On the variety of two dimensional real associative algebras

TL;DR

This paper classifies two-dimensional real associative algebras by decomposing them into Jordan and Lie parts, analyzing their isomorphism classes, components, and contractions within a nonstandard analysis framework.

Contribution

It introduces a novel approach using nonstandard analysis to classify and analyze the structure of two-dimensional real associative algebras.

Findings

Classification of isomorphism classes of 2D real associative algebras

Decomposition into Jordan and Lie algebra components

Identification of variety components and contractions

Abstract

This paper consists of a description of the variety of two dimensional associative algebras within the framework of Nonstandard Analysis. By decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra, we calculate the isomorphism classes of two dimensional real associative algebras over the field of real numbers and determine the components and the contractions of the variety.