The algebraic and geometric classification of right alternative and semi-alternative algebras

TL;DR

This paper classifies complex 3-dimensional right alternative and semi-alternative algebras algebraically and geometrically, revealing new examples and providing insights into related algebraic structures.

Contribution

It provides the first classifications of certain non-associative algebras in low dimensions and simplifies an existing problem from the Dniester Notebook.

Findings

First non-associative right alternative algebra in dimension 3

First non-associative assosymmetric algebra in dimension 3

First non-$(-1,1)$-semi-alternative algebra in dimension 4

Abstract

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 $\mathfrak{perm}$, binary $\mathfrak{perm}$, 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 $\mathfrak{perm}$ non-$\mathfrak{perm}$ algebras appears in dimension $4.$ As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas.