JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. ABSTRACT. Fix f E C[X]. Define R = C[A, B, H] subject to the relations HA-AH = A, HB-BH =-B, AB-BA = f(H).We study these algebras (for different f) and in particular show how they are similar to (and different from) U(sl(2)), the enveloping algebra of sl(2, C). There is a notion of highest weight modules and a category a' for such R. For each n > 0, if f (x) = (x + 1)n+l _ Xn+1 , then R has precisely n simple modules in each finite dimension, and every finite-dimensional R-module is semisimple. INTRODUCTION Fix f E C[X]. Define R = C[A, B, H] subject to the relations [H, A] = A, [H, B] = -B, AB -BA = f(H).This paper studies these algebras (for different f) and in particular shows how they are similar to (and different from) U(sl(2)), the enveloping algebra of sl(2, C). For example, each R is a noetherian domain of Gelfand-Kirillov dimension 3 and has finite-dimensional simple modules of arbitrarily large finite dimension (whenever f 0 C). Furthermore, there is a theory of Verma modules, highest weight modules, and a category a for R. One reason for studying the rings R is to construct examples of noetherian rings which have a rich structure that can be understood in detail and which exhibit some new features. Perhaps the most interesting part of the paper is ?5, which analyzes the finite-dimensional R-modules. The results there show that the ideas involved in the study of enveloping algebras of semisimple Lie algebras have a wider applicability. One striking result is the following. Let n > 0, and set f(x) = (x + 1 )n+l -xn+l . Then for each d > 0, R has precisely n simple modules of dimension d, and every finite-dimensional R-module is semisimple (Example 5.10). The case n = 1 gives R U(sl(2)). For general R not every finite-dimensional module is semisimple. It is this feature which makes the structure of R a little more interesting than that of U(sl(2)).Nevertheless, there are many similarities to U(sl(2)), as is clear from the following brief description of the papers' contents. . Description of R as a skew polynomial ring over the enveloping algebra of the 2-dimensional nonabelian Lie algebra. R is a noetherian domain of GK-dimension 3. The center of R is a polynomial ring in one variable. R is a subalgebra of the second Weyl algebra.?2. Definition of highest weight modules, V(A), and the unique simple quotient of V(A), L(i). Every finite-dimensional simple R-module occurs among the L(i). Description of which L(A) are finite-dimensional in terms of properties of f. The number of finite-dimensional simples of dimension n is < deg(f). Central characters and homomorphisms between the V(A). ?3. The primitive ideals of R are all of the form Ann L(A); {Ann V(A)} = {minimal primitive...

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by Dx the sheaf of differential operators on X, and D(X)=Γ(X,Dx), the ring of global differential operators on X. The following is established: THEOREM. D(X) is a finitely generated k‐algebra, and a noetherian ring. Furthermore, D(X) has a unique minimal non‐zero ideal J, and D(X)/J is a finite‐dimensional k‐algebra. Let X ˜ denoted the normalisation of X, and π: X ˜ →X the projection map. The main technique is to compare D(X) with D( X ˜ ). THEOREM. The following are equivalent: (i) π is injective, (ii) D(X) is a simple ring, (iii) D(X) is Morita equivalent to D( X ˜ ), (iv) the categories DX‐Mod and DX˜‐Mod are equivalent, (v) gr D(X) is noetherian, (vi) the global homological dimension of D(X) is 1. For higher‐dimensional varieties the techniques produce examples of varieties X for which D(X) is right but not left noetherian.

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This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential graded algebra. This is compared to current ideas on noncommutative algebraic geometry.

In 1982 E.K. Sklyanin 13] de ned a family of graded algebras A(E; ), depending on an elliptic curve E and a point 2 E which is not 4-torsion. Basic properties of these algebras were established in 16], and a study of their representation theory was begun in 7]. The present paper classi es the nite dimensional simple A-modules when is a point of in nite order. Sklyanin 14] de nes for each k 2 N a representation of A in a certain k-dimensional subspace of theta functions of order 2(k ? 1): We prove that these are irreducible representations, and that any other simple module is obtained by twisting one of these by an automorphism of A. The automorphism group of A is explicitly computed. The method of proof relies on results in 7]. In particular, it is proved that every nite dimensional simple module is a quotient of a line module. An important part of the analysis is a determination of the 1-critical A-modules, and the fact that such a module is (equivalent to) a quotient of a line module by a shifted line module.

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denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curve E and a point z 6 E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension of A. It is known that A is a finite module over its center if and only if z is of finite order. Generators and defining relations for the center Z(A) are given. If S = Proj(Z(A)) and d is the sheaf of d~s-algebras defined by d(Scr)) = A [ f-1 ]o then the center .~e of d is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result that Sloe Y' ~ p2. However, for the 4-dimensional Sklyanin algebra there is not such a simple result: although Slm: ~e is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus of d .