On commutative isotopes of Jordan algebra of a symmetric bilinear nondegenerate form on a two-dimensional vector space
Vita Glizburg, Sergey Pchelintsev
TL;DR
The paper addresses the classification of simple 3-dimensional commutative algebras with nil-index 2 over an algebraically closed field of characteristic not equal to 2. It develops and applies Albert’s isotopy framework, connecting simple unital algebras to Jordan-type structures via main isotopes, with detailed analysis of the Jordan algebra $J_2$, the nonunital $C_2$, and the families $C(\alpha,\beta,\gamma)$ and $C_3$. The main result shows that any simple unital 3-dimensional commutative algebra with nil-index 2 is isotopic to $J_2$, while cyclotomic $C_3$ is isotopic to the standard isotope $C(-2)$, situating all such algebras within a small isotopy landscape. The work also establishes nonexistence of isotopically simple strongly degenerate commutative algebras, and discusses open questions and specific isotopy relations among various small-dimensional algebras.
Abstract
In the article we study the simple unital communitative three-dimensional algebras over an algebraically closed field of characteristic not equal to 2. It is proved that every simple unital communitative three-dimensional algebra of nil-rank 2 is isotopic to Jordan algebra of a symmetric bilinear nondegenerate form on a two-dimensional vector space.