Complete classification of algebras of level two

TL;DR

This paper provides a complete classification of nonassociative algebras of level two, detailing their degenerations, levels, and the relationship with their generation type, especially for those with a square zero ideal of codimension one.

Contribution

It offers the first comprehensive classification of all nonassociative algebras of level two, including their degeneration structures and level estimations based on generation type.

Findings

Classified all nonassociative algebras of level two.

Established bounds on algebra levels via generation type.

Described degenerations and levels for algebras with specific ideal structures.

Abstract

The main result of the paper is the classification of all (nonassociative) algebras of level two, i.e. such algebras that maximal chains of nontrivial degenerations starting at them have length two. During this classification we obtain an estimation of the level of an algebra via its generation type, i.e. the maximal dimension of its one generated subalgebra. Also we describe all degenerations and levels of algebras of the generation type $1$ with a square zero ideal of codimension $1$.