The Algebraic and Geometric Classification of Four Dimensional Superalgebras

Abstract

The algebraic and geometric classification of k-algbras, of dimension four or less, was started by Gabriel in “Finite representation type is open” [12]. Several years later Mazzola continued in this direction with his paper “The algebraic and geometric classification of associative algebras of dimension five” [21]. The problem we attempt in this thesis, is to extend the results of Gabriel to the setting of super (or Z2-graded) algebras — our main efforts being devoted to the case of superalgebras of dimension four. We give an algebraic classification for superalgebras of dimension four with non-trivial Z2-grading. By combining these results with Gabriel’s we obtain a complete algebraic classification of four dimensional superalgebras. This completes the classification of four dimensional Yetter-Drinfeld module algebras over Sweedler’s Hopf algebra H4 given by Chen and Zhang in “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. The geometric classification problem leads us to define a new variety, Salgn — the variety of n-dimensional superalgebras—and study some of its properties. The geometry of Salgn is influenced by the geometry of the variety Algn yet it is also more complicated, an important difference being that Salgn is disconnected. While we make significant progress on the geometric classification of four dimensional superalgebras, it is not complete. We discover twenty irreducible components of Salg4 — however there could be up to two further irreducible components.