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

TL;DR

This paper provides a comprehensive analysis of the moduli space of 1|2-dimensional complex associative algebras, including its construction, deformation theory, and the structure of its components.

Contribution

It introduces a detailed construction of the moduli space using extensions and develops miniversal deformations to understand its structure.

Findings

Complete description of the moduli space structure.

Explicit construction of miniversal deformations.

Insight into how the space is assembled via jump deformations.

Abstract

In this paper, we study the moduli space of $1|2$-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a $2|1$-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.