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 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 2 -graded (or super) algebras with a nontrivial grading over any field k with ch k = 2, up to isomorphism. A complete list of nonisomorphic 2 -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 2 -graded algebras, we define a new variety, Salg n , which possesses many different properties to Alg 4 .
Let k be a field and H 4 be Sweedler's 4-dimensional algebra over k. It is well known that H 4 has a family of triangular structures R t indexed by the ground field k and each triangular structure R t makes the H 4 -module category H 4 M a braided monoidal category, denoted H 4 M R t . In this paper, we study the Azumaya algebras in the categories H 4 M R t . We obtain the structure theorems for Azumaya algebras in each braided monoidal category H 4 M R t , t ∈ k. Utilizing the structure theorems we obtain a scalar invariant for each Azumaya algebra in the aforementioned categories. IntroductionIn [22], Wall introduced the Brauer-Wall group BW(k) of central simple graded algebras (CSGAs) over a field k generalizing the Brauer group of central simple algebras over k. The significance of the group BW(k) can be illustrated by its structure relations to quadratic forms, Clifford invariants, Hasse invariants and Witt invariants etc. The key to
<p><b>The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12].</b></p> <p>Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras.</p> <p>This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties.</p> <p>The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.</p>
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