Classification and versal deformations of $L_{\infty}$ algebras on a $2\vert 1$-dimensional space

Abstract

This article explores $\mathbb{Z}_2$-graded $L_\infty$ algebra structures on a $2|1$-dimensional vector space. The reader should note that our convention on the parities is the opposite of the usual one, because we define our structures on the symmetric coalgebra of the parity reversion of a space, so our $2|1$-dimensional \linf algebras correspond to the usual $1|2$-dimensional algebras. We give a complete classification of all structures with a nonzero degree 1 term. We also classify all degree 2 codifferentials, which is the same as a classification of all $1|2$-dimensional $\mathbb{Z}_2$-graded Lie algebras. For each of these algebra structures, we calculate the cohomology and a miniversal deformation.