The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras

TL;DR

This paper provides a comprehensive algebraic and geometric classification of certain low-dimensional nilpotent Leibniz algebras, revealing their structural properties and the nature of their algebraic varieties.

Contribution

It offers the first complete classification of 4- and 5-dimensional nilpotent symmetric Leibniz algebras, including their geometric properties and irreducible components.

Findings

The variety of 4-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property.

There are no rigid nilpotent algebras among these classifications.

The classification clarifies the structure and properties of these algebraic varieties.

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).