Geometric classification of 4-dimensional superalgebras

Abstract

In this paper, we give a geometric classification of 4-dimensional superalgebras over an algebraic closed field. It turns out that the number of irreducible components of the variety of 4-dimensional superalgebras Salg$_4$ under the Zariski topology is between 20 and 22. One of the significant differences between the variety Alg$_n$ and the variety Salg$_n$ is that Salg$_n$ is disconnected while Alg$_n$ is connected. Under certain conditions on $n$, one can show that the variety Salg$_n$ is the disjoint union of $n$ connected subvrieties. We shall present the degeneration diagrams of the 4 disjoint connected subvarieties Salg$^i_4$ of Salg$_4$.