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Model Theory and Modules
, 1988
"... 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 73 (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
InfiniteDimensional Modules in the Representation Theory of FiniteDimensional Algebras
, 1998
"... this article. Throughout, we restrict to studying finitedimensional associative algebras (with 1) over an algebraically closed field K, and write D for duality with the field. Except where stated, all modules are left modules. We write Amod for the category of finitedimensional Amodules, and AM ..."
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Cited by 17 (0 self)
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this article. Throughout, we restrict to studying finitedimensional associative algebras (with 1) over an algebraically closed field K, and write D for duality with the field. Except where stated, all modules are left modules. We write Amod for the category of finitedimensional Amodules, and AMod for the category of all
Stable equivalence preserves representation type
 COMMENTARII MATHEMATICI HELVETICI
, 1997
"... ..."
Canonical Matrices for Linear Matrix Problems
 Bielefeld University
, 1999
"... We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskiĭ’s algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, ..."
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Cited by 11 (6 self)
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We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskiĭ’s algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set Cmn of indecomposable canonical m × n matrices. Considering Cmn as a subset in the affine space of mbyn matrices, we prove that either Cmn consists of a finite number of points and straight lines for every m × n, or Cmn contains a 2dimensional plane for a certain m × n. AMS classification: 15A21; 16G60. Keywords: Canonical forms; Canonical matrices; Reduction; Classification; Tame and wild matrix problems. All matrices are considered over an algebraically closed field k; k m×n denotes the set of mbyn matrices over k. The article consists of three sections. In Section 1 we present Belitskiĭ’s algorithm [2] (see also [3]) in a form, which is convenient for linear algebra. In particular, the algorithm permits to reduce pairs of nbyn matrices to a canonical form by transformations of simultaneous similarity: (A, B) ↦ → (S −1 AS, S −1 BS); another solution of this classical problem was given by Friedland [15]. This section uses rudimentary linear algebra (except for the proof of Theorem 1.1) and may be interested for the general reader. This is the author’s version of a work that was published in Linear Algebra Appl. 317 (2000) 53–102. 1 In Section 2 we determine a broad class of matrix problems, which includes the problems of classifying representations of quivers, partially ordered sets and finite dimensional algebras. In Section 3 we get the following geometric characterization of the set of canonical matrices in the spirit of [17]: if a matrix problem does not ‘contain ’ the canonical form problem for pairs of matrices under simultaneous similarity, then its set of indecomposable canonical m × n matrices in the affine space k m×n consists of a finite number of points and straight lines (contrary to [17], these lines are unpunched). A detailed introduction is given at the beginning of every section. Each introduction may be read independently. 1 Belitskiĭ’s algorithm
Halfquantum groups at roots of unity, path algebras and representation type
 INTERNAT MATH. RES. NOTICES
, 1997
"... Let G be a simple Lie algebra of type A, D or E and q a primitive root of unity of order n ≥ 5. We show that the finite dimensional halfquantum group u + q (G) is of wild representation type, except for G = sl2. Moreover, the algebra u + q (G) is an admissible quotient of the path algebra of the Ca ..."
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Cited by 9 (1 self)
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Let G be a simple Lie algebra of type A, D or E and q a primitive root of unity of order n ≥ 5. We show that the finite dimensional halfquantum group u + q (G) is of wild representation type, except for G = sl2. Moreover, the algebra u + q (G) is an admissible quotient of the path algebra of the Cayley graph of the abelian group (Z/nZ) t with respect to the columns of the t × t Cartan matrix of G.
Module varieties over canonical algebras
 J. Algebra
"... Abstract. The main purpose of this paper is the study of module varieties over the class of canonical algebras, providing a rich source of examples of varieties with interesting properties. Our main tool is a stratication of module varieties, which was recently introduced by Richmond. This straticat ..."
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Abstract. The main purpose of this paper is the study of module varieties over the class of canonical algebras, providing a rich source of examples of varieties with interesting properties. Our main tool is a stratication of module varieties, which was recently introduced by Richmond. This stratication does not require a precise knowledge of the module category. If it is nite, then it provides a method to classify irreducible components. We determine the canonical algebras for which this stratication is nite. In this case, we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension. For an innite family of examples we give easy combinatorial criteria for irreducibility, CohenMacaulay and normality.
Tameness and complexity of finite group schemes
 Bull. London Math. Soc
"... Abstract. Using a representationtheoretic interpretation of support varieties due to FriedlanderPevtsova [18], we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard [23], whose proof is flawed. ..."
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Abstract. Using a representationtheoretic interpretation of support varieties due to FriedlanderPevtsova [18], we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard [23], whose proof is flawed.