The algebraic and geometric classifications of complex 3-dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex 3-dimensional perm, binary perm, associative, (−1,1)-, binary (−1,1)-, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension 3; the first example of non-associative assosymmetric algebras appears in dimension 3; the first example of non-assosymmetric semi-alternative algebras appears in dimension 4; the first example of binary (−1,1)-algebras, which is non-(−1,1)-, appears in dimension 4; the first example of right alternative algebras, which is not binary (−1,1)-, appears in dimension 4; the first example of binary perm non-perm algebras appears in dimension 4. As a byproduct, we give an easier answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas.
Original languageEnglish
Pages (from-to)792-824
Number of pages33
JournalJournal of Algebra
Volume687
Early online date15 Oct 2025
DOIs
StatePublished - 1 Feb 2026

    Research areas

  • Algebraic classification, Geometric classification, Right alternative algebras, Semi-alternative algebras

ID: 142567065