Degenerations of Jordan algebras

Abstract

We describe all degenerations of complex $4$-dimensional Jordan algebras and of complex $5$-dimensional nilpotent Jordan algebras. The proofs do not use other papers on degenerations, and all degenerations and non-degenerations were formally verified in the proof assistant Lean~4. The graph of primary degenerations of $4$-dimensional Jordan algebras has $73$ vertices and $160$ edges. The variety of $5$-dimensional nilpotent Jordan algebras is stratified into $63$ normal forms, five of which are one- or two-parameter families; together with two special members of a family this gives $65$ objects, and the graph of primary degenerations has $154$ edges. In particular, we describe all irreducible components of the corresponding varieties: the variety of $4$-dimensional Jordan algebras has $10$ irreducible components, and the variety of $5$-dimensional nilpotent Jordan algebras has dimension $22$ and $5$ irreducible components.