[Paper Review] Four dimensional Jordan algebras

María Eugenia Martin|arXiv (Cornell University)|Sep 14, 2012

Advanced Topics in AlgebraMathematics4 references18 citations

TL;DR

This paper provides a complete classification of four-dimensional Jordan algebras over an algebraically closed field of characteristic not equal to two, identifying 73 pairwise non-isomorphic algebras. The classification is achieved through structural analysis using invariants such as annihilator dimension, automorphism group dimension, and nilpotency type, with explicit multiplication tables and cohomological techniques to verify non-isomorphism.

ABSTRACT

In this paper, we classify four-dimensional Jordan algebras over an algebraically closed field of characteristic different of two. We establish the list of 73 non-isomorphic Jordan algebras.

Motivation & Objective

• To provide a complete algebraic classification of all four-dimensional Jordan algebras over an algebraically closed field with characteristic ≠ 2.
• To extend prior classifications of nilpotent and special Jordan algebras to include both unital and non-unital, associative and non-associative cases.
• To establish a foundational list of 73 non-isomorphic Jordan algebras for further study of the variety $Jor_4$.
• To verify the non-isomorphism of algebras using structural invariants and cohomological tools.

Proposed method

• Systematic classification based on the radical structure and nilpotency type, using the lower central series and power ideals.
• Enumeration of indecomposable Jordan algebras of dimension less than four as building blocks.
• Construction of explicit multiplication tables for all 73 algebras, including both associative and non-associative cases.
• Use of invariants such as $\dim \operatorname{Ann}(\mathcal{J})$, $\dim \operatorname{Aut}(\mathcal{J})$, and $\dim \mathcal{J}^2$ to distinguish isomorphism classes.
• Application of second cohomology groups $H^2(\mathcal{J}, \mathcal{J})$ to verify non-isomorphism between algebras with similar invariants.
• Comparison of radical structures, such as $\operatorname{Rad}(\mathcal{J}_{58}) = \mathcal{B}_3 \oplus \mathbb{k}n_2$ and $\operatorname{Rad}(\mathcal{J}_{60}) = \mathcal{T}_4$, to distinguish isomorphic candidates.

Experimental results

Research questions

• RQ1 How many non-isomorphic four-dimensional Jordan algebras exist over an algebraically closed field of characteristic ≠ 2?
• RQ2 Which four-dimensional Jordan algebras are nilpotent, and what are their nilpotency types and structural invariants?
• RQ3 Are all four-dimensional Jordan algebras over this field special, and how can this be verified?
• RQ4 Which structural invariants (e.g., annihilator, automorphism group, radical) suffice to distinguish non-isomorphic algebras?
• RQ5 Can cohomological methods be used to resolve remaining isomorphism ambiguities in the classification?

Key findings

• The paper establishes a complete list of 73 pairwise non-isomorphic four-dimensional Jordan algebras over an algebraically closed field of characteristic ≠ 2.
• All 73 algebras are special Jordan algebras, consistent with known results on nilpotent and low-dimensional nonassociative Jordan algebras.
• The classification includes both associative and non-associative, unital and non-unital algebras, with explicit multiplication tables provided.
• The automorphism group dimensions range from 1 to 16, and annihilator dimensions range from 0 to 4, with $\mathcal{J}_{73}$ having trivial annihilator and maximal automorphism group.
• Nilpotent algebras are fully classified, with $\mathcal{J}_{61}$ to $\mathcal{J}_{73}$ covering all nilpotent cases, including associative types like $\mathcal{T}_4 \oplus \mathbb{k}n_4$ and $\mathcal{B}_3 \oplus \mathcal{B}_3$.
• Cohomological invariants and radical structure comparisons resolve isomorphism ambiguities, confirming that $\mathcal{J}_{55}$, $\mathcal{J}_{56}$, $\mathcal{J}_{58}$, $\mathcal{J}_{59}$, and $\mathcal{J}_{60}$ are all pairwise non-isomorphic.