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42
Model Theory and Modules
, 2006
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ..."
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Cited by 64 (20 self)
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The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ModR, the full subcategory of finitely presented modules will be denoted modR, the
Isomorphism Problems And Groups Of Automorphisms For Generalized Weyl Algebras
 Trans. Amer. Math. Soc
"... . 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 m ..."
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. 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 kautomorphism 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 kautomorphism group of the infinitedimensional 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...
Indecomposable representations of generalized Weyl algebras
"... 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 Le ..."
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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 nonzero 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 2dimens...
Topological and Geometric aspects of the Ziegler Spectrum
, 1998
"... 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 ..."
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Cited by 6 (5 self)
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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 modR 7. The GabrielZariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pureinjectives 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 Rlinear 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 ; ...
The Carlitz algebras
"... The Carlitz Fqalgebra C = Cν, ν ∈ N, is generated by an algebraically closed field K (which contains a nondiscrete 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 ..."
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Cited by 4 (2 self)
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The Carlitz Fqalgebra C = Cν, ν ∈ N, is generated by an algebraically closed field K (which contains a nondiscrete 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 Cmodules 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 Cmodule is a sum of eigenspaces of the element Y X (the set of eigenvalues for Y X is given explicitly for each simple Cmodule). This fact is crucial in finding the group AutFq(C) of Fqalgebra 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.
WHITTAKER MODULES FOR GENERALIZED WEYL ALGEBRAS
"... 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 alg ..."
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Cited by 4 (0 self)
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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 φstable 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 wellknown results on Whittaker modules and their associated annihilators for U(sl2). 1.
Generalized Verma Modules
, 1999
"... this paper for all missing technical details. In particular, we define a generalization of the Shapovalov form on a GVM M(; p), see Chapter 5. We calculate the determinant of this form. Using the determinant formula we generalize the BGG Theorem about the embeddings of GVMs. In particular, these res ..."
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Cited by 4 (0 self)
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this paper for all missing technical details. In particular, we define a generalization of the Shapovalov form on a GVM M(; p), see Chapter 5. We calculate the determinant of this form. Using the determinant formula we generalize the BGG Theorem about the embeddings of GVMs. In particular, these results covers all known generalizations of the BGG Theorem for ffstratified modules, obtained in Chapter 5 and in [43] in the case of an affine algebra of type A
Constructing irreducible representations of quantum groups Uq(f(K)), preprint
"... 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 ..."
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Cited by 3 (2 self)
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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.
Futorny V., Weyl algebra modules
"... 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 locallyfinite weight module categories and describe indecomposable modules in tame blocks. 1 Introduction. The nth Weyl algebra An is the ..."
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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 locallyfinite 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)
Pure injective envelopes of finite length modules over a Generalised Weyl Algebra
"... We investigate certain pureinjective modules over generalised Weyl algebras. We consider pureinjective hulls of finite length modules, the elementary duals of these, torsionfree pureinjective modules and the closure in the Ziegler spectrum of the category of finite length modules supported on a n ..."
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We investigate certain pureinjective modules over generalised Weyl algebras. We consider pureinjective hulls of finite length modules, the elementary duals of these, torsionfree pureinjective 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 finitedimensional algebras and hereditary orders.