The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

. We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of typeA Kleinian singularities [14] and the algebras similar to U(sl 2 ) introduced in [31]. For the former, we generalize results of Dixmier [11, 12] on the first Weyl algebra and the minimal primitive factors of U(sl 2 ) by finding sets of generators for the group of automorphisms. 1. Introduction Let k be an algebraically closed field of characteristic 0 and consider the Weyl algebra A 1 (k) = h@; x : @x \Gamma x@ = 1i. Dixmier [11] showed that the k-automorphism group of A 1 (k) is generated by the automorphisms e ad x n and e ad @ n where n 1 and 2 k. Adapting his methods in [12], he found an analogous set of generators for the k-automorphism group of the infinite-dimensional primitive factor B := U(sl 2 )=(C \Gamma ) of the universal enveloping algebra of the Lie algebra sl 2 , where C is the Casimir element and 2 k. He also showed that B ' B 0 if and...

The aim here is to emphasise the topological and geometric structure that the Ziegler spectrum carries and to illustrate how this structure may be used in the analysis of particular examples. There is not space here for me to give a survey of what is known about the Ziegler spectrum so there are a number of topics that I will just mention in order to give some indication of what lies beyond what is discussed here. 1. The Ziegler spectrum 2. Various dimensions 3. These dimensions for artin algebras 4. These dimensions in general 5. Duality 6. The complexity of morphisms in mod-R 7. The Gabriel-Zariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pure-injectives Suppose that M is a submodule of N . Consider a finite system \Sigma n i=1 x i r ij = a j (j = 1; :::m) of R-linear equations over M : that is, the r ij are in R, the 1 a j are in M and the x i are indeterminates. Suppose that there is a solution b 1 ; ...

For a class of generalized Weyl algebras which includes the Weyl algebras A n the tame criteria is given for the problem of describing indecomposable weight and generalized weight modules with supports from a fixed orbit. In tame cases all the indecomposable modules are described. Introduction Let an algebra A =\Omega n 1 A i be the tensor product over an algebraically closed field K of generalized Weyl algebras A i = D i (oe i ; a i ) of degree 1 (see subsec. 1.1 and 1.4) with basic polynomial rings D i = K[H i ] in one variable, where oe i is any automorphism of D i and a i is any non-zero element of D i : Then A is a generalized Weyl algebra of degree n: If a i = H i and oe i (H i ) = H i \Gamma 1 for i = 1; : : : ; n the algebra A is isomorphic to the Weyl algebra A n of degree n (subsec. 1.4). Remark that considered generalized Weyl algebras of degree 1 include the quantum Weyl algebra A 1 (q); the quantum plane, the algebra of functions A(S 2 q ) on the quantum 2-dimens...

The Carlitz Fq-algebra C = Cν, ν ∈ N, is generated by an algebraically closed field K (which contains a non-discrete locally compact field of positive characteristic p> 0, i.e. K ≃ Fq[[x,x −1]], q = p ν), by the (power of the) Frobenius map X = Xν: f ↦ → f q, and by the Carlitz derivative Y = Yν. It is proved that the Krull and global dimensions of C are 2, a classification of simple C-modules and ideals are given, there are only countably many ideals, they commute (IJ = JI), and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple C-module is a sum of eigenspaces of the element Y X (the set of eigenvalues for Y X is given explicitly for each simple C-module). This fact is crucial in finding the group AutFq(C) of Fq-algebra automorphisms of C and in proving that two distinct Carlitz rings are not isomorphic (Cν ̸ ≃ Cµ if ν ̸ = µ). The centre of C is found explicitly, it is a UFD that contains countably many elements.

Abstract. As generalizations of Uq(sl2), a class of algebras Uq(f(K)) were introduced and studied in [7]. For some special parameters f(K) = a(K m − K −m), a ̸ = 0, m ∈ N, Uq(f(K)) are Hopf algebras and hence quantum groups in the sense of Drinfeld ([3]). In this paper, we realize these algebras as Hyperbolic algebras ([12]). As an application of this realization, we obtain a natural construction of irreducible weight representations of Uq(f(K)) using methods in spectral theory as developed in [12]. Based on the knowledge of the center of Uq(f(K)), we will investigate the Whittaker model of the center of Uq(f(K)) following [8]. As a result, the structure of Whittaker representations is determined and all irreducible Whittaker representations are explicitly constructed. We also prove that the annihilator of a Whittaker representation is centrally generated.

We investigate weight modules for finite and infinite Weyl algebras, classifying all such simple modules. We also study the representation type of the blocks of locally-finite weight module categories and describe indecomposable modules in tame blocks. 1 Introduction. The nth Weyl algebra An is the unital associative algebra over a field K with generators xi,∂i, i = 1,2,...,n, which satisfy the defining relations [xi,xj] = 0 = [∂i,∂j] (1) [∂i,xj] = δi,j1, (2)

We investigate certain pure-injective modules over generalised Weyl algebras. We consider pure-injective hulls of finite length modules, the elementary duals of these, torsionfree pure-injective modules and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finite-dimensional algebras and hereditary orders.

Abstract. We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl2 and of Heisenberg Lie algebras, Smith’s generalizations of U(sl2), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(φ, t) are in bijection with the φ-stable left ideals of R. We determine the annihilator AnnA(w) of the cyclic generator w of V.We also describe the annihilator ideal AnnA(V) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant’s well-known results on Whittaker modules and their associated annihilators for U(sl2). 1.

We study isomorphisms between generalized Weyl algebras, giving a complete answer to the quantum case of this problem for R = k[h]. 1