Conservative algebras of $2$-dimensional algebras, IV

TL;DR

This paper studies the algebra $W(2)$, a key example of conservative algebras introduced by Kantor, exploring its structure and subalgebras to deepen understanding of these non-classical algebraic systems.

Contribution

It advances the understanding of the structure and subalgebras of the algebra $W(2)$, a fundamental example of conservative algebras, continuing a series of investigations.

Findings

Analysis of the structure of $W(2)$

Identification of principal subalgebras

Insights into 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, dedicated to the study of the algebra $W(2)$ and its principal subalgebras.