On the Classification of Two-Dimensional Algebras

1. Problem Statement

The paper addresses the classification of non-trivial two-dimensional algebras over an arbitrary base field $F$. Specifically, it aims to:

1. Correct inaccuracies in a previously published classification theorem (referenced as [3]) regarding the isomorphism conditions between specific classes of algebras, particularly in cases where the characteristic of the field is not 2 or 3, and in characteristics 2 and 3.
2. Determine the exact number of non-isomorphic two-dimensional algebras over a finite field $\mathbb{F}_q$ (where $q = p^n$) based on the corrected classification.

The author notes a discrepancy between the number of algebras derived from the previous classification and a formula computed in a separate study [1], prompting a re-examination of the isomorphism relations within the canonical classes.

2. Methodology

The paper employs a rigorous algebraic approach involving:

• Structure Constants and Matrices: Algebras are represented by $2 \times 2$ matrices of structure constants. An algebra is defined by the multiplication rules of basis elements $e_1, e_2$.
• Isomorphism Analysis: The author analyzes the action of the general linear group $GL(2, F)$ on the space of structure constants. Two algebras are isomorphic if their structure constant matrices can be transformed into one another via a change of basis $g \in GL(2, F)$.
• Correction of Canonical Forms: The paper focuses on the "fifth subset" of algebras in the previous classification. By explicitly calculating the transformed parameters ($\alpha', \beta'$) under a general change of basis $g$, the author derives the precise conditions under which two algebras in the same family are isomorphic.
• Polynomial Invariants: The isomorphism conditions are reduced to the solvability of specific polynomial equations over the field $F$. The author identifies that the transformation of the parameter $\beta_1$ is governed by a rational function $f(a, t)$, which forms a semigroup action.
• Orbit Counting: For finite fields, the number of non-isomorphic algebras in a class is determined by counting the number of orbits of the parameter space under the action of the transformation group. This involves analyzing the roots of specific polynomials (e.g., cubic and quadratic equations) and the properties of the field's multiplicative group (specifically regarding cubic roots).

3. Key Contributions

The primary contributions of the paper are the corrections to the classification theorem and the refined counting formulas:

A. Correction of Isomorphism Relations

The author identifies that the description of isomorphisms in classes $A_{10}, A_{11}, A_{8,2}, A_{9,2}, A_{10,3},$ and $A_{11,3}$ in the original work [3] was incomplete or inaccurate.

• New Isomorphism Criteria: The paper establishes that algebras within these classes are isomorphic if their parameters are related by a specific rational transformation $\beta'_1(t)$.
  • For example, in characteristic $\neq 2, 3$, the class $A_{10}$ involves a transformation $\beta'_1(t) = \frac{( \beta_1^2 t^3 + 6\beta_1 t^2 + 3\beta_1 t + \beta_1 - 2)^2}{(\beta_1 t^2 + \beta_1 t + 1)^3}$.
• Exclusion Conditions: The paper clarifies that these transformations are valid only when specific polynomials (derived from the denominator and numerator of the transformation) have no roots in the field $F$. If roots exist, the algebra falls into a different canonical class.
• Characteristic Specifics:
  • Char 2: The transformation polynomials change (e.g., involving terms like $t^3+t+1$).
  • Char 3: The polynomials simplify further, and specific cases (like $A_{9,3}$) require distinct handling regarding cubic roots.

B. Refined Classification Theorem

The paper presents a corrected version of Theorem 1.1, listing the canonical forms for algebras in characteristics $\neq 2,3$, $=2$, and $=3$. It explicitly defines the equivalence relations (denoted by $\simeq$) for the corrected classes, replacing the previous, less precise descriptions.

4. Results

The paper provides the exact count of non-isomorphic two-dimensional algebras over a finite field $\mathbb{F}_q$ for all characteristics.

A. Characteristic $\neq 2, 3$

The total number of non-isomorphic algebras (including the trivial one) is:

• $q^4 + q^3 + 4q^2 + 4q + 8 + |A_{10}|$ if $3 \mid (q-2)$.
• $q^4 + q^3 + 4q^2 + 4q + 10 + |A_{10}|$ otherwise.
• Here, $|A_{10}|$ depends on the number of orbits of the rational function action on the parameter set. The paper notes that if $3 \mid (q-2)$, the count for specific sub-classes changes (e.g., $|A_{11}|$ becomes 2 instead of 4).

B. Characteristic $= 2$

The total number is:

• $q^4 + q^3 + 4q^2 + 3q + 7 + |A_{8,2}|$ if $q = 2^{2k-1}$.
• $q^4 + q^3 + 4q^2 + 3q + 8 + |A_{8,2}|$ if $q = 2^{2k}$.
• The term $|A_{8,2}|$ accounts for the orbits of the transformation in characteristic 2.

C. Characteristic $= 3$

The total number is:

• $q^4 + q^3 + 4q^2 + 4q + 8 + |A_{9,3}|$.

Note on Previous Discrepancy: The original formula cited in the introduction ($q^4 + q^3 + 4q^2 + 4q + 7$ for char $\neq 2,3$) is shown to be incorrect due to the miscounting of isomorphism classes in the original classification. The corrected formulas generally yield higher counts (e.g., $+8$ or $+10$ instead of $+7$) depending on the field properties.

5. Significance

• Resolution of Inconsistencies: The paper resolves a conflict between two existing methods of counting 2D algebras, proving that the previous classification missed specific isomorphism nuances.
• Mathematical Rigor: By explicitly deriving the transformation laws for structure constants, the paper provides a robust framework for classifying low-dimensional algebras that can be extended to other characteristics or dimensions.
• Orbit Theory Application: The work highlights the importance of analyzing the orbits of rational functions over finite fields. The discovery that the transformation forms a semigroup action ($f(f(a, s), t) = f(a, f(s, t))$) offers a deeper algebraic structure to the classification problem.
• Foundation for Further Research: The corrected canonical forms and exact counts serve as a definitive reference for researchers studying the structure of non-associative algebras, particularly in finite fields, which are relevant in coding theory, cryptography, and combinatorics.

In summary, Bekbaev's paper corrects the foundational classification of 2D algebras, providing precise isomorphism criteria and exact enumeration formulas that depend on the arithmetic properties (specifically the existence of roots for certain polynomials) of the underlying finite field.