The algebraic and geometric classification of derived Jordan and bicommutative algebras

We developed a new proper method for classifying $n$-dimensional derived Jordan algebras, and apply it to the classification of $3$-dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of $3$-dimensional metabelian commutative algebras and $3$-dimensional derived commutative associative algebras. After that, we introduced a method of classifying $n$-dimensional bicommutative algebras, based on the classification of $n$-dimensional derived commutative associative algebras, and applied it to the classification of $3$-dimensional bicommutative algebras. The second part of the paper is dedicated to the geometric classification of $3$-dimensional metabelian commutative, derived commutative associative, derived Jordan and bicommutative algebras.

💡 Research Summary

The paper introduces the notion of a “derived Ω‑algebra”: for a variety Ω defined by polynomial identities, an algebra A is called derived if its square A² (the subspace generated by all products) satisfies the identities of Ω. This concept unifies several important classes—metabelian (2‑step solvable) algebras, derived commutative associative algebras, derived Jordan algebras, and bicommutative algebras—under a single framework.

The authors first develop an algebraic classification method based on the action of the general linear group GL₃(ℂ) on the space of structure constants. By choosing suitable basis changes they reduce the constants to normal forms and study the resulting orbits. For metabelian commutative algebras they distinguish the cases dim A² = 2 and dim A² = 1, encode the multiplication by a triple of symmetric bilinear forms (B₁,B₂,B₃), and obtain a complete list (Theorem A1) consisting of seven families M₀₁,…,M₀₇(α), where α is a parameter and M₀₁, M₀₂, M₀₃ turn out to be associative.

Next, they treat derived commutative associative algebras. Since every metabelian commutative algebra is automatically derived, the classification includes the families from Theorem A1, all three‑dimensional commutative associative algebras (Proposition 2), and eight new families A_{α,β,γ}⁰ⁱ (i = 1…8) with complex parameters α, β, γ (Theorem A2). Certain parameter values are identified via isomorphisms (e.g., α ↔ α⁻¹).

Derived Jordan algebras are obtained by endowing a derived commutative associative algebra with the Jordan product x∘y = xy + yx. The authors verify that the derived condition forces the square to be associative, and they list twelve non‑isomorphic 3‑dimensional derived Jordan algebras (Theorem A3), some of which coincide with known Jordan algebras while others are genuinely new.

Bicommutative algebras satisfy the identities (ab)c = a(bc) = b(ac). It is known that the Jordan product of a bicommutative algebra is derived commutative associative. Exploiting this, the authors transfer the classification from the derived commutative associative case and obtain nine families of 3‑dimensional bicommutative algebras (Theorem A4), again with a parameter α accounting for a continuous family.

The second part of the paper addresses the geometric classification, i.e., the study of deformations in the sense of Gerstenhaber. For each of the four varieties the authors compute the second cohomology groups, describe orbit closures, and determine the irreducible components of the corresponding algebraic varieties. Their results are:

• Metabelian commutative algebras: a 8‑dimensional variety, two irreducible components, one rigid algebra (Theorem G1).
• Derived commutative associative algebras: a 12‑dimensional variety, two irreducible components, one rigid algebra (Theorem G2).
• Derived Jordan algebras: a 12‑dimensional variety, seven irreducible components, five rigid algebras (Theorem G3).
• Bicommutative algebras: a 10‑dimensional variety, four irreducible components, one rigid algebra (Theorem G4).

The dimensions reflect the number of independent structure constants after factoring out the GL₃‑action and the identities defining each variety. The number of irreducible components indicates the existence of distinct continuous families of algebras that cannot be deformed into each other, while rigid algebras are isolated points in the deformation space, representing “stiff” structures that admit no non‑trivial deformations.

Overall, the paper makes three major contributions: (1) it introduces a unifying derived‑algebra framework that links metabelian, derived associative, derived Jordan, and bicommutative structures; (2) it provides a systematic, GLₙ‑orbit‑based method for obtaining normal forms of structure constants, which is scalable to higher dimensions; (3) it combines algebraic classification with deformation theory to give a complete picture of both the discrete isomorphism classes and the continuous geometric landscape of 3‑dimensional algebras in these varieties. These results lay a solid foundation for future work on higher‑dimensional classifications, applications to physics (where Jordan algebras model observables) and non‑associative cryptographic constructions, and deeper investigations into the interplay between algebraic identities and geometric deformation spaces.