Degenerations of Leibniz and anticommutative algebras

TL;DR

This paper classifies all degenerations of three-dimensional anticommutative and Leibniz algebras over the complex numbers, identifying irreducible components and rigid algebras in these algebraic varieties.

Contribution

It provides a complete description of degenerations, irreducible components, and rigid algebras for three-dimensional anticommutative and Leibniz algebras over a9.

Findings

All degenerations of a9 and Leibniz algebras are described.

Irreducible components of the algebra varieties are identified.

Rigid algebras within these varieties are classified.

Abstract

We describe all degenerations of three dimensional anticommutative algebras $\mathfrak{Acom}_3$ and of three dimensional Leibniz algebras $\mathfrak{Leib}_3$ over $\mathbb{C}.$ In particular, we describe all irreducible components and rigid algebras in the corresponding varieties