Factorization structures with a 2-dimensional factor

TL;DR

This paper studies algebraic structures called factorization structures with a two-dimensional factor, introducing quantum duplicates to model quantizations in quantum mechanics, and classifies certain finite-dimensional algebras.

Contribution

It provides a complete description of quantum duplicates of finite set algebras and classifies all 4-dimensional algebras that factorize into two parts.

Findings

Complete description of quantum duplicates of finite set algebras

Classification of all 4-dimensional algebras as products of two factors

Abstract

We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model relies on the classification of factorization structures with a two-dimensional factor. In the present paper, main properties of this particular kind of structures are determined, and we present a complete description of quantum duplicates of finite set algebras. As an application, we obtain a classification (up to isomorphism) of all the algebras of dimension 4 (over an arbitrary field) that can be factorized as a product of two factors.