[Paper Review] 2-dimensional algebras. Application to Jordan, G-associative and Hom-associative algebras

Elisabeth Remm, Michel Goze|arXiv (Cornell University)|May 6, 2012

Advanced Topics in Algebra4 references3 citations

TL;DR

This paper classifies 2-dimensional algebras over a field 𝕂 up to isomorphism, focusing on characteristic 2 via the Fano plane and symmetric group Σ₃ actions. It applies this classification to derive new results on Jordan, G-associative, and Hom-associative algebras, identifying explicit families of Hom-associative algebras through determinant conditions and parameterized families in 𝕂⁶.

ABSTRACT

We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric group S_3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.

Motivation & Objective

• To classify 2-dimensional algebras over a field 𝕂 up to isomorphism, especially in characteristic 2.
• To identify minimal invariant families of algebras rather than exhaustive isomorphism lists, improving usability for higher-dimensional classifications.
• To apply the classification to specific classes: Jordan, G-associative, and Hom-associative algebras.
• To determine conditions under which 2D algebras satisfy the Hom-associative identity using linear systems and determinant criteria.

Proposed method

• Decomposes the multiplication tensor μ into symmetric and skew-symmetric parts for characteristic ≠ 2.
• Uses the action of GL(2,𝔽₂) ≅ Σ₃ on a 3-dimensional plane to classify algebras in characteristic 2 via Fano plane orbits.
• Applies isomorphism invariance to reduce classification to orbit analysis under group actions.
• Derives Hom-associative conditions via the identity (xy)f(z) = f(x)(yz) and constructs linear systems for endomorphisms f.
• Computes the determinant H(A) of a 4×4 matrix HA_A to determine when a commutative algebra is Hom-associative.
• Uses algebraic geometry to describe the set of 2D commutative Hom-associative algebras as a hypersurface in 𝕂⁶.

Experimental results

Research questions

• RQ1 Which 2-dimensional algebras over a field 𝕂 are isomorphic, and how can they be classified using group actions and tensor decomposition?
• RQ2 How can the classification of 2D algebras be adapted to identify minimal invariant families useful for higher-dimensional nonassociative algebras?
• RQ3 What are the necessary and sufficient conditions for a 2D algebra to be Hom-associative, and how can these be expressed algebraically?
• RQ4 How do the results on 2D algebras recover or re-derive known classifications of Jordan and G-associative algebras?
• RQ5 What is the geometric structure of the set of 2D commutative Hom-associative algebras, and how is it encoded via a determinant condition?

Key findings

• The classification of 2D algebras over 𝕂 is achieved via orbit analysis under GL(2,𝔽₂) ≅ Σ₃ in characteristic 2, using the Fano plane structure.
• The algebra A⁴_{1,β₄} is Hom-associative for β₄ ≠ ±2, with nontrivial diagonal endomorphisms f, and is nonassociative when β₄ ≠ 2 or -2.
• The algebra A⁵₀ is Hom-associative with f mapping into 𝕂{e₁}, and nontrivial solutions exist only when α₂ = ±1, yielding two distinct noncommutative Hom-associative algebras.
• For commutative Hom-associative algebras, the condition H(A) = 0 defines a hypersurface in 𝕂⁶, with explicit roots given for H(A⁶) = 0.
• The algebras A⁹ through A¹⁷ are all Hom-associative, as H(Aⁱ) = 0 for i = 9 to 17.
• The algebras A⁷ and A⁸ are not Hom-associative, as H(A⁷) = -1/4 and H(A⁸) = -9/64, both nonzero.