Conservative algebras of $2$-dimensional algebras, V

TL;DR

This paper explores the structure of the algebra $W(2)$, a key example of conservative algebras introduced by Kantor, focusing on its subalgebras and their properties.

Contribution

It advances the understanding of $W(2)$ by analyzing its principal subalgebras, contributing to the broader study of conservative algebras.

Findings

Detailed description of principal subalgebras of $W(2)$

Identification of properties distinguishing these subalgebras

Establishment of the role of $W(2)$ in conservative algebra theory

Abstract

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like $W(n)$ in the theory of conservative algebras plays a similar role to the role of $\mathfrak{gl}_n$ in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra $W(n)$ for some $n \in \mathbb{N}.$ The present paper is a part of a series of papers, which dedicated to the study of the algebra $W(2)$ and its principal subalgebras.