Three-Dimensional Associative Algebras

• Three-Dimensional Associative Algebras are finite-dimensional structures with a bilinear, associative multiplication classified via codifferentials.
• The moduli space is stratified into isolated rigid types and smooth orbifold families, with deformation behavior captured by jump and continuous transitions.
• Versal deformation theory and Hochschild cohomology offer practical tools for computing local deformation spaces and understanding algebra extensions.

A three-dimensional associative algebra is a finite-dimensional algebra of dimension three over a field (typically ℂ), endowed with a bilinear multiplication that satisfies the associativity law. The paper of these algebras plays a central role in understanding the structure, classification, and deformation theory of associative systems in small dimensions. The subject integrates concepts from codifferential algebra, moduli space stratification, cohomology, and deformation theory, revealing a remarkably intricate moduli space with explicit connections to classical and modern approaches.

1. Classification via Structure Constants and Codifferentials

Every three-dimensional complex associative algebra can be specified, up to isomorphism, by a codifferential $d$ on a vector space $V$ with $\dim V = 3$. Explicitly, an algebra structure corresponds to an odd coderivation $d \in C^2(V) = \operatorname{Hom}(T^2(V), V)$ satisfying the Maurer–Cartan equation

$[d, d] = 0,$

where $[\ , \ ]$ is the Gerstenhaber bracket. The condition translates to quadratic equations in the $27$ structure constants parametrizing the multiplication in a fixed basis.

The complete classification yields exactly 22 non-equivalent types of three-dimensional complex associative algebras. Each type is realized (up to equivalence under the action of the general linear group $G = \mathrm{GL}(V)$) by a canonical codifferential, labeled as $d_1, d_2, \ldots, d_{22}$. The types can be interpreted as:

• Direct sums of lower-dimensional associative algebras (“pure” types),
• Non-pure types where certain products vanish in all bases,
• Algebras realized as extensions of a 1-dimensional algebra by a 2-dimensional algebra; every 3-dimensional associative algebra possesses a 2-dimensional ideal, reflecting the absence of simple algebras in this dimension.

For instance, $d_1 = \psi_{111}^1$ encodes the structure $C_1 \oplus C_0 \oplus C_0$ (one non-trivial one-dimensional and two trivial algebras) in the language of Peirce.

2. Moduli Space and Orbit Structure

The moduli space of three-dimensional associative algebras is constructed as the quotient of the set of codifferentials satisfying $[d, d]=0$ by the group $G$ of invertible linear transformations: $d' = g^{-1} d(g\cdot, g\cdot)g,\quad g \in G.$ This quotient is highly singular and non-Hausdorff. The set of structure constants forms an algebraic (quadratic) variety in $\mathbb{C}^{27}$, and orbits under $G$ correspond to isomorphism classes.

The geometric pathological nature of the quotient is resolved by stratifying the moduli space according to deformation theory, with each stratum corresponding to equivalence classes of codifferentials having the same deformation behavior.

Some key points:

• Level 0: Algebras that have no nontrivial deformations (rigid types, isolated in the moduli space).
• Higher levels: Algebras that deform (possibly via nontrivial “jump” deformations) to algebras of lower level.
• Orbifold strata: For example, the family $d_{22}(x:y)$ is parameterized by the orbifold $\mathbb{P}^1/\Sigma_2$ (projective line modulo coordinate exchange).

The graphical topology of the moduli space, as documented in the referenced Figure 1, comprises isolated points and a “curve” (the orbifold), with "jump" deformations producing non-Hausdorff gluing between strata.

3. Versal Deformations and Cohomological Control

Versal deformation theory is the key to describing local neighborhoods in the moduli space. For a fixed codifferential $d$, the classical infinitesimal deformation is

$$d_t = d + t\psi, \quad \text{where} \quad [d, \psi] = 0.$$

To construct the full miniversal deformation, introduce parameters $t_i$ for a prebasis $\{\delta_i\}$ of the second Hochschild cohomology $H^2(d)$, as well as additional variables $x_j$ associated with higher-degree corrections: $$d^\infty = d + \sum_i t_i\delta_i + \sum_j x_j\gamma_j.$$ The condition $[d^\infty, d^\infty] = 0$ imposes polynomial constraints (“relations on the base”) among the parameters, specifying the local deformation space.

In explicit cases (such as $d_1$), the deformation matrix is written, and computational algebra systems (e.g., Maple) are employed to solve the resulting system and determine the deformation structure. Some algebras have only smooth deformations (remaining within their family), while others admit jump deformations to different types. The interplay between $H^2(d)$ and $H^3(d)$ governs the miniversality and obstruction theory for extending deformations.

An archetype of such phenomena is the family $d_{22}(x:y)$: for generic $(x:y)$ its deformation space is smooth along the family, but for special parameter values, jump phenomena occur due to changes in cohomology dimensions.

4. Stratification and Gluing: Level Hierarchy

The detailed gluing of the moduli space is organized by a “level” hierarchy:

• Level 0: Rigid algebras (e.g., $d_7$, $d_8$, $d_{10}$, $d_{11}$, $d_{13}$, $d_{14}$, $d_{20}$) are isolated and have no nontrivial deformations.
• Higher Levels: Algebras from which level $k$ algebras may be reached by deformation are assigned to level $k+1$.
• Jump deformations: Non-smooth, nontrivial deformations that connect distinct strata, effecting the non-Hausdorff topology of the moduli space.
• Orbifold structure: Families (such as $d_{22}(x:y)$) capture 1-dimensional strata parameterized by symmetric quotients of projective spaces.

This stratified structure is reflected in the organization of the moduli space and is conjecturally connected to projective orbifolds of higher dimension (e.g., $\mathbb{P}^n/\Sigma_{n+1}$ in higher dimensions). Hochschild cohomology not only classifies infinitesimal deformations but also rigorously encodes which strata can be reached from each rigid algebra by deformation.

5. Connections and Implications

This classification fully recovers and refines Peirce’s 19th-century work, offering new insight into non-pure cases and the precise nature of algebra extension and degeneration. The use of codifferentials and cohomology presents a more modern and intrinsic language for describing associative algebra structures, which aligns with advances in deformation theory and moduli space investigations.

Independently, the techniques developed are broadly applicable to the paper of moduli spaces for other algebraic structures (e.g., Lie algebras), and the explicit organization of jump/smooth deformations has consequences for the geometric and representation-theoretic analysis of finite-dimensional algebras.

6. Representative Formulas and Explicit Matrices

Key expressions from the classification and deformation analysis include:

• Codifferential condition:

$[d, d] = 0$

• Equivalence under automorphism:

$$d'(g(a), g(b)) = g(d(a,b)), \quad g \in \mathrm{GL}(V)$$

• Infinitesimal deformation:

$d_t = d + t\psi, \quad [d, \psi] = 0$

• Miniversal deformation:

$d^\infty = d + \sum t_i\delta_i + \sum x_j\gamma_j$

• Example of a miniversal deformation matrix ($d_1$):

$$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & t_8 & 0 & t_6 & 0 & t_4 & t_3 \ 0 & 0 & 0 & t_7 & 0 & t_5 & 0 & t_2 & t_1 \end{pmatrix}$$

These formulas supply the computational backbone for explicit classification and for algorithmically determining the web of deformations and moduli strata.

7. Summary and Further Directions

• Every three-dimensional complex associative algebra is equivalently determined by a codifferential $d$ satisfying $[d, d]=0$ and is classified into 22 explicit types.
• The moduli space of such algebras is identified with the space of these codifferentials modulo automorphism, stratified by their deformation theory.
• Versal deformation theory, grounded in Hochschild cohomology, provides the local geometry and connects disparate strata by smooth and jump deformations.
• This structure yields an organized moduli space, typically consisting of isolated points (rigid types) and families parameterized by projective orbifolds, with a detailed gluing diagram derived from explicit deformation connections.
• The approach and techniques generalize to broader classes of algebras and moduli problems, informing both classification theory and geometric invariant theory in associative and related algebraic settings.

The synthesis of structure constants, cohomological deformation, and moduli stratification presented here reveals the nuanced geometry of three-dimensional associative algebras and the computational mechanisms by which equivalence, rigidity, and deformation are precisely determined (0807.3178).