The moduli space of $2|1$-dimensional complex associative algebras

TL;DR

This paper provides a comprehensive analysis of the moduli space of 2|1-dimensional complex associative algebras, including construction, classification, and deformation theory, revealing the structure and connections within this algebraic space.

Contribution

It introduces a complete description of the moduli space of 2|1-dimensional complex associative algebras, including explicit construction and miniversal deformations.

Findings

Complete classification of the moduli space

Construction of miniversal deformations

Description of jump deformations connecting algebras

Abstract

In this paper, we study the moduli space of $2|1$-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a $1|2$-dimensional complex space. We construct the moduli space by considering extensions of lower dimensional algebras. We also construct miniversal deformations of these algebras. This gives a complete description of how the moduli space is glued together via jump deformations.