Classification of 4-dimensional graded algebras

Abstract

Let k be an algebraically closed field. The algebraic and geometric classification of finite dimensional algebras over k with ch(k) not equal 2 was initiated by Gabriel in [6], where a complete list of nonisomorphic 4-dimensional k-algebras was given and the number of irreducible components of the variety Alg(4) was discovered to be 5. The classification of 5-dimensional k-algebras was done by Mazzola in [10]. The number of irreducible components of the variety Alg(5) is 10. With the dimension n increasing, the algebraic and geometric classification of n-dimensional k-algebras becomes more and more difficult. However, a lower and a upper bound for the number of irreducible components of Alg(n) can be given (see [11]). In this article, we classify 4-dimensional Z(2)-graded (or super) algebras with a nontrivial grading over any field k with ch(k) not equal 2, up to isomorphism. A complete list of nonisomorphic Z(2)-graded algebras over an algebraically closed field k with ch(k) not equal 2 is obtained. The main result in this article is twofold. On one hand, it completes the classification of 4-dimensional Yetter-Drinfeld module algebras over Sweedler's 4-dimensional Hopf algebra H-4 initiated in [3]. On the other hand, it establishes the basis for the geometric classification of 4-dimensional super algebras. In approaching the geometric classification of n-dimensional Z(2)-graded algebras, we define a new variety, Salg(n), which possesses many different properties to Alg(4).