deformation is equivalent to d5. Otherwise, the deformation is equivalent to d1.

The cohomology is given by

The matrix of this codifferential is

The algebra is the unique unital extension of the simple 1-dimensional algebra by the trivial nilpotent 2-dimensional algebra, and it is commutative.

The matrix of the versal deformation is

Given that h3 = 12, it is not surprising that there are a lot of relations on the base of the versal deformation. Removing duplicate relations, we obtain the following 8 relations:

The solutions to these relations are given by

For the first solution to the relations, the deformation is equivalent to d1, except on some hypersurfaces, where it is either equivalent to d5 or d7. For the second solution, there are two parameters to the deformation, but the versal deformation is equivalent to d2 (unless both parameters vanish). In fact, the transformation g given by the matrix

satisfies , whenever not both t1 and t4 vanish. This is an example of a two-parameter family of deformations that are all equivalent to the same deformation.

It is difficult to compute the cohomology in general. We have h0 = 3, h1 = 3, h2 = 6, h3 = 12, h4 = 24, and it seems likely that hn = 3 ∗ 2n−1 when n > 1. Determining a basis for Hn in general appears to be a difficult problem.

The matrix of this codifferential is

The algebra is the direct of the simple 1-dimensional algebra and the trivial nilpotent 2-dimensional algebra, and it is commutative but not unital.

The matrix of the versal deformation is

This means that the deformations of d18 are precisely the elements in the moduli space which are direct sums of the simple 1-dimensional algebra, and an arbitrary nontrivial 2-dimensional algebra, so the deformation theory of this codifferential captures the complete deformation theory of the moduli space of 2-dimensional algebras.

There are 16 nontrivial relations on the base of the versal deformation, corresponding to the fact that h3 = 16, and the solution to these relations decomposes into 7 distinct solutions. The deformations which occur are d1, d3, d4, d5, d6 and d8, the six elements of the moduli space which correspond to the six nontrivial 2-dimensional algebras.

It is easy to compute the cohomology of this codifferential. We have h0 = 3, and hn = 2n+1 for n > 0. In fact, a basis of Hn is given by those elements in Cn whose indices are either 1 or 2, which accounts for the dimension. This is a pattern which should be repeated in every dimension. Namely, the extension of the simple 1-dimensional algebra by a trivial m-dimensional algebra should have Hn coinciding with Cn of the m-dimensional subspace, and therefore we will have hn = mn+1. Moreover, the codifferential will deform to those elements which are direct sums of the simple algebra and a nontrivial element of the m-dimensional moduli space.

Actually, we can say a similar thing about the direct sum of the 1-dimensional simple algebra and any m-dimensional algebra. Its cohomology coincides with the cohomology of the m-dimensional algebra, and its deformations correspond to the deformations of the m-dimensional algebra. However, many of the extensions of the 1-dimensional simple algebra by a trivial m-dimensional algebra are not decomposable as direct sums of the simple algebra and an m-dimensional algebra.

The matrix of this codifferential is

This is an extension of the trivial 1-dimensional algebra by the trivial 2-dimensional nilpotent algebra. It is nilpotent, and therefore not unital, but it is commutative.

The matrix of the versal deformation is

There are no relations on the base of the versal deformation, which means that the infinitesimal deformation is versal. However, given that there are 3 deformation parameters, it is not surprising that the codifferential has a lot of deformations. It deforms to d1, d5, d6, d7, d8 and d16.

It appears that hn = 3 for all n, but the basis of Hn looks rather difficult to compute.

The matrix of this codifferential is

This is a projective family of extensions of the trivial 1-dimensional algebra by the trivial 2-dimensional nilpotent algebra. By projective, we mean that the codifferential labeled by (p : q) is isomorphic to the element labeled by (tp : tq) for any nonzero t. Moreover, d20(p : q) ∼ d20(q : p), so that actually the family is parameterized by the projective orbifold 1/Σ2, where the action of Σ2 on 1 is given by permuting the projective coordinates. Note that we include the generic point (0 : 0) of 1 in our consideration. We will describe this special point in more detail below.

The algebra d20(p : q) is nilpotent, and it is not commutative except when p = q, in other words, the special points d20(0 : 0) and d20(1 : 1).

There are some special points for which the cohomology of the codifferential is not generic, the points d20(0 : 0), d20(1 : 0), and the two orbifold points d20(1 : 1) and d20(1 : −1). (An element of an orbifold is called an orbifold point if its isotropy group is nontrivial.) We will describe the deformation theory of these special points separately.

Generic case

Let us suppose that (p : q) is not one of the four spacial points listed above. Then h2 = 3 and h3 = 3, so there are three parameters to the versal deformation and 3 relations on the base. The versal deformation is a bit complicated, so we do not give the general form. However, the relations on the base are given by

When we solve these relations, we obtain two solutions. The first is given by t1 = t2 = 0, and the second by the two conditions p = q + t3 and 3t3 = −t1(q + t3). For a solution to the relations to be “local”, it is necessary that the solution is well defined if all the t parameters can be set equal to zero. However, in the second solution, if we substitute t3 = 0 in the first equation we obtain p = q, which is not local unless p = q, which only happens at a special point. Thus generically it is not true. As a consequence, generically only the first solution applies and in this case, the versal deformation has a very simple format:

This means that generically d20(p : q) only has deformations along the family!

Generically, h0 = 1 and the center is spanned by v2. We have h1 = 2, h2 = h3 = 3 and h4 = 4. It seems likely that it would be very difficult to compute the cohomology for this codifferential.

The special point d20(1 : −1)

In this case, h2 = 3 as before, but h3 = 4, and the versal deformation has four relations on the base, with exactly one solution, for which two of the parameters vanish. The solution gives a codifferential d20(1 : −1) = d20(1 : −1) + , which is equivalent to . This is also just a deformation along the family. Thus the special point (1 : −1) is not too special. The cohomology is not generic though, because h3 = 4 and h4 = 5 which are not the generic numbers. The center is still spanned by v2.

The special point d20(1 : 0)

In this case, h2 = 5 and h3 = 8, so there are 5 parameters to the versal deformation, and 8 relations on the base. We will not give them here, but mention that one of the relations has a denominator 1 − t5. It is interesting to note that while there is no guarantee in our method of computation of the versal deformation that the form of the versal deformation or the relations on the base should be given by rational functions of the parameters (theoretically, they are given by formal power series in the parameters), computationally, in the cases we have studied it has always been possible to find a basis of the cohomology for which both of these conditions hold.

In the case here, there are four solutions of the relations on the following base:

For the first solution, the versal deformation is d20(1 : 0) = d20(1 : t5), which gives a deformation along the family.

For the second solution, the versal deformation has matrix

Except on the curves t1 = t4 and t1 = 2t4, the deformation is equivalent to d4. On the punctured curve t1 = t4, the deformation is equivalent to d14 and on the punctured curve t1 = 2t4 it is equivalent to d13.

For the third solution, the versal deformation has matrix

Except on the curves t3 = 0 and t3 = −t2, the deformation is equivalent to d3. On the punctured curve t3 = 0 the deformation is equivalent to d15 and on the punctured curve t3 = −t2 it is equivalent to d12.

For the fourth solution, the versal deformation has matrix

Except on the curves t2 = 0 and t4 = 0, the deformation is equivalent to d2. On the punctured curve t2 = 0 the deformation is equivalent to d14 and on the punctured curve t4 = 0 it is equivalent to d15.

The center of d20(1 : 0) is spanned by v2, h1 = 2, and we have h4 = 8, which is not generic.

The special point d20(1 : 1)

We have h2 = 5 and h3 = 7, so there are 5 parameters and 7 relations on the base of the versal deformation. There are two solutions to the relations:

The first solution gives the deformation d20(1 : 1 + t2), which is just a deformation along the family. The second solution is a bit complicated, so we will not give the precise form. Since there are 4 nonzero parameters in the expression for the versal deformation, it is not surprising that the deformation can be equivalent to several different types, d1, d5, d6, d7, d8, d16 and d19. Notice that d19 is another nilpotent algebra.

The generic point d20(0 : 0)

Unfortunately, the point (0 : 0) is traditionally called the generic point in the literature, because its behavior is anything but generic. In projective geometry, the closure of the generic point is the entire space, which is reflected here in that d20(0 : 0) has jump deformations to any point d20(x : y) (except itself).

We have h2 = 9 and h3 = 17, so there are 9 parameters and 17 relations on the base of the versal deformation, giving 9 solutions to the relations on the base, which we will omit here for brevity. The versal deformation can be equivalent to d1, d2, d3, d4, d5, d6, d7, d8, d12, d13, d14, d15, d16, d17, d18, d19, and it has jump deformations to d20(x : y) for any (x : y) except (0 : 0).Most of these jumps follow from the fact that it jumps to d20(x : y), because an algebra always deforms to any algebra to which an algebra that it jumps to deforms.

This algebra is commutative. We have h1 = 5 and h4 = 33, which are not generic.

The matrix of this codifferential is

This is an extension of the trivial 1-dimensional algebra by the trivial 2-dimensional nilpotent algebra. It is nilpotent and its center is spanned by v2.

We have h2 = 5 and h3 = 8, so there are 5 parameters and 8 relations on the base of the versal deformation. We do not give the formula for the versal deformation or the relations on the base, but give the solutions to the relations below:

The third solution is not local, since t4 = −1.

The first solution has matrix

The second solution has matrix

This gives a jump deformation to d20(1 : −1) and smooth deformations nearby. Whenever a codifferential has a jump deformation to another codifferential, it always has smooth deformations to codifferentials which are near the codifferential to which it jumps.

If we restrict our study of the deformations of the nilpotent elements to just the nilpotent elements, we see a completely parallel pattern to the moduli space of 3-dimensional Lie algebras.

#### Unital algebras

It is well known that finite dimensional unital algebras only deform to unital algebras. Let us identify the unital algebras and their deformations which occur in this moduli space. It is not hard to see that if an algebra is unital, then in terms of the decomposition of the algebra as a semidirect sum of its radical and its semisimple part, the identity of the algebra must be equal to the identity in the semisimple part. Moreover, no nilpotent algebra can be unital. Therefore, it is easy to identify the unital algebras in the moduli space. They are d1, d2, d5, d7 and d17. We have jump deformations

while d2, the only noncommutative unital algebra, neither deforms to any other algebra, nor does any unital algebra deform to it. (It is possible for a nonunital algebra to deform to a unital one, and in fact, there are algebras which deform to d2.)

It is known that any n-dimensional algebra can be realized as an ideal in an n + 1-dimensional unital algebra in a natural manner, by simply adding a new basis element which becomes the identity in the extended algebra. Thus, every 2-dimensional algebra can be realized as an ideal of a 3-dimensional algebra in this form. One might imagine that one could therefore identify the 2-dimensional algebras as forming a subset of the unital algebras in a natural manner, but this is not the case, as the process of constructing a unital extension does not produce an injective map of algebras.

For example, of the six 2-dimensional algebras d1, . . . , d6, the 2-dimensional algebra d1 extends to the 3- dimensional algebra d1, the 2-dimensional algebras d2 and d3 extend to the 3-dimensional algebra d2, the 2- dimensional algebras d4 and d5 extend to the 3-dimensional algebra d5, and the nilpotent 2-dimensional algebra extends to the 3-dimensional d7. Thus it is not possible to reconstruct the 2-dimensional algebras from the unital 3-dimensional algebras in a simple manner.

#### Commutative algebras

If an algebra deforms to a commutative algebra, then it must be commutative. On the other hand, commutative algebras can deform to noncommutative ones, as is well known. The commutative algebras which occur in this moduli space are d1, d5, d6, d7, d8, d16, d17, d18, d20(1 : 1) and d20(0 : 0) (and of course, the zero algebra). All commutative algebras deform to d1, which is just 3, while every algebra except d1 and d6 also deforms to d5. There is a long sequence of jumps

The algebras d18, d16 and d8 have jumps to both d5 and d6, while d20(1 : 1) jumps to d16 and d18. The algebra d20(0 : 0) has jump deformations to every other commutative algebra (except the trivial one).

#### Conclusions

We gave in this paper a logical classification of the elements in the moduli space of 3-dimensional associative algebras and computed all the versal deformations. The only elements which will play a role in constructing higher dimensional algebras are the nilpotent algebras, since every algebra is either semisimple, an extension of a semisimple algebra by a nilpotent algebra, or an extension of a trivial 1-dimensional algebra by a nilpotent algebra, so in some sense, it is not so important what order the non-nilpotent algebras are presented in.We have been careful to construct the algebras in such a manner that the deformations always are in a “downward direction,” that is, the label of an algebra is never lower than the label of an algebra to which it deforms.

One sees from our description of the algebras that there are certain families of algebras, which have counterparts in higher dimensions. Up to dimension 3, the only complex simple algebra is , but in dimension 4, the matrix algebra , which is simple, adds some new features to the description of the moduli space. We are hoping that a systematic construction of the moduli spaces of higher dimension is possible. However, the computational complexity will increase dramatically in higher dimensions, making it difficult to compute the versal deformations, so that there are practical limitations to our approach.

We have shown that for 3-dimensional complex associative algebras, the conjecture of Fialowski-Penkava, that the moduli space decomposes into strata given by projective orbifolds, connected by jump deformations holds. It remains to be seen whether the conjecture will hold in higher dimensions.

#### Acknowledgments

The research of the authors was partially supported by OTKA grants K77757 and NK72523, by grants from the University of Wisconsin-Eau Claire, and the Max Planck Institute for Mathematics, in Bonn, Germany, where part of the work on this paper took place.

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