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

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

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

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

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.

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 *A ^{n}* = 0 for some

**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.

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* : *V* ⊗ *V* → *V* . 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* = *V _{e}* ⊕

Given an ordinary associative algebra, we can view the underlying space V as being _{2}-graded, with *V* = *V _{e}*. Then its parity reversion

Denote the tensor (co)-algebra of *W* by , where *W ^{k}* is the

Define , where π : *V* → *W* 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 *φ* ∈ *C ^{k}*(

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.

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* = *M* ⊕*W* as a -vector space, *M* is an ideal in the algebra *V* , and *W* = *V/M* is the quotient algebra. Suppose that *δ* ∈ *C ^{2}*(

Let *C ^{k,l}* = Hom(

where λ ∈ *C*^{1,1} and *ψ* ∈ *C*^{0,2}. Note that in this notation, *μ* ∈ *C*^{2,0}. 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 *D _{μ}* by

Then the compatibility condition means that λ ∈ *Z*^{1,1}. If we define *D _{δ+λ}*(

If *β* ∈ *C*^{0,1}, then *g* = exp(*β*) : *Τ* (*V*) → *Τ*(*V*) is given by *g*(*m*,*w*) = (*m* + *β*(*w*), *w*). Furthermore *g*^{∗} = exp(−ad_{β}) : *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 *D _{μ}*-coboundary, so in . If λ satisfies the MC-equation for some

Let , so are codifferentials on *Τ* (*M*) and *Τ* (*W*), respectively. The group *G _{δ,μ}* is the subgroup of

Next, consider the group *G _{δ,μ,λ}* consisting of the automorphisms

Thus the group *G _{δ,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 *β* ∈ *C*^{0,1} and *g* ∈ *G _{δ,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 ψ* ∈ *C*^{0,2}, *under the action of the group G _{δ,μ}*;

*(2) equivalence classes of under the action of the group G _{δ,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* = *M* ⊕ *W*, 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 *G _{δ,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 *L _{k}* and

(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 *L _{k}* and

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 *L _{k}* and

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 *L _{k}* could be nonzero since we can assume they are diagonal matrices. Similarly, at most

When *μ* ≠ 0, the matrices *L _{k}* and

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.

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*.

Let The group *G _{M,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 *L _{k}* and

These extensions give the nonequivalent codifferentials *d*_{2}, *d*_{3}, *d*_{4}, *d*_{5} and *d*_{6}.

Let . The group *G _{M,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 *d*_{7} and *d*_{8}. It is interesting to note that this is the stable case even though *n* < 2*m*.

*Extensions by the trivial nilpotent algebra*

In this case, *L*_{1} is given by the identity matrix, diag(1, 0), or the zero matrix. If *L*_{1} is the identity matrix or the zero matrix, then *R*_{1} has the same possibilities, but when *L*_{1} is given by the matrix diag(1, 0), we obtain one additional case, where *R*_{1} is given by the matrix diag(0, 1).

This gives a total of 10 possibilities for λ, which gives rise to the codifferentials *d*_{9} through *d*_{18}. 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 *G _{δ,μ}* coincides with

Thus we have . Also, if , for some t ≠ 0, then is equivalent to λ under the action of *G _{δ,μ}*, so the λ’s form a projective family labeled by projective coordinates (

Any element of *C*^{0,1} is of the form . Therefore, unless *p* = −*q*, it does not matter, up to equivalence, what the value of *c*_{2} is. It may seem natural to choose *c*_{2} = 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 *c*_{2} = 1 yields an equivalent codifferential to the one resulting from choosing *c*_{2} = 0. However, at exactly this point, the two codifferentials are not equivalent. Since every value of *c*_{2} except *c*_{2} = 0 yields an equivalent codifferential to the one with *c*_{2} = 1, the codifferential given by choosing *c*_{2} = 0 has a jump deformation to the one given by *c*_{2} = 1. Thus the element with *c*_{2} = 1 should belong to the family instead of the one with *c*_{2} = 0.

Moreover we have , and since [λ, *ψ*] = 0, this forces *c*_{1} = 0 unless *p* = *q*. In this case, when *p* = *q* = 1, we obtain a special case for *c*_{1} = 1. This gives the element *d*_{19}, while the generic case *c*_{1} = 0 and *p* and *q* being arbitrary, with *c*_{2} = 1, gives a projective family *d*_{20}(*p* : *q*). When *p* = 1, *q* = −1 and *c*_{2} = 0, we obtain the special element *d*_{21}.

In the case when *p* = *q* = 0, we obtain no condition on either *c*_{2} or *c*_{1}. The group *G _{μ,δ,λ}* coincides with the group

*Extensions by the nontrivial nilpotent algebra*

Suppose that . Then

as before.

From the MC equation, we obtain that = 0 and *c*_{2} = 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 *c*_{1}. Nevertheless, the case when *c*_{1} = 0 will be treated separately.

If *c*_{1} = 0 and *q* = 1, then . It can be shown that *d* ∼ *d*_{20}(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 *G _{M,W}*. Thus the two codifferentials do not give equivalent extensions, but give isomorphic algebra structures on

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

Then *g*^{∗}(*d*) = *d*_{20}(*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 *d*_{20}(0 : 0). Thus the extensions by the nontrivial nilpotent algebra just give some special cases of the family *d*_{20}(*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 *d*_{20}(*p* : *q*).

*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 *m _{t}* is associative up to first order. From an algebraic point of view, this means that we assume that

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 *C ^{n}*(

**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 *C ^{n}*(

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* = (*i*_{1}, . . . , *i _{k}*), denote its

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 [*d _{t}*,

For associative algebras, we require that *d* and *ψ* lie in *C*^{2}(*W*). Since in this paper our algebras are ordinary algebras, so that the parity of an element in *C*^{n}(*W*) is *n* + 1, thus elements of *C*^{2}(*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 *W _{A}* =

Any A-linear map *f* : *T _{A}*(

given by . Moreover if *δ _{A}* is a codifferential, then so is

If *d _{A}* is a deformation of

where . The condition for *d _{A}* to be a codifferential is the Maurer-Cartan equation,

If m^{2} = 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: *d _{t}* =

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 *d _{t}* to be a formal deformation reduces to

An automorphism of *W _{A}* over

The map *g _{A}* is induced by its restriction to

so we can express *g _{A}* in the form

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

Two deformations *d _{A}* and

An infinitesimal deformation *d _{A}* with base

**Theorem 6** (see [1,2]). *If* , *then there is a universal infinitesimal deformation d ^{inf} of d, given by*

*where* *is the base of this deformation*.

A formal deformation *d _{A}* with base

**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 *d _{t}* is a one parameter family of deformations, then two things can occur. First, it may happen that

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 *d _{t}* of smooth deformations of

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 Σ_{n+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.

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 *h*^{0} = 3 and *h ^{n}* = 0 otherwise, so this algebra is rigid, as is always the case with semisimple algebras. Recall that

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 *h*^{0} = 1, and its center is spanned by *v*_{1} + *v*_{2}, which is also the identity of this algebra, so this algebra is unital. Since *h ^{n}* = 0 for

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 *d*_{3}. We have *h*^{0} = 1 and its center is spanned by *v*_{3}. Since this algebra is a direct sum of two rigid algebras, it is not surprising that *h ^{n}* = 0 for

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 *d*_{2}. We have *h*^{0} = 1 and its center is spanned by *v*_{3}. Since this is the opposite algebra to *d*_{3}, the dimensions of its cohomology spaces *H ^{n}* are the same as for

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 *d*_{4}, so the algebra *d*_{5} is both commutative and unital. Since the 2-dimensional algebra *d*_{4} deforms to the 2-dimensional semisimple algebra, it is obvious that the 3-dimensional algebra *d*_{5} deforms to the semisimple algebra *d*_{1}. In fact, the versal deformation of *d*_{5} is given by

If *g* is given by the matrix

then , as long as *t* ≠ 0. In general, *h ^{n}* = 1 for

In particular, *d*_{5} 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, *h*^{0} = 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 *d*_{5}, 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, *h ^{n}* = 1 for

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 *d*_{1} and *d*_{5}. In fact, since *d*_{5} deforms to *d*_{1}, the fact that *d*_{7} deforms to *d*_{5} implies that it will deform to *d*_{1}.

In fact, , on the surface given by , except when *v* = *u*, *v* = –*u*/2 and *v* = −2*u*, where it is equivalent to *d*_{5}, unless *u* = *v* = 0.

In general, it appears that *h ^{n}* = 2 for

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 *t*_{1} = 0 and . Along the curve *t*_{1} = 0 (except when *t*_{2} = 0), it is equivalent to *d*_{5}, and on the punctured curve , the deformation is equivalent to *d*_{6}. Note that all three of the codifferentials to which *d*_{8} 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 *h ^{n}* = 2 when

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 *d*_{10}. 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 *h*^{1} = 3, reflecting that the space of outer derivations is 3-dimensional. We have and *h ^{n}* = 0 unless

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 *d*_{9}.

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

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 *h*^{0} = 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 *v*_{3}. Its opposite algebra is *d*_{13}.

The versal deformation is given by

and the versal deformation is equivalent to *d*_{3} 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 *v*_{3}. Its opposite algebra is *d*_{12}.

The versal deformation is given by

and the versal deformation is equivalent to *d*_{4} 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 *d*_{2} and the trivial 1-dimensional algebra. It is neither unital nor commutative, and its center is spanned by *v*_{1}. Its opposite algebra is *d*_{15}.

The versal deformation is given by

However, in this case, there are relations on the base of the versal deformation. The relation is *t*_{2}(*t*_{2}−*t*_{1}) = 0, which has solutions *t*_{2} = 0 or *t*_{2} = *t*_{1}. For the first solution, *t*_{2} = 0, the deformation is equivalent to *d*_{4}. For the second solution, *t*_{2} = *t*_{1}, the deformation is equivalent to *d*_{2}. Notice how different the deformation situation is when there are relations. Neither *d*_{2} nor *d*_{4} 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 *d*_{2} and the trivial 1-dimensional algebra. It is neither unital nor commutative, and its center is spanned by *v*_{1}. Its opposite algebra is *d*_{14}.

The versal deformation is given by

As in the previous case, there are relations on the base of the versal deformation. The relation is *t*_{2}(*t*_{2} − *t*_{1}) = 0, which has solutions *t*_{2} = 0 or *t*_{2} = *t*_{1}. For the first solution, *t*_{2} = 0, the deformation is equivalent to *d*_{3}. For the second solution, *t*_{2} = *t*_{1}, the deformation is equivalent to *d*_{2}.

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 *d*_{4} and the trivial 1-dimensional algebra. It is not unital, but it is commutative.

The versal deformation is given by

On the punctured curve *t*_{1} = 0, the deformation is equivalent to *d*_{6}, while on the punctured curve *t*_{2} = 0, the deformation is equivalent to *d*_{5}. Otherwise, the deformation is equivalent to *d*_{1}.

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 *h*^{3} = 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 *d*_{1}, except on some hypersurfaces, where it is either equivalent to *d*_{5} or *d*_{7}. For the second solution, there are two parameters to the deformation, but the versal deformation is equivalent to *d*_{2} (unless both parameters vanish). In fact, the transformation *g* given by the matrix

satisfies , whenever not both *t*_{1} and *t*_{4} 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 *h*^{0} = 3, *h*^{1} = 3, *h*^{2} = 6, *h*^{3} = 12, *h*^{4} = 24, and it seems likely that *h*^{n} = 3 ∗ 2^{n−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 *d*_{18} 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 *h*^{3} = 16, and the solution to these relations decomposes into 7 distinct solutions. The deformations which occur are *d*_{1}, *d*_{3}, *d*_{4}, *d*_{5}, *d*_{6} and *d*_{8}, 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 *h*^{0} = 3, and *h*^{n} = 2^{n+1} for *n* > 0. In fact, a basis of *H ^{n}* is given by those elements in

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 *d*_{1}, *d*_{5}, *d*_{6}, *d*_{7}, *d*_{8} and *d*_{16}.

It appears that *h*^{n} = 3 for all *n*, but the basis of *H*^{n} 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, *d*_{20}(*p* : *q*) ∼ *d*_{20}(*q* : *p*), so that actually the family is parameterized by the projective orbifold ^{1}/*Σ*_{2}, where the action of *Σ*_{2} 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 *d*_{20}(*p* : *q*) is nilpotent, and it is not commutative except when *p* = *q*, in other words, the special points *d*_{20}(0 : 0) and *d*_{20}(1 : 1).

There are some special points for which the cohomology of the codifferential is not generic, the points *d*_{20}(0 : 0), *d*_{20}(1 : 0), and the two orbifold points *d*_{20}(1 : 1) and *d*_{20}(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 *h*^{2} = 3 and *h*^{3} = 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 *t*_{1} = *t*_{2} = 0, and the second by the two conditions *p* = *q* + *t*_{3} and 3*t*_{3} = −*t*_{1}(*q* + *t*_{3}). 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 *t*_{3} = 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 *d*_{20}(*p* : *q*) only has deformations along the family!

Generically, *h*^{0} = 1 and the center is spanned by *v*_{2}. We have *h*^{1} = 2, *h*^{2} = *h*^{3} = 3 and *h*^{4} = 4. It seems likely that it would be very difficult to compute the cohomology for this codifferential.

*The special point* *d*_{20}(1 : −1)

In this case, *h*^{2} = 3 as before, but *h*^{3} = 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 *d*_{20}(1 : −1)^{∞} = *d*_{20}(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 *h*^{3} = 4 and *h*^{4} = 5 which are not the generic numbers. The center is still spanned by *v*_{2}.

*The special point* *d*_{20}(1 : 0)

In this case, *h*^{2} = 5 and *h*^{3} = 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 − *t*_{5}. 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 *d*_{20}(1 : 0)^{∞} = *d*_{20}(1 : *t*_{5}), which gives a deformation along the family.

For the second solution, the versal deformation has matrix

Except on the curves *t*_{1} = *t*_{4} and *t*_{1} = 2*t*_{4}, the deformation is equivalent to *d*_{4}. On the punctured curve *t*_{1} = *t*_{4}, the deformation is equivalent to *d*_{14} and on the punctured curve *t*_{1} = 2*t*_{4} it is equivalent to *d*_{13}.

For the third solution, the versal deformation has matrix

Except on the curves *t*_{3} = 0 and *t*_{3} = −*t*_{2}, the deformation is equivalent to *d*_{3}. On the punctured curve *t*_{3} = 0 the deformation is equivalent to *d*_{15} and on the punctured curve *t*_{3} = −*t*_{2} it is equivalent to *d*_{12}.

For the fourth solution, the versal deformation has matrix

Except on the curves *t*_{2} = 0 and *t*_{4} = 0, the deformation is equivalent to *d*_{2}. On the punctured curve *t*_{2} = 0 the deformation is equivalent to *d*_{14} and on the punctured curve *t*_{4} = 0 it is equivalent to *d*_{15}.

The center of *d*_{20}(1 : 0) is spanned by *v*_{2}, *h*^{1} = 2, and we have *h*^{4} = 8, which is not generic.

*The special point* *d*_{20}(1 : 1)

We have *h*^{2} = 5 and *h*^{3} = 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 *d*_{20}(1 : 1 + *t*_{2}), 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, *d*_{1}, *d*_{5}, *d*_{6}, *d*_{7}, *d*_{8}, *d*_{16} and *d*_{19}. Notice that *d*_{19} is another nilpotent algebra.

*The generic point* *d*_{20}(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 *d*_{20}(0 : 0) has jump deformations to any point *d*_{20}(*x* : *y*) (except itself).

We have *h*^{2} = 9 and *h*^{3} = 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 *d*_{1}, *d*_{2}, *d*_{3}, *d*_{4}, *d*_{5}, *d*_{6}, *d*_{7}, *d*_{8}, *d*_{12}, *d*_{13}, *d*_{14}, *d*_{15}, *d*_{16}, *d*_{17}, *d*_{18}, *d*_{19}, and it has jump deformations to *d*_{20}(*x* : *y*) for any (*x* : *y*) except (0 : 0).Most of these jumps follow from the fact that it jumps to *d*_{20}(*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 *h*^{1} = 5 and *h*^{4} = 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 *v*_{2}.

We have *h*^{2} = 5 and *h*^{3} = 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 *t*_{4} = −1.

The first solution has matrix

and gives a jump to *d*_{11}.

The second solution has matrix

This gives a jump deformation to *d*_{20}(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.

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 *d*_{1}, *d*_{2}, *d*_{5}, *d*_{7} and *d*_{17}. We have jump deformations

while *d*_{2}, 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 *d*_{2}.)

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 *d*_{1}, . . . , *d*_{6}, the 2-dimensional algebra *d*_{1} extends to the 3- dimensional algebra *d*_{1}, the 2-dimensional algebras *d*_{2} and *d*_{3} extend to the 3-dimensional algebra *d*_{2}, the 2- dimensional algebras *d*_{4} and *d*_{5} extend to the 3-dimensional algebra *d*_{5}, and the nilpotent 2-dimensional algebra extends to the 3-dimensional *d*_{7}. Thus it is not possible to reconstruct the 2-dimensional algebras from the unital 3-dimensional algebras in a simple manner.

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 *d*_{1}, *d*_{5}, *d*_{6}, *d*_{7}, *d*_{8}, *d*_{16}, *d*_{17}, *d*_{18}, *d*_{20}(1 : 1) and *d*_{20}(0 : 0) (and of course, the zero algebra). All commutative algebras deform to *d*_{1}, which is just ^{3}, while every algebra except *d*_{1} and *d*_{6} also deforms to *d*_{5}. There is a long sequence of jumps

The algebras *d*_{18}, *d*_{16} and *d*_{8} have jumps to both *d*_{5} and *d*_{6}, while *d*_{20}(1 : 1) jumps to *d*_{16} and *d*_{18}. The algebra *d*_{20}(0 : 0) has jump deformations to every other commutative algebra (except the trivial one).

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.

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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