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) ≠ 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 Alg4 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 Alg5 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 ℤ2-graded (or super) algebras with a nontrivial grading over any field k with ch(k) ≠ 2, up to isomorphism. A complete list of nonisomorphic ℤ2-graded algebras over an algebraically closed field k with ch(k) ≠ 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 ℤ2-graded algebras, we define a new variety, Salg n , which possesses many different properties to Alg4.