Some Finite Dimensional Algebras Related to Elliptic Curves

The Koszul dual of a Sklyanin algebra A is a nite dimensional graded algebra depending on an elliptic curve, a translation automorphism, and an integer n 3; it may be deened as Ext A (k;k). The representation theory and structure of these algebras is studied by using the functor Ext A (?; k) to transfer results from the Sklyanin algebras to the nite dimensional algebras. We show that their representation theory is closely related to the elliptic curve and the automorphism. 0. Introduction Given an elliptic curve E over an algebraically closed eld k, a translation auto-morphism of E, and an integer n 3, we deene in Section 10 a nite dimensional algebra B n (E;) depending on this data. Its Hilbert series is the same as that of the exterior algebra (k n). In particular, B n (E;) is local. It is also of wild representation type, and a Frobenius algebra (in fact, symmetric when n = 3). The construction is such that E is naturally embedded in P(B 1), the projective space of 1-dimensional subspaces of the degree one component of B n (E;). These properties of B = B n (E;) are proved in an indirect fashion; the starting point is that B is a Koszul algebra, and its properties are consequences of properties of its Koszul dual A n (E;). Since this paper is aimed at those whose main interest is nite dimensional algebras, we will treat B n (E;) as the primary object. However, A n (E;) is the object of primary interest to the author. It is a Sklyanin algebra, and has been the object of intense study over the past 6 or 7 years. A survey of what is known about A 4 (E;) may be found in 24]. Each A = A n (E;) is a connected graded algebra whose deening relations are homogeneous of degree two; its Koszul dual is, by deenition, Ext A (k; k) endowed with the Yoneda product. The contravariant functor Ext A (?; k), sending graded A-modules to graded B-modules, is the vehicle used for transferring properties from A to B. The basic properties of A n (E;) are reviewed in Section 8. The key result, due to Tate and van den Bergh 35], is that A n (E;) is a quantum polynomial ring (Deenition 8.5); the terminology suggests that A n (E;) is a non-commutative deformation …