Conservative algebras of $2$-dimensional algebras, II

TL;DR

This paper investigates the structure of conservative algebras of 2-dimensional vector spaces, specifically automorphisms, ideals, and idempotents, expanding understanding of these unique algebra classes beyond well-known types.

Contribution

It provides a detailed analysis of automorphisms, ideals, and idempotents for the conservative algebra W(2) and related algebraic structures on 2-dimensional spaces, which are not part of classical algebra classes.

Findings

Automorphisms of W(2) are characterized.

One-sided ideals and idempotents of W(2) are described.

Similar structural results are obtained for related commutative algebras.

Abstract

In 1990 Kantor 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). We describe automorphisms, one-sided ideals, and idempotents of $W(2).$ Also similar problems are solved for the algebra $W_2$ of all commutative algebras on the 2-dimensional vector space and for the algebra $S_2$ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.